Physics Book - User contributions [en]

Coefficient of Restitution

2015-12-06T03:27:18Z

Mmoreno30:

Claimed by Maria Moreno

The coefficient of restitution measures the amount of kinetic energy that remains and how much energy is dissipated after a collision. It also shows the degree of elasticity of a collision. It is defined as the difference in initial velocities divided by the difference in final velocities. The coefficient of restitution of a collision is usually between 0 and 1 except in rare cases when energy is released by a collision.
[[File:coe2.png|right]]

==The Main Idea==

The coefficient of restitution is a ratio that describes the degree of elasticity of a collision. It is used to solve problems dealing with collisions that are not perfectly elastic or inelastic.
The equation that describes the coefficient of restitution involved dividing the difference in the final velocities by the difference in the initial velocity.

Consider objects A and B with initial velocities vAi and vBi and final velocities vAf and vBf.
The coefficient of restitution, <math>e</math>, is determined with the following formula:

<math> e = \frac{v_{Bf}-v_{Af}}{v_{Ai}-v_{Bi}} </math> 
The coefficient is a unitless form of measurement as it is the ratio of two velocities and the units cancel.

===A Mathematical Model===
Consider a perfectly elastic collision between objects A and B where vAi and vBi refer to the initial velocities of A and B and vAf and vBf refer to the final velocities of A and B.

Start with the conservation of momentum principle which states that 
<math>\vec{p_i} = \vec{p_f}</math> and 
<math>m_A\vec{v_{Ai}}+m_A\vec{v_{Bi}}=m_A\vec{v_{Af}}+m_A\vec{v_{Bf}}</math> (1) 

Remember that for a perfectly elastic collision kinetic energy is also conserved meaning: 
<math>KE_i = KE_f</math> and 
<math>\frac{1}{2}m_A v^2_{Ai}+\frac{1}{2}m_A v^2_Bi=\frac{1}{2}m_A v^2_{Af}+\frac{1}{2}m_A v^2_{Bf}</math> (2) 

Dividing (2) by (1) yields: 

<math>\vec{v_{Ai}}+\vec{v_{Bi}}=\vec{v_{Af}}+\vec{v_{Bf}}</math> 
<math>1=\frac{v_{Ai}+v_{Bi}}{v_{Af}+v_{Bf}}</math> 

<math>e=\frac{v_{Ai}+v_{Bi}}{v_{Af}+v_{Bf}}</math> 

For perfectly elastic collision <math>e</math>=1.

===Inelastic Collisions===

An inelastic collision is a type of collision where Kinetic Energy is not conserved. In this case the difference in initial velocities will not equal the difference in final velocities. This means that the coefficient of restitution will not equal 1.

Consider the same collision as previously described, except this collision is inelastic and Kinetic Energy is not conserved. Since some of the kinetic energy is dissipated: 
<math>\frac{1}{2}m_A v^2_{Ai}+\frac{1}{2}m_A v^2_Bi<\frac{1}{2}m_A v^2_{Af}+\frac{1}{2}m_A v^2_{Bf}</math> 
<math>\frac{v_{Ai}+v_{Bi}}{v_{Af}+v_{Bf}}<1</math> 

<math>e=\frac{v_{Ai}+v_{Bi}}{v_{Af}+v_{Bf}}<1</math> 

A perfectly inelastic collision (for example when two objects collide and then stick together) has a coefficient of restitution of 0.

===A Computational Model===
Here is a link to a glowscript simulation
http://www.glowscript.org/#/user/mmoreno181/folder/phys/program/cor 
Here is the code for the simulation: 
<code>
ball1=sphere(pos=vec(0,0,0), color=color.red, radius=.5)
ball2=sphere(pos=vec(10,0,0), color=color.blue, radius=.5)

#initialize velocity
v1i=vec(.02,0,0)
v2i=vec(-.1,0,0)
v1f=vec(-.07,0,0)

#set coefficient of resitution
e=.5

#find v2f
v2f=v1f-(e*(v2i-v1i))

#find the time that the balls collide
tmeet= (ball2.pos.x-ball1.pos.x)/(v1i.x-v2i.x)

t=0

#animate
while t<150:
rate(100)
if t < tmeet: #if it's before the collision
ball1.pos=ball1.pos+v1i
ball2.pos=ball2.pos+v2i
else:
ball1.pos=ball1.pos+v1f
ball2.pos=ball2.pos+v2f

t=t+1
</code>

==Examples==
===Simple===
A ball is traveling at <math>4 \frac{m}{s}</math> and collides with a ball. After the collision the ball travels at <math>2.4\frac{m}{s}</math> in the opposite direction. Find the coefficient of restitution of the collision. 
Start with the definition of the coefficient of restitution: 
<math>e=\frac{v_{wall_f}-v_{ball_f}}{v_{ball_i}-v_{wall_i}}</math> 
Then plug in values and solve: 
<math>e=\frac{0-v_{ball_f}}{v_{ball_i}-0}</math> 
<math>e=\frac{-(-2.4)}{4}</math> 
<math>e=.6</math>

===Middling===
A hockey puck travels at a constant velocity <math>5\frac{m}{s}</math> and is hit by another puck with a velocity of <math>8\frac{m}{s}</math>. After the collision, it travels in the opposite direction at 4m/s. This collision has a coefficient of restitution of .55. Find the velocity of the second puck after the collision. 
Start with the definition of the coefficient of restitution. We are calling the velocity of the first puck <math>v_1</math> and the velocity of the second puck <math>v_2</math>. 
<math>e=\frac{v_{1f}-v_{2f}}{v_{2i}-v_{1i}}</math> 
Solve for <math>v_{2f}</math> 

<math>v_{2f}=v_{1f}-e(v_{2i}-v_{1i})</math> 
Plug in the values stated in the problem. 
<math>v_{2f}=4-.55(5-8)</math>
<math>v_{2f}=2.35\frac{m}{s}</math>

===Difficult===
[[File:Cor1.jpg|right]]
Bouncing Ball Example:

Consider a ball dropped from initial height <math>h_0</math>. When it bounces it reaches a height <math>h_f</math>. Find the coefficient of restitution of this collision. 
<math>e=\frac{v_{floor_f}-v_{ball_f}}{v_{ball_i}-v_{floor_i}}</math> 
The floor's velocity is always 0. 
<math>e=\frac{v_{ball_f}}{v_{ball_i}}</math> 
Recall the law of conservation of energy: 
<math> \Delta KE = \Delta PE</math> 
For the ball from h0 to h=0
<math> \frac{1}{2}mv_{ball_i}^2=mgh_0</math> 
<math> \frac{1}{2}v_{ball_i}^2=gh_0</math> 
<math> v_{ball_i}=2\sqrt{gh_0} </math> 
The same concept can be applied to the second bounce. 
<math>v_{ball_f}=2\sqrt{gh_f}</math> 
Plugging this in to the coefficient of restitution expression we get: 
<math>e=\frac{2\sqrt{mgh_f}}{2\sqrt{mgh_0}}</math> 
<math>e=\frac{\sqrt{h_f}}{\sqrt{h_0}}</math> 

==Connectedness==
'''How is this topic connected to something that you are interested in?''' 
This topic has major connections to sports. As a former high school volleyball player, it is very interesting to think about the collisions that occurred in the game and how the coefficient of restitution is related to those collisions. 
[[File:sportsgraph.png|right|thumb|This graph shows the initial and final velocities of many different balls from different sports.]]
'''How is it connected to your major (Mechanical Engineering)?''' 
The coefficient of restitution is a key concept in the study of Dynamics which is the foundation of many aspects of mechanical engineering. It also has to do with applications of mechanical engineering regarding materials. 

[[File:sportschart.png|right|thumb|This chart shows the relationship between different coefficients of restitution and the initial and final velocities of sports balls.]]

'''Is there an interesting industrial application?'''
In sports the coefficient of restitution is an important concept especially in sports like golf or tennis where collisions are a frequent and essential part of the game. In these sports sometimes limitations are put on how high the coefficient of restitution can be for certain clubs or bats. The maximum coefficient of restitution allowed in golf for clubs is .82. There have been studies done on the coefficients for different sports balls. These studies usually follow a procedure where different balls are dropped and left to bounce. Then data regarding their initial and final velocities or how high they bounced is recorded. The images below show some of the data from one of these experiments.

== See also ==

http://physicsbook.gatech.edu/Elastic_Collisions 
http://physicsbook.gatech.edu/Inelastic_Collisions 
http://physicsbook.gatech.edu/Collisions 
http://physicsbook.gatech.edu/Conservation_of_Momentum 
http://physicsbook.gatech.edu/Conservation_of_Energy 

===External links===
http://www.quintic.com/education/case_studies/coefficient_restitution.htm

==References==

McGill, David J., and Wilton W. King. Engineering mechanics : an introduction to dynamics. Bloomington, IN: Tichenor Pub, 2003. Print. 
"Quintic 4 Education Sports Video Analysis Software Education Package - The Coefficient of Restitution." Quintic 4 Education Sports Video Analysis Software Education Package - The Coefficient of Restitution. Quintic 4 Education, n.d. Web. 05 Dec. 2015. 
"Coefficient of Restitution (COR)." Yale. Web. 5 Dec. 2015. 

[[Category:Collisions]]

Coefficient of Restitution

2015-12-06T03:22:22Z

Mmoreno30:

Claimed by Maria Moreno

The coefficient of restitution measures the amount of kinetic energy that remains and how much energy is dissipated after a collision. It also shows the degree of elasticity of a collision. It is defined as the ratio of the difference in initial velocities divided by the difference in final velocities. The coefficient of restitution of a collision is usually between 0 and 1 except in rare cases when energy is released by a collision.
[[File:coe2.png|right]]

==The Main Idea==

The coefficient of restitution is a ratio that describes the degree of elasticity of a collision. It is used to solve problems dealing with collisions that are not perfectly elastic or inelastic.
The equation that describes the coefficient of restitution involved dividing the difference in the final velocities by the difference in the initial velocity.

Consider objects A and B with initial velocities vAi and vBi and final velocities vAf and vBf.
The coefficient of restitution, <math>e</math>, is determined with the following formula:

<math> e = \frac{v_{Bf}-v_{Af}}{v_{Ai}-v_{Bi}} </math> 
The coefficient is a unitless form of measurement as it is the ratio of two velocities and the units cancel.

===A Mathematical Model===
Consider a perfectly elastic collision between objects A and B where vAi and vBi refer to the initial velocities of A and B and vAf and vBf refer to the final velocities of A and B.

Start with the conservation of momentum principle which states that 
<math>\vec{p_i} = \vec{p_f}</math> and 
<math>m_A\vec{v_{Ai}}+m_A\vec{v_{Bi}}=m_A\vec{v_{Af}}+m_A\vec{v_{Bf}}</math> (1) 

Remember that for a perfectly elastic collision kinetic energy is also conserved meaning: 
<math>KE_i = KE_f</math> and 
<math>\frac{1}{2}m_A v^2_{Ai}+\frac{1}{2}m_A v^2_Bi=\frac{1}{2}m_A v^2_{Af}+\frac{1}{2}m_A v^2_{Bf}</math> (2) 

Dividing (2) by (1) yields: 

<math>\vec{v_{Ai}}+\vec{v_{Bi}}=\vec{v_{Af}}+\vec{v_{Bf}}</math> 
<math>1=\frac{v_{Ai}+v_{Bi}}{v_{Af}+v_{Bf}}</math> 

<math>e=\frac{v_{Ai}+v_{Bi}}{v_{Af}+v_{Bf}}</math> 

For perfectly elastic collision <math>e</math>=1.

===Inelastic Collisions===

An inelastic collision is a type of collision where Kinetic Energy is not conserved. In this case the difference in initial velocities will not equal the difference in final velocities. This means that the coefficient of restitution will not equal 1.

Consider the same collision as previously described, except this collision is inelastic and Kinetic Energy is not conserved. Since some of the kinetic energy is dissipated: 
<math>\frac{1}{2}m_A v^2_{Ai}+\frac{1}{2}m_A v^2_Bi<\frac{1}{2}m_A v^2_{Af}+\frac{1}{2}m_A v^2_{Bf}</math> 
<math>\frac{v_{Ai}+v_{Bi}}{v_{Af}+v_{Bf}}<1</math> 

<math>e=\frac{v_{Ai}+v_{Bi}}{v_{Af}+v_{Bf}}<1</math> 

A perfectly inelastic collision (for example when two objects collide and then stick together) has a coefficient of restitution of 0.

===A Computational Model===
Here is a link to a glowscript simulation
http://www.glowscript.org/#/user/mmoreno181/folder/phys/program/cor 
Here is the code for the simulation: 
<code>
ball1=sphere(pos=vec(0,0,0), color=color.red, radius=.5)
ball2=sphere(pos=vec(10,0,0), color=color.blue, radius=.5)

#initialize velocity
v1i=vec(.02,0,0)
v2i=vec(-.1,0,0)
v1f=vec(-.07,0,0)

#set coefficient of resitution
e=.5

#find v2f
v2f=v1f-(e*(v2i-v1i))

#find the time that the balls collide
tmeet= (ball2.pos.x-ball1.pos.x)/(v1i.x-v2i.x)

t=0

#animate
while t<150:
rate(100)
if t < tmeet: #if it's before the collision
ball1.pos=ball1.pos+v1i
ball2.pos=ball2.pos+v2i
else:
ball1.pos=ball1.pos+v1f
ball2.pos=ball2.pos+v2f

t=t+1
</code>

==Examples==
===Simple===
A ball is traveling at <math>4 \frac{m}{s}</math> and collides with a ball. After the collision the ball travels at <math>2.4\frac{m}{s}</math> in the opposite direction. Find the coefficient of restitution of the collision. 
Start with the definition of the coefficient of restitution: 
<math>e=\frac{v_{wall_f}-v_{ball_f}}{v_{ball_i}-v_{wall_i}}</math> 
Then plug in values and solve: 
<math>e=\frac{0-v_{ball_f}}{v_{ball_i}-0}</math> 
<math>e=\frac{-(-2.4)}{4}</math> 
<math>e=.6</math>

===Middling===
A hockey puck travels at a constant velocity <math>5\frac{m}{s}</math> and is hit by another puck with a velocity of <math>8\frac{m}{s}</math>. After the collision, it travels in the opposite direction at 4m/s. This collision has a coefficient of restitution of .55. Find the velocity of the second puck after the collision. 
Start with the definition of the coefficient of restitution. We are calling the velocity of the first puck <math>v_1</math> and the velocity of the second puck <math>v_2</math>. 
<math>e=\frac{v_{1f}-v_{2f}}{v_{2i}-v_{1i}}</math> 
Solve for <math>v_{2f}</math> 

<math>v_{2f}=v_{1f}-e(v_{2i}-v_{1i})</math> 
Plug in the values stated in the problem. 
<math>v_{2f}=4-.55(5-8)</math>
<math>v_{2f}=2.35\frac{m}{s}</math>

===Difficult===
[[File:Cor1.jpg|right]]
Bouncing Ball Example:

Consider a ball dropped from initial height <math>h_0</math>. When it bounces it reaches a height <math>h_f</math>. Find the coefficient of restitution of this collision. 
<math>e=\frac{v_{floor_f}-v_{ball_f}}{v_{ball_i}-v_{floor_i}}</math> 
The floor's velocity is always 0. 
<math>e=\frac{v_{ball_f}}{v_{ball_i}}</math> 
Recall the law of conservation of energy: 
<math> \Delta KE = \Delta PE</math> 
For the ball from h0 to h=0
<math> \frac{1}{2}mv_{ball_i}^2=mgh_0</math> 
<math> \frac{1}{2}v_{ball_i}^2=gh_0</math> 
<math> v_{ball_i}=2\sqrt{gh_0} </math> 
The same concept can be applied to the second bounce. 
<math>v_{ball_f}=2\sqrt{gh_f}</math> 
Plugging this in to the coefficient of restitution expression we get: 
<math>e=\frac{2\sqrt{mgh_f}}{2\sqrt{mgh_0}}</math> 
<math>e=\frac{\sqrt{h_f}}{\sqrt{h_0}}</math> 

==Connectedness==
'''How is this topic connected to something that you are interested in?''' 
This topic has major connections to sports. As a former high school volleyball player, it is very interesting to think about the collisions that occurred in the game and how the coefficient of restitution is related to those collisions. 
[[File:sportsgraph.png|right|thumb|This graph shows the initial and final velocities of many different balls from different sports.]]
'''How is it connected to your major (Mechanical Engineering)?''' 
The coefficient of restitution is a key concept in the study of Dynamics which is the foundation of many aspects of mechanical engineering. It also has to do with applications of mechanical engineering regarding materials. 

[[File:sportschart.png|right|thumb|This chart shows the relationship between different coefficients of restitution and the initial and final velocities of sports balls.]]

'''Is there an interesting industrial application?'''
In sports the coefficient of restitution is an important concept especially in sports like golf or tennis where collisions are a frequent and essential part of the game. In these sports sometimes limitations are put on how high the coefficient of restitution can be for certain clubs or bats. The maximum coefficient of restitution allowed in golf for clubs is .82. There have been studies done on the coefficients for different sports balls. These studies usually follow a procedure where different balls are dropped and left to bounce. Then data regarding their initial and final velocities or how high they bounced is recorded. The images below show some of the data from one of these experiments.

== See also ==

http://physicsbook.gatech.edu/Elastic_Collisions 
http://physicsbook.gatech.edu/Inelastic_Collisions 
http://physicsbook.gatech.edu/Collisions 
http://physicsbook.gatech.edu/Conservation_of_Momentum 
http://physicsbook.gatech.edu/Conservation_of_Energy 

===External links===
http://www.quintic.com/education/case_studies/coefficient_restitution.htm

==References==

McGill, David J., and Wilton W. King. Engineering mechanics : an introduction to dynamics. Bloomington, IN: Tichenor Pub, 2003. Print. 
"Quintic 4 Education Sports Video Analysis Software Education Package - The Coefficient of Restitution." Quintic 4 Education Sports Video Analysis Software Education Package - The Coefficient of Restitution. Quintic 4 Education, n.d. Web. 05 Dec. 2015. 
"Coefficient of Restitution (COR)." Yale. Web. 5 Dec. 2015. 

[[Category:Collisions]]

Coefficient of Restitution

2015-12-06T03:20:45Z

Mmoreno30: /* The Main Idea */

Claimed by Maria Moreno

The coefficient of restitution measures the amount of Kinetic Energy remains and how much energy is lost after a collision. It also shows the degree of elasticity of a collision. It is defined as the ratio of the difference in initial velocities divided by the difference in final velocities. The coefficient of restitution of a collision is usually between 0 and 1 except in rare cases when energy is released by a collision.
[[File:coe2.png|right]]

==The Main Idea==

The coefficient of restitution is a ratio that describes the degree of elasticity of a collision. It is used to solve problems dealing with collisions that are not perfectly elastic or inelastic.
The equation that describes the coefficient of restitution involved dividing the difference in the final velocities by the difference in the initial velocity.

Consider objects A and B with initial velocities vAi and vBi and final velocities vAf and vBf.
The coefficient of restitution, <math>e</math>, is determined with the following formula:

<math> e = \frac{v_{Bf}-v_{Af}}{v_{Ai}-v_{Bi}} </math> 
The coefficient is a unitless form of measurement as it is the ratio of two velocities and the units cancel.

===A Mathematical Model===
Consider a perfectly elastic collision between objects A and B where vAi and vBi refer to the initial velocities of A and B and vAf and vBf refer to the final velocities of A and B.

Start with the conservation of momentum principle which states that 
<math>\vec{p_i} = \vec{p_f}</math> and 
<math>m_A\vec{v_{Ai}}+m_A\vec{v_{Bi}}=m_A\vec{v_{Af}}+m_A\vec{v_{Bf}}</math> (1) 

Remember that for a perfectly elastic collision kinetic energy is also conserved meaning: 
<math>KE_i = KE_f</math> and 
<math>\frac{1}{2}m_A v^2_{Ai}+\frac{1}{2}m_A v^2_Bi=\frac{1}{2}m_A v^2_{Af}+\frac{1}{2}m_A v^2_{Bf}</math> (2) 

Dividing (2) by (1) yields: 

<math>\vec{v_{Ai}}+\vec{v_{Bi}}=\vec{v_{Af}}+\vec{v_{Bf}}</math> 
<math>1=\frac{v_{Ai}+v_{Bi}}{v_{Af}+v_{Bf}}</math> 

<math>e=\frac{v_{Ai}+v_{Bi}}{v_{Af}+v_{Bf}}</math> 

For perfectly elastic collision <math>e</math>=1.

===Inelastic Collisions===

An inelastic collision is a type of collision where Kinetic Energy is not conserved. In this case the difference in initial velocities will not equal the difference in final velocities. This means that the coefficient of restitution will not equal 1.

Consider the same collision as previously described, except this collision is inelastic and Kinetic Energy is not conserved. Since some of the kinetic energy is dissipated: 
<math>\frac{1}{2}m_A v^2_{Ai}+\frac{1}{2}m_A v^2_Bi<\frac{1}{2}m_A v^2_{Af}+\frac{1}{2}m_A v^2_{Bf}</math> 
<math>\frac{v_{Ai}+v_{Bi}}{v_{Af}+v_{Bf}}<1</math> 

<math>e=\frac{v_{Ai}+v_{Bi}}{v_{Af}+v_{Bf}}<1</math> 

A perfectly inelastic collision (for example when two objects collide and then stick together) has a coefficient of restitution of 0.

===A Computational Model===
Here is a link to a glowscript simulation
http://www.glowscript.org/#/user/mmoreno181/folder/phys/program/cor 
Here is the code for the simulation: 
<code>
ball1=sphere(pos=vec(0,0,0), color=color.red, radius=.5)
ball2=sphere(pos=vec(10,0,0), color=color.blue, radius=.5)

#initialize velocity
v1i=vec(.02,0,0)
v2i=vec(-.1,0,0)
v1f=vec(-.07,0,0)

#set coefficient of resitution
e=.5

#find v2f
v2f=v1f-(e*(v2i-v1i))

#find the time that the balls collide
tmeet= (ball2.pos.x-ball1.pos.x)/(v1i.x-v2i.x)

t=0

#animate
while t<150:
rate(100)
if t < tmeet: #if it's before the collision
ball1.pos=ball1.pos+v1i
ball2.pos=ball2.pos+v2i
else:
ball1.pos=ball1.pos+v1f
ball2.pos=ball2.pos+v2f

t=t+1
</code>

==Examples==
===Simple===
A ball is traveling at <math>4 \frac{m}{s}</math> and collides with a ball. After the collision the ball travels at <math>2.4\frac{m}{s}</math> in the opposite direction. Find the coefficient of restitution of the collision. 
Start with the definition of the coefficient of restitution: 
<math>e=\frac{v_{wall_f}-v_{ball_f}}{v_{ball_i}-v_{wall_i}}</math> 
Then plug in values and solve: 
<math>e=\frac{0-v_{ball_f}}{v_{ball_i}-0}</math> 
<math>e=\frac{-(-2.4)}{4}</math> 
<math>e=.6</math>

===Middling===
A hockey puck travels at a constant velocity <math>5\frac{m}{s}</math> and is hit by another puck with a velocity of <math>8\frac{m}{s}</math>. After the collision, it travels in the opposite direction at 4m/s. This collision has a coefficient of restitution of .55. Find the velocity of the second puck after the collision. 
Start with the definition of the coefficient of restitution. We are calling the velocity of the first puck <math>v_1</math> and the velocity of the second puck <math>v_2</math>. 
<math>e=\frac{v_{1f}-v_{2f}}{v_{2i}-v_{1i}}</math> 
Solve for <math>v_{2f}</math> 

<math>v_{2f}=v_{1f}-e(v_{2i}-v_{1i})</math> 
Plug in the values stated in the problem. 
<math>v_{2f}=4-.55(5-8)</math>
<math>v_{2f}=2.35\frac{m}{s}</math>

===Difficult===
[[File:Cor1.jpg|right]]
Bouncing Ball Example:

Consider a ball dropped from initial height <math>h_0</math>. When it bounces it reaches a height <math>h_f</math>. Find the coefficient of restitution of this collision. 
<math>e=\frac{v_{floor_f}-v_{ball_f}}{v_{ball_i}-v_{floor_i}}</math> 
The floor's velocity is always 0. 
<math>e=\frac{v_{ball_f}}{v_{ball_i}}</math> 
Recall the law of conservation of energy: 
<math> \Delta KE = \Delta PE</math> 
For the ball from h0 to h=0
<math> \frac{1}{2}mv_{ball_i}^2=mgh_0</math> 
<math> \frac{1}{2}v_{ball_i}^2=gh_0</math> 
<math> v_{ball_i}=2\sqrt{gh_0} </math> 
The same concept can be applied to the second bounce. 
<math>v_{ball_f}=2\sqrt{gh_f}</math> 
Plugging this in to the coefficient of restitution expression we get: 
<math>e=\frac{2\sqrt{mgh_f}}{2\sqrt{mgh_0}}</math> 
<math>e=\frac{\sqrt{h_f}}{\sqrt{h_0}}</math> 

==Connectedness==
'''How is this topic connected to something that you are interested in?''' 
This topic has major connections to sports. As a former high school volleyball player, it is very interesting to think about the collisions that occurred in the game and how the coefficient of restitution is related to those collisions. 
[[File:sportsgraph.png|right|thumb|This graph shows the initial and final velocities of many different balls from different sports.]]
'''How is it connected to your major (Mechanical Engineering)?''' 
The coefficient of restitution is a key concept in the study of Dynamics which is the foundation of many aspects of mechanical engineering. It also has to do with applications of mechanical engineering regarding materials. 

[[File:sportschart.png|right|thumb|This chart shows the relationship between different coefficients of restitution and the initial and final velocities of sports balls.]]

'''Is there an interesting industrial application?'''
In sports the coefficient of restitution is an important concept especially in sports like golf or tennis where collisions are a frequent and essential part of the game. In these sports sometimes limitations are put on how high the coefficient of restitution can be for certain clubs or bats. The maximum coefficient of restitution allowed in golf for clubs is .82. There have been studies done on the coefficients for different sports balls. These studies usually follow a procedure where different balls are dropped and left to bounce. Then data regarding their initial and final velocities or how high they bounced is recorded. The images below show some of the data from one of these experiments.

== See also ==

http://physicsbook.gatech.edu/Elastic_Collisions 
http://physicsbook.gatech.edu/Inelastic_Collisions 
http://physicsbook.gatech.edu/Collisions 
http://physicsbook.gatech.edu/Conservation_of_Momentum 
http://physicsbook.gatech.edu/Conservation_of_Energy 

===External links===
http://www.quintic.com/education/case_studies/coefficient_restitution.htm

==References==

McGill, David J., and Wilton W. King. Engineering mechanics : an introduction to dynamics. Bloomington, IN: Tichenor Pub, 2003. Print. 
"Quintic 4 Education Sports Video Analysis Software Education Package - The Coefficient of Restitution." Quintic 4 Education Sports Video Analysis Software Education Package - The Coefficient of Restitution. Quintic 4 Education, n.d. Web. 05 Dec. 2015. 
"Coefficient of Restitution (COR)." Yale. Web. 5 Dec. 2015. 

[[Category:Collisions]]

Coefficient of Restitution

2015-12-06T03:20:14Z

Mmoreno30: /* A Mathematical Model */

Claimed by Maria Moreno

The coefficient of restitution measures the amount of Kinetic Energy remains and how much energy is lost after a collision. It also shows the degree of elasticity of a collision. It is defined as the ratio of the difference in initial velocities divided by the difference in final velocities. The coefficient of restitution of a collision is usually between 0 and 1 except in rare cases when energy is released by a collision.
[[File:coe2.png|right]]

==The Main Idea==

The coefficient of restitution is a ratio that describes the degree of elasticity of a collision. It is used to solve problems dealing with collisions that are not perfectly elastic or inelastic.
The equation that describes the coefficient of restitution involved dividing the difference in the final velocities by the difference in the initial velocity.

Consider objects A and B with initial velocities vAi and vBi and final velocities vAf and vBf.
The coefficient of restitution, e is determined with the following formula:

<math> e = \frac{v_{Bf}-v_{Af}}{v_{Ai}-v_{Bi}} </math> 
The coefficient is a unitless form of measurement as it is the ratio of two velocities and the units cancel.

===A Mathematical Model===
Consider a perfectly elastic collision between objects A and B where vAi and vBi refer to the initial velocities of A and B and vAf and vBf refer to the final velocities of A and B.

Start with the conservation of momentum principle which states that 
<math>\vec{p_i} = \vec{p_f}</math> and 
<math>m_A\vec{v_{Ai}}+m_A\vec{v_{Bi}}=m_A\vec{v_{Af}}+m_A\vec{v_{Bf}}</math> (1) 

Remember that for a perfectly elastic collision kinetic energy is also conserved meaning: 
<math>KE_i = KE_f</math> and 
<math>\frac{1}{2}m_A v^2_{Ai}+\frac{1}{2}m_A v^2_Bi=\frac{1}{2}m_A v^2_{Af}+\frac{1}{2}m_A v^2_{Bf}</math> (2) 

Dividing (2) by (1) yields: 

<math>\vec{v_{Ai}}+\vec{v_{Bi}}=\vec{v_{Af}}+\vec{v_{Bf}}</math> 
<math>1=\frac{v_{Ai}+v_{Bi}}{v_{Af}+v_{Bf}}</math> 

<math>e=\frac{v_{Ai}+v_{Bi}}{v_{Af}+v_{Bf}}</math> 

For perfectly elastic collision <math>e</math>=1.

===Inelastic Collisions===

An inelastic collision is a type of collision where Kinetic Energy is not conserved. In this case the difference in initial velocities will not equal the difference in final velocities. This means that the coefficient of restitution will not equal 1.

Consider the same collision as previously described, except this collision is inelastic and Kinetic Energy is not conserved. Since some of the kinetic energy is dissipated: 
<math>\frac{1}{2}m_A v^2_{Ai}+\frac{1}{2}m_A v^2_Bi<\frac{1}{2}m_A v^2_{Af}+\frac{1}{2}m_A v^2_{Bf}</math> 
<math>\frac{v_{Ai}+v_{Bi}}{v_{Af}+v_{Bf}}<1</math> 

<math>e=\frac{v_{Ai}+v_{Bi}}{v_{Af}+v_{Bf}}<1</math> 

A perfectly inelastic collision (for example when two objects collide and then stick together) has a coefficient of restitution of 0.

===A Computational Model===
Here is a link to a glowscript simulation
http://www.glowscript.org/#/user/mmoreno181/folder/phys/program/cor 
Here is the code for the simulation: 
<code>
ball1=sphere(pos=vec(0,0,0), color=color.red, radius=.5)
ball2=sphere(pos=vec(10,0,0), color=color.blue, radius=.5)

#initialize velocity
v1i=vec(.02,0,0)
v2i=vec(-.1,0,0)
v1f=vec(-.07,0,0)

#set coefficient of resitution
e=.5

#find v2f
v2f=v1f-(e*(v2i-v1i))

#find the time that the balls collide
tmeet= (ball2.pos.x-ball1.pos.x)/(v1i.x-v2i.x)

t=0

#animate
while t<150:
rate(100)
if t < tmeet: #if it's before the collision
ball1.pos=ball1.pos+v1i
ball2.pos=ball2.pos+v2i
else:
ball1.pos=ball1.pos+v1f
ball2.pos=ball2.pos+v2f

t=t+1
</code>

==Examples==
===Simple===
A ball is traveling at <math>4 \frac{m}{s}</math> and collides with a ball. After the collision the ball travels at <math>2.4\frac{m}{s}</math> in the opposite direction. Find the coefficient of restitution of the collision. 
Start with the definition of the coefficient of restitution: 
<math>e=\frac{v_{wall_f}-v_{ball_f}}{v_{ball_i}-v_{wall_i}}</math> 
Then plug in values and solve: 
<math>e=\frac{0-v_{ball_f}}{v_{ball_i}-0}</math> 
<math>e=\frac{-(-2.4)}{4}</math> 
<math>e=.6</math>

===Middling===
A hockey puck travels at a constant velocity <math>5\frac{m}{s}</math> and is hit by another puck with a velocity of <math>8\frac{m}{s}</math>. After the collision, it travels in the opposite direction at 4m/s. This collision has a coefficient of restitution of .55. Find the velocity of the second puck after the collision. 
Start with the definition of the coefficient of restitution. We are calling the velocity of the first puck <math>v_1</math> and the velocity of the second puck <math>v_2</math>. 
<math>e=\frac{v_{1f}-v_{2f}}{v_{2i}-v_{1i}}</math> 
Solve for <math>v_{2f}</math> 

<math>v_{2f}=v_{1f}-e(v_{2i}-v_{1i})</math> 
Plug in the values stated in the problem. 
<math>v_{2f}=4-.55(5-8)</math>
<math>v_{2f}=2.35\frac{m}{s}</math>

===Difficult===
[[File:Cor1.jpg|right]]
Bouncing Ball Example:

Consider a ball dropped from initial height <math>h_0</math>. When it bounces it reaches a height <math>h_f</math>. Find the coefficient of restitution of this collision. 
<math>e=\frac{v_{floor_f}-v_{ball_f}}{v_{ball_i}-v_{floor_i}}</math> 
The floor's velocity is always 0. 
<math>e=\frac{v_{ball_f}}{v_{ball_i}}</math> 
Recall the law of conservation of energy: 
<math> \Delta KE = \Delta PE</math> 
For the ball from h0 to h=0
<math> \frac{1}{2}mv_{ball_i}^2=mgh_0</math> 
<math> \frac{1}{2}v_{ball_i}^2=gh_0</math> 
<math> v_{ball_i}=2\sqrt{gh_0} </math> 
The same concept can be applied to the second bounce. 
<math>v_{ball_f}=2\sqrt{gh_f}</math> 
Plugging this in to the coefficient of restitution expression we get: 
<math>e=\frac{2\sqrt{mgh_f}}{2\sqrt{mgh_0}}</math> 
<math>e=\frac{\sqrt{h_f}}{\sqrt{h_0}}</math> 

==Connectedness==
'''How is this topic connected to something that you are interested in?''' 
This topic has major connections to sports. As a former high school volleyball player, it is very interesting to think about the collisions that occurred in the game and how the coefficient of restitution is related to those collisions. 
[[File:sportsgraph.png|right|thumb|This graph shows the initial and final velocities of many different balls from different sports.]]
'''How is it connected to your major (Mechanical Engineering)?''' 
The coefficient of restitution is a key concept in the study of Dynamics which is the foundation of many aspects of mechanical engineering. It also has to do with applications of mechanical engineering regarding materials. 

[[File:sportschart.png|right|thumb|This chart shows the relationship between different coefficients of restitution and the initial and final velocities of sports balls.]]

'''Is there an interesting industrial application?'''
In sports the coefficient of restitution is an important concept especially in sports like golf or tennis where collisions are a frequent and essential part of the game. In these sports sometimes limitations are put on how high the coefficient of restitution can be for certain clubs or bats. The maximum coefficient of restitution allowed in golf for clubs is .82. There have been studies done on the coefficients for different sports balls. These studies usually follow a procedure where different balls are dropped and left to bounce. Then data regarding their initial and final velocities or how high they bounced is recorded. The images below show some of the data from one of these experiments.

== See also ==

http://physicsbook.gatech.edu/Elastic_Collisions 
http://physicsbook.gatech.edu/Inelastic_Collisions 
http://physicsbook.gatech.edu/Collisions 
http://physicsbook.gatech.edu/Conservation_of_Momentum 
http://physicsbook.gatech.edu/Conservation_of_Energy 

===External links===
http://www.quintic.com/education/case_studies/coefficient_restitution.htm

==References==

McGill, David J., and Wilton W. King. Engineering mechanics : an introduction to dynamics. Bloomington, IN: Tichenor Pub, 2003. Print. 
"Quintic 4 Education Sports Video Analysis Software Education Package - The Coefficient of Restitution." Quintic 4 Education Sports Video Analysis Software Education Package - The Coefficient of Restitution. Quintic 4 Education, n.d. Web. 05 Dec. 2015. 
"Coefficient of Restitution (COR)." Yale. Web. 5 Dec. 2015. 

[[Category:Collisions]]

Coefficient of Restitution

2015-12-06T03:19:52Z

Mmoreno30: /* A Computational Model */

Claimed by Maria Moreno

The coefficient of restitution measures the amount of Kinetic Energy remains and how much energy is lost after a collision. It also shows the degree of elasticity of a collision. It is defined as the ratio of the difference in initial velocities divided by the difference in final velocities. The coefficient of restitution of a collision is usually between 0 and 1 except in rare cases when energy is released by a collision.
[[File:coe2.png|right]]

==The Main Idea==

The coefficient of restitution is a ratio that describes the degree of elasticity of a collision. It is used to solve problems dealing with collisions that are not perfectly elastic or inelastic.
The equation that describes the coefficient of restitution involved dividing the difference in the final velocities by the difference in the initial velocity.

Consider objects A and B with initial velocities vAi and vBi and final velocities vAf and vBf.
The coefficient of restitution, e is determined with the following formula:

<math> e = \frac{v_{Bf}-v_{Af}}{v_{Ai}-v_{Bi}} </math> 
The coefficient is a unitless form of measurement as it is the ratio of two velocities and the units cancel.

===A Mathematical Model===
Consider a perfectly elastic collision between objects A and B where vAi and vBi refer to the initial velocities of A and B and vAf and vBf refer to the final velocities of A and B.

Start with the conservation of momentum principle which states that 
<math>\vec{p_i} = \vec{p_f}</math> and 
<math>m_A\vec{v_{Ai}}+m_A\vec{v_{Bi}}=m_A\vec{v_{Af}}+m_A\vec{v_{Bf}}</math> (1) 

Remember that for a perfectly elastic collision kinetic energy is also conserved meaning: 
<math>KE_i = KE_f</math> and 
<math>\frac{1}{2}m_A v^2_{Ai}+\frac{1}{2}m_A v^2_Bi=\frac{1}{2}m_A v^2_{Af}+\frac{1}{2}m_A v^2_{Bf}</math> (2) 

Dividing (2) by (1) yields: 

<math>\vec{v_{Ai}}+\vec{v_{Bi}}=\vec{v_{Af}}+\vec{v_{Bf}}</math> 
<math>1=\frac{v_{Ai}+v_{Bi}}{v_{Af}+v_{Bf}}</math> 

<math>e=\frac{v_{Ai}+v_{Bi}}{v_{Af}+v_{Bf}}</math> 

For perfectly elastic collision e=1.

===Inelastic Collisions===

An inelastic collision is a type of collision where Kinetic Energy is not conserved. In this case the difference in initial velocities will not equal the difference in final velocities. This means that the coefficient of restitution will not equal 1.

Consider the same collision as previously described, except this collision is inelastic and Kinetic Energy is not conserved. Since some of the kinetic energy is dissipated: 
<math>\frac{1}{2}m_A v^2_{Ai}+\frac{1}{2}m_A v^2_Bi<\frac{1}{2}m_A v^2_{Af}+\frac{1}{2}m_A v^2_{Bf}</math> 
<math>\frac{v_{Ai}+v_{Bi}}{v_{Af}+v_{Bf}}<1</math> 

<math>e=\frac{v_{Ai}+v_{Bi}}{v_{Af}+v_{Bf}}<1</math> 

A perfectly inelastic collision (for example when two objects collide and then stick together) has a coefficient of restitution of 0.

===A Computational Model===
Here is a link to a glowscript simulation
http://www.glowscript.org/#/user/mmoreno181/folder/phys/program/cor 
Here is the code for the simulation: 
<code>
ball1=sphere(pos=vec(0,0,0), color=color.red, radius=.5)
ball2=sphere(pos=vec(10,0,0), color=color.blue, radius=.5)

#initialize velocity
v1i=vec(.02,0,0)
v2i=vec(-.1,0,0)
v1f=vec(-.07,0,0)

#set coefficient of resitution
e=.5

#find v2f
v2f=v1f-(e*(v2i-v1i))

#find the time that the balls collide
tmeet= (ball2.pos.x-ball1.pos.x)/(v1i.x-v2i.x)

t=0

#animate
while t<150:
rate(100)
if t < tmeet: #if it's before the collision
ball1.pos=ball1.pos+v1i
ball2.pos=ball2.pos+v2i
else:
ball1.pos=ball1.pos+v1f
ball2.pos=ball2.pos+v2f

t=t+1
</code>

==Examples==
===Simple===
A ball is traveling at <math>4 \frac{m}{s}</math> and collides with a ball. After the collision the ball travels at <math>2.4\frac{m}{s}</math> in the opposite direction. Find the coefficient of restitution of the collision. 
Start with the definition of the coefficient of restitution: 
<math>e=\frac{v_{wall_f}-v_{ball_f}}{v_{ball_i}-v_{wall_i}}</math> 
Then plug in values and solve: 
<math>e=\frac{0-v_{ball_f}}{v_{ball_i}-0}</math> 
<math>e=\frac{-(-2.4)}{4}</math> 
<math>e=.6</math>

===Middling===
A hockey puck travels at a constant velocity <math>5\frac{m}{s}</math> and is hit by another puck with a velocity of <math>8\frac{m}{s}</math>. After the collision, it travels in the opposite direction at 4m/s. This collision has a coefficient of restitution of .55. Find the velocity of the second puck after the collision. 
Start with the definition of the coefficient of restitution. We are calling the velocity of the first puck <math>v_1</math> and the velocity of the second puck <math>v_2</math>. 
<math>e=\frac{v_{1f}-v_{2f}}{v_{2i}-v_{1i}}</math> 
Solve for <math>v_{2f}</math> 

<math>v_{2f}=v_{1f}-e(v_{2i}-v_{1i})</math> 
Plug in the values stated in the problem. 
<math>v_{2f}=4-.55(5-8)</math>
<math>v_{2f}=2.35\frac{m}{s}</math>

===Difficult===
[[File:Cor1.jpg|right]]
Bouncing Ball Example:

Consider a ball dropped from initial height <math>h_0</math>. When it bounces it reaches a height <math>h_f</math>. Find the coefficient of restitution of this collision. 
<math>e=\frac{v_{floor_f}-v_{ball_f}}{v_{ball_i}-v_{floor_i}}</math> 
The floor's velocity is always 0. 
<math>e=\frac{v_{ball_f}}{v_{ball_i}}</math> 
Recall the law of conservation of energy: 
<math> \Delta KE = \Delta PE</math> 
For the ball from h0 to h=0
<math> \frac{1}{2}mv_{ball_i}^2=mgh_0</math> 
<math> \frac{1}{2}v_{ball_i}^2=gh_0</math> 
<math> v_{ball_i}=2\sqrt{gh_0} </math> 
The same concept can be applied to the second bounce. 
<math>v_{ball_f}=2\sqrt{gh_f}</math> 
Plugging this in to the coefficient of restitution expression we get: 
<math>e=\frac{2\sqrt{mgh_f}}{2\sqrt{mgh_0}}</math> 
<math>e=\frac{\sqrt{h_f}}{\sqrt{h_0}}</math> 

==Connectedness==
'''How is this topic connected to something that you are interested in?''' 
This topic has major connections to sports. As a former high school volleyball player, it is very interesting to think about the collisions that occurred in the game and how the coefficient of restitution is related to those collisions. 
[[File:sportsgraph.png|right|thumb|This graph shows the initial and final velocities of many different balls from different sports.]]
'''How is it connected to your major (Mechanical Engineering)?''' 
The coefficient of restitution is a key concept in the study of Dynamics which is the foundation of many aspects of mechanical engineering. It also has to do with applications of mechanical engineering regarding materials. 

[[File:sportschart.png|right|thumb|This chart shows the relationship between different coefficients of restitution and the initial and final velocities of sports balls.]]

'''Is there an interesting industrial application?'''
In sports the coefficient of restitution is an important concept especially in sports like golf or tennis where collisions are a frequent and essential part of the game. In these sports sometimes limitations are put on how high the coefficient of restitution can be for certain clubs or bats. The maximum coefficient of restitution allowed in golf for clubs is .82. There have been studies done on the coefficients for different sports balls. These studies usually follow a procedure where different balls are dropped and left to bounce. Then data regarding their initial and final velocities or how high they bounced is recorded. The images below show some of the data from one of these experiments.

== See also ==

http://physicsbook.gatech.edu/Elastic_Collisions 
http://physicsbook.gatech.edu/Inelastic_Collisions 
http://physicsbook.gatech.edu/Collisions 
http://physicsbook.gatech.edu/Conservation_of_Momentum 
http://physicsbook.gatech.edu/Conservation_of_Energy 

===External links===
http://www.quintic.com/education/case_studies/coefficient_restitution.htm

==References==

McGill, David J., and Wilton W. King. Engineering mechanics : an introduction to dynamics. Bloomington, IN: Tichenor Pub, 2003. Print. 
"Quintic 4 Education Sports Video Analysis Software Education Package - The Coefficient of Restitution." Quintic 4 Education Sports Video Analysis Software Education Package - The Coefficient of Restitution. Quintic 4 Education, n.d. Web. 05 Dec. 2015. 
"Coefficient of Restitution (COR)." Yale. Web. 5 Dec. 2015. 

[[Category:Collisions]]

Coefficient of Restitution

2015-12-06T03:19:35Z

Mmoreno30: /* A Computational Model */

Claimed by Maria Moreno

The coefficient of restitution measures the amount of Kinetic Energy remains and how much energy is lost after a collision. It also shows the degree of elasticity of a collision. It is defined as the ratio of the difference in initial velocities divided by the difference in final velocities. The coefficient of restitution of a collision is usually between 0 and 1 except in rare cases when energy is released by a collision.
[[File:coe2.png|right]]

==The Main Idea==

The coefficient of restitution is a ratio that describes the degree of elasticity of a collision. It is used to solve problems dealing with collisions that are not perfectly elastic or inelastic.
The equation that describes the coefficient of restitution involved dividing the difference in the final velocities by the difference in the initial velocity.

Consider objects A and B with initial velocities vAi and vBi and final velocities vAf and vBf.
The coefficient of restitution, e is determined with the following formula:

<math> e = \frac{v_{Bf}-v_{Af}}{v_{Ai}-v_{Bi}} </math> 
The coefficient is a unitless form of measurement as it is the ratio of two velocities and the units cancel.

===A Mathematical Model===
Consider a perfectly elastic collision between objects A and B where vAi and vBi refer to the initial velocities of A and B and vAf and vBf refer to the final velocities of A and B.

Start with the conservation of momentum principle which states that 
<math>\vec{p_i} = \vec{p_f}</math> and 
<math>m_A\vec{v_{Ai}}+m_A\vec{v_{Bi}}=m_A\vec{v_{Af}}+m_A\vec{v_{Bf}}</math> (1) 

Remember that for a perfectly elastic collision kinetic energy is also conserved meaning: 
<math>KE_i = KE_f</math> and 
<math>\frac{1}{2}m_A v^2_{Ai}+\frac{1}{2}m_A v^2_Bi=\frac{1}{2}m_A v^2_{Af}+\frac{1}{2}m_A v^2_{Bf}</math> (2) 

Dividing (2) by (1) yields: 

<math>\vec{v_{Ai}}+\vec{v_{Bi}}=\vec{v_{Af}}+\vec{v_{Bf}}</math> 
<math>1=\frac{v_{Ai}+v_{Bi}}{v_{Af}+v_{Bf}}</math> 

<math>e=\frac{v_{Ai}+v_{Bi}}{v_{Af}+v_{Bf}}</math> 

For perfectly elastic collision e=1.

===Inelastic Collisions===

An inelastic collision is a type of collision where Kinetic Energy is not conserved. In this case the difference in initial velocities will not equal the difference in final velocities. This means that the coefficient of restitution will not equal 1.

Consider the same collision as previously described, except this collision is inelastic and Kinetic Energy is not conserved. Since some of the kinetic energy is dissipated: 
<math>\frac{1}{2}m_A v^2_{Ai}+\frac{1}{2}m_A v^2_Bi<\frac{1}{2}m_A v^2_{Af}+\frac{1}{2}m_A v^2_{Bf}</math> 
<math>\frac{v_{Ai}+v_{Bi}}{v_{Af}+v_{Bf}}<1</math> 

<math>e=\frac{v_{Ai}+v_{Bi}}{v_{Af}+v_{Bf}}<1</math> 

A perfectly inelastic collision (for example when two objects collide and then stick together) has a coefficient of restitution of 0.

===A Computational Model===
Here is a link to a glowscript simulation
http://www.glowscript.org/#/user/mmoreno181/folder/phys/program/cor 
Here is the code for the simulation: 
<code>
ball1=sphere(pos=vec(0,0,0), color=color.red, radius=.5)
ball2=sphere(pos=vec(10,0,0), color=color.blue, radius=.5)
#initialize velocity
v1i=vec(.02,0,0)
v2i=vec(-.1,0,0)
v1f=vec(-.07,0,0)

#set coefficient of resitution
e=.5

#find v2f
v2f=v1f-(e*(v2i-v1i))

#find the time that the balls collide
tmeet= (ball2.pos.x-ball1.pos.x)/(v1i.x-v2i.x)

t=0

#animate
while t<150:
rate(100)
if t < tmeet: #if it's before the collision
ball1.pos=ball1.pos+v1i
ball2.pos=ball2.pos+v2i
else:
ball1.pos=ball1.pos+v1f
ball2.pos=ball2.pos+v2f

t=t+1
</code>

==Examples==
===Simple===
A ball is traveling at <math>4 \frac{m}{s}</math> and collides with a ball. After the collision the ball travels at <math>2.4\frac{m}{s}</math> in the opposite direction. Find the coefficient of restitution of the collision. 
Start with the definition of the coefficient of restitution: 
<math>e=\frac{v_{wall_f}-v_{ball_f}}{v_{ball_i}-v_{wall_i}}</math> 
Then plug in values and solve: 
<math>e=\frac{0-v_{ball_f}}{v_{ball_i}-0}</math> 
<math>e=\frac{-(-2.4)}{4}</math> 
<math>e=.6</math>

===Middling===
A hockey puck travels at a constant velocity <math>5\frac{m}{s}</math> and is hit by another puck with a velocity of <math>8\frac{m}{s}</math>. After the collision, it travels in the opposite direction at 4m/s. This collision has a coefficient of restitution of .55. Find the velocity of the second puck after the collision. 
Start with the definition of the coefficient of restitution. We are calling the velocity of the first puck <math>v_1</math> and the velocity of the second puck <math>v_2</math>. 
<math>e=\frac{v_{1f}-v_{2f}}{v_{2i}-v_{1i}}</math> 
Solve for <math>v_{2f}</math> 

<math>v_{2f}=v_{1f}-e(v_{2i}-v_{1i})</math> 
Plug in the values stated in the problem. 
<math>v_{2f}=4-.55(5-8)</math>
<math>v_{2f}=2.35\frac{m}{s}</math>

===Difficult===
[[File:Cor1.jpg|right]]
Bouncing Ball Example:

Consider a ball dropped from initial height <math>h_0</math>. When it bounces it reaches a height <math>h_f</math>. Find the coefficient of restitution of this collision. 
<math>e=\frac{v_{floor_f}-v_{ball_f}}{v_{ball_i}-v_{floor_i}}</math> 
The floor's velocity is always 0. 
<math>e=\frac{v_{ball_f}}{v_{ball_i}}</math> 
Recall the law of conservation of energy: 
<math> \Delta KE = \Delta PE</math> 
For the ball from h0 to h=0
<math> \frac{1}{2}mv_{ball_i}^2=mgh_0</math> 
<math> \frac{1}{2}v_{ball_i}^2=gh_0</math> 
<math> v_{ball_i}=2\sqrt{gh_0} </math> 
The same concept can be applied to the second bounce. 
<math>v_{ball_f}=2\sqrt{gh_f}</math> 
Plugging this in to the coefficient of restitution expression we get: 
<math>e=\frac{2\sqrt{mgh_f}}{2\sqrt{mgh_0}}</math> 
<math>e=\frac{\sqrt{h_f}}{\sqrt{h_0}}</math> 

==Connectedness==
'''How is this topic connected to something that you are interested in?''' 
This topic has major connections to sports. As a former high school volleyball player, it is very interesting to think about the collisions that occurred in the game and how the coefficient of restitution is related to those collisions. 
[[File:sportsgraph.png|right|thumb|This graph shows the initial and final velocities of many different balls from different sports.]]
'''How is it connected to your major (Mechanical Engineering)?''' 
The coefficient of restitution is a key concept in the study of Dynamics which is the foundation of many aspects of mechanical engineering. It also has to do with applications of mechanical engineering regarding materials. 

[[File:sportschart.png|right|thumb|This chart shows the relationship between different coefficients of restitution and the initial and final velocities of sports balls.]]

'''Is there an interesting industrial application?'''
In sports the coefficient of restitution is an important concept especially in sports like golf or tennis where collisions are a frequent and essential part of the game. In these sports sometimes limitations are put on how high the coefficient of restitution can be for certain clubs or bats. The maximum coefficient of restitution allowed in golf for clubs is .82. There have been studies done on the coefficients for different sports balls. These studies usually follow a procedure where different balls are dropped and left to bounce. Then data regarding their initial and final velocities or how high they bounced is recorded. The images below show some of the data from one of these experiments.

== See also ==

http://physicsbook.gatech.edu/Elastic_Collisions 
http://physicsbook.gatech.edu/Inelastic_Collisions 
http://physicsbook.gatech.edu/Collisions 
http://physicsbook.gatech.edu/Conservation_of_Momentum 
http://physicsbook.gatech.edu/Conservation_of_Energy 

===External links===
http://www.quintic.com/education/case_studies/coefficient_restitution.htm

==References==

McGill, David J., and Wilton W. King. Engineering mechanics : an introduction to dynamics. Bloomington, IN: Tichenor Pub, 2003. Print. 
"Quintic 4 Education Sports Video Analysis Software Education Package - The Coefficient of Restitution." Quintic 4 Education Sports Video Analysis Software Education Package - The Coefficient of Restitution. Quintic 4 Education, n.d. Web. 05 Dec. 2015. 
"Coefficient of Restitution (COR)." Yale. Web. 5 Dec. 2015. 

[[Category:Collisions]]

Coefficient of Restitution

2015-12-06T03:17:22Z

Mmoreno30: /* A Computational Model */

Claimed by Maria Moreno

The coefficient of restitution measures the amount of Kinetic Energy remains and how much energy is lost after a collision. It also shows the degree of elasticity of a collision. It is defined as the ratio of the difference in initial velocities divided by the difference in final velocities. The coefficient of restitution of a collision is usually between 0 and 1 except in rare cases when energy is released by a collision.
[[File:coe2.png|right]]

==The Main Idea==

The coefficient of restitution is a ratio that describes the degree of elasticity of a collision. It is used to solve problems dealing with collisions that are not perfectly elastic or inelastic.
The equation that describes the coefficient of restitution involved dividing the difference in the final velocities by the difference in the initial velocity.

Consider objects A and B with initial velocities vAi and vBi and final velocities vAf and vBf.
The coefficient of restitution, e is determined with the following formula:

<math> e = \frac{v_{Bf}-v_{Af}}{v_{Ai}-v_{Bi}} </math> 
The coefficient is a unitless form of measurement as it is the ratio of two velocities and the units cancel.

===A Mathematical Model===
Consider a perfectly elastic collision between objects A and B where vAi and vBi refer to the initial velocities of A and B and vAf and vBf refer to the final velocities of A and B.

Start with the conservation of momentum principle which states that 
<math>\vec{p_i} = \vec{p_f}</math> and 
<math>m_A\vec{v_{Ai}}+m_A\vec{v_{Bi}}=m_A\vec{v_{Af}}+m_A\vec{v_{Bf}}</math> (1) 

Remember that for a perfectly elastic collision kinetic energy is also conserved meaning: 
<math>KE_i = KE_f</math> and 
<math>\frac{1}{2}m_A v^2_{Ai}+\frac{1}{2}m_A v^2_Bi=\frac{1}{2}m_A v^2_{Af}+\frac{1}{2}m_A v^2_{Bf}</math> (2) 

Dividing (2) by (1) yields: 

<math>\vec{v_{Ai}}+\vec{v_{Bi}}=\vec{v_{Af}}+\vec{v_{Bf}}</math> 
<math>1=\frac{v_{Ai}+v_{Bi}}{v_{Af}+v_{Bf}}</math> 

<math>e=\frac{v_{Ai}+v_{Bi}}{v_{Af}+v_{Bf}}</math> 

For perfectly elastic collision e=1.

===Inelastic Collisions===

An inelastic collision is a type of collision where Kinetic Energy is not conserved. In this case the difference in initial velocities will not equal the difference in final velocities. This means that the coefficient of restitution will not equal 1.

Consider the same collision as previously described, except this collision is inelastic and Kinetic Energy is not conserved. Since some of the kinetic energy is dissipated: 
<math>\frac{1}{2}m_A v^2_{Ai}+\frac{1}{2}m_A v^2_Bi<\frac{1}{2}m_A v^2_{Af}+\frac{1}{2}m_A v^2_{Bf}</math> 
<math>\frac{v_{Ai}+v_{Bi}}{v_{Af}+v_{Bf}}<1</math> 

<math>e=\frac{v_{Ai}+v_{Bi}}{v_{Af}+v_{Bf}}<1</math> 

A perfectly inelastic collision (for example when two objects collide and then stick together) has a coefficient of restitution of 0.

===A Computational Model===
Here is a link to a glowscript simulation
http://www.glowscript.org/#/user/mmoreno181/folder/phys/program/cor 
Here is the code for the simulation: 
<code>
ball1=sphere(pos=vec(0,0,0), color=color.red, radius=.5)
ball2=sphere(pos=vec(10,0,0), color=color.blue, radius=.5)

#initialize velocity
ball1=sphere(pos=vec(0,0,0), color=color.red, radius=.5)
ball2=sphere(pos=vec(10,0,0), color=color.blue, radius=.5)

#initialize velocity
v1i=vec(.02,0,0)
v2i=vec(-.1,0,0)
v1f=vec(-.07,0,0)

#set coefficient of resitution
e=.5

#find v2f
v2f=v1f-(e*(v2i-v1i))

#find the time that the balls collide
tmeet= (ball2.pos.x-ball1.pos.x)/(v1i.x-v2i.x)

t=0

#animate
while t<150:
rate(100)
if t < tmeet: #if it's before the collision
ball1.pos=ball1.pos+v1i
ball2.pos=ball2.pos+v2i
else:
ball1.pos=ball1.pos+v1f
ball2.pos=ball2.pos+v2f

t=t+1
</code>

==Examples==
===Simple===
A ball is traveling at <math>4 \frac{m}{s}</math> and collides with a ball. After the collision the ball travels at <math>2.4\frac{m}{s}</math> in the opposite direction. Find the coefficient of restitution of the collision. 
Start with the definition of the coefficient of restitution: 
<math>e=\frac{v_{wall_f}-v_{ball_f}}{v_{ball_i}-v_{wall_i}}</math> 
Then plug in values and solve: 
<math>e=\frac{0-v_{ball_f}}{v_{ball_i}-0}</math> 
<math>e=\frac{-(-2.4)}{4}</math> 
<math>e=.6</math>

===Middling===
A hockey puck travels at a constant velocity <math>5\frac{m}{s}</math> and is hit by another puck with a velocity of <math>8\frac{m}{s}</math>. After the collision, it travels in the opposite direction at 4m/s. This collision has a coefficient of restitution of .55. Find the velocity of the second puck after the collision. 
Start with the definition of the coefficient of restitution. We are calling the velocity of the first puck <math>v_1</math> and the velocity of the second puck <math>v_2</math>. 
<math>e=\frac{v_{1f}-v_{2f}}{v_{2i}-v_{1i}}</math> 
Solve for <math>v_{2f}</math> 

<math>v_{2f}=v_{1f}-e(v_{2i}-v_{1i})</math> 
Plug in the values stated in the problem. 
<math>v_{2f}=4-.55(5-8)</math>
<math>v_{2f}=2.35\frac{m}{s}</math>

===Difficult===
[[File:Cor1.jpg|right]]
Bouncing Ball Example:

Consider a ball dropped from initial height <math>h_0</math>. When it bounces it reaches a height <math>h_f</math>. Find the coefficient of restitution of this collision. 
<math>e=\frac{v_{floor_f}-v_{ball_f}}{v_{ball_i}-v_{floor_i}}</math> 
The floor's velocity is always 0. 
<math>e=\frac{v_{ball_f}}{v_{ball_i}}</math> 
Recall the law of conservation of energy: 
<math> \Delta KE = \Delta PE</math> 
For the ball from h0 to h=0
<math> \frac{1}{2}mv_{ball_i}^2=mgh_0</math> 
<math> \frac{1}{2}v_{ball_i}^2=gh_0</math> 
<math> v_{ball_i}=2\sqrt{gh_0} </math> 
The same concept can be applied to the second bounce. 
<math>v_{ball_f}=2\sqrt{gh_f}</math> 
Plugging this in to the coefficient of restitution expression we get: 
<math>e=\frac{2\sqrt{mgh_f}}{2\sqrt{mgh_0}}</math> 
<math>e=\frac{\sqrt{h_f}}{\sqrt{h_0}}</math> 

==Connectedness==
'''How is this topic connected to something that you are interested in?''' 
This topic has major connections to sports. As a former high school volleyball player, it is very interesting to think about the collisions that occurred in the game and how the coefficient of restitution is related to those collisions. 
[[File:sportsgraph.png|right|thumb|This graph shows the initial and final velocities of many different balls from different sports.]]
'''How is it connected to your major (Mechanical Engineering)?''' 
The coefficient of restitution is a key concept in the study of Dynamics which is the foundation of many aspects of mechanical engineering. It also has to do with applications of mechanical engineering regarding materials. 

[[File:sportschart.png|right|thumb|This chart shows the relationship between different coefficients of restitution and the initial and final velocities of sports balls.]]

'''Is there an interesting industrial application?'''
In sports the coefficient of restitution is an important concept especially in sports like golf or tennis where collisions are a frequent and essential part of the game. In these sports sometimes limitations are put on how high the coefficient of restitution can be for certain clubs or bats. The maximum coefficient of restitution allowed in golf for clubs is .82. There have been studies done on the coefficients for different sports balls. These studies usually follow a procedure where different balls are dropped and left to bounce. Then data regarding their initial and final velocities or how high they bounced is recorded. The images below show some of the data from one of these experiments.

== See also ==

http://physicsbook.gatech.edu/Elastic_Collisions 
http://physicsbook.gatech.edu/Inelastic_Collisions 
http://physicsbook.gatech.edu/Collisions 
http://physicsbook.gatech.edu/Conservation_of_Momentum 
http://physicsbook.gatech.edu/Conservation_of_Energy 

===External links===
http://www.quintic.com/education/case_studies/coefficient_restitution.htm

==References==

McGill, David J., and Wilton W. King. Engineering mechanics : an introduction to dynamics. Bloomington, IN: Tichenor Pub, 2003. Print. 
"Quintic 4 Education Sports Video Analysis Software Education Package - The Coefficient of Restitution." Quintic 4 Education Sports Video Analysis Software Education Package - The Coefficient of Restitution. Quintic 4 Education, n.d. Web. 05 Dec. 2015. 
"Coefficient of Restitution (COR)." Yale. Web. 5 Dec. 2015. 

[[Category:Collisions]]

Coefficient of Restitution

2015-12-06T03:16:15Z

Mmoreno30: /* A Computational Model */

Claimed by Maria Moreno

The coefficient of restitution measures the amount of Kinetic Energy remains and how much energy is lost after a collision. It also shows the degree of elasticity of a collision. It is defined as the ratio of the difference in initial velocities divided by the difference in final velocities. The coefficient of restitution of a collision is usually between 0 and 1 except in rare cases when energy is released by a collision.
[[File:coe2.png|right]]

==The Main Idea==

The coefficient of restitution is a ratio that describes the degree of elasticity of a collision. It is used to solve problems dealing with collisions that are not perfectly elastic or inelastic.
The equation that describes the coefficient of restitution involved dividing the difference in the final velocities by the difference in the initial velocity.

Consider objects A and B with initial velocities vAi and vBi and final velocities vAf and vBf.
The coefficient of restitution, e is determined with the following formula:

<math> e = \frac{v_{Bf}-v_{Af}}{v_{Ai}-v_{Bi}} </math> 
The coefficient is a unitless form of measurement as it is the ratio of two velocities and the units cancel.

===A Mathematical Model===
Consider a perfectly elastic collision between objects A and B where vAi and vBi refer to the initial velocities of A and B and vAf and vBf refer to the final velocities of A and B.

Start with the conservation of momentum principle which states that 
<math>\vec{p_i} = \vec{p_f}</math> and 
<math>m_A\vec{v_{Ai}}+m_A\vec{v_{Bi}}=m_A\vec{v_{Af}}+m_A\vec{v_{Bf}}</math> (1) 

Remember that for a perfectly elastic collision kinetic energy is also conserved meaning: 
<math>KE_i = KE_f</math> and 
<math>\frac{1}{2}m_A v^2_{Ai}+\frac{1}{2}m_A v^2_Bi=\frac{1}{2}m_A v^2_{Af}+\frac{1}{2}m_A v^2_{Bf}</math> (2) 

Dividing (2) by (1) yields: 

<math>\vec{v_{Ai}}+\vec{v_{Bi}}=\vec{v_{Af}}+\vec{v_{Bf}}</math> 
<math>1=\frac{v_{Ai}+v_{Bi}}{v_{Af}+v_{Bf}}</math> 

<math>e=\frac{v_{Ai}+v_{Bi}}{v_{Af}+v_{Bf}}</math> 

For perfectly elastic collision e=1.

===Inelastic Collisions===

An inelastic collision is a type of collision where Kinetic Energy is not conserved. In this case the difference in initial velocities will not equal the difference in final velocities. This means that the coefficient of restitution will not equal 1.

Consider the same collision as previously described, except this collision is inelastic and Kinetic Energy is not conserved. Since some of the kinetic energy is dissipated: 
<math>\frac{1}{2}m_A v^2_{Ai}+\frac{1}{2}m_A v^2_Bi<\frac{1}{2}m_A v^2_{Af}+\frac{1}{2}m_A v^2_{Bf}</math> 
<math>\frac{v_{Ai}+v_{Bi}}{v_{Af}+v_{Bf}}<1</math> 

<math>e=\frac{v_{Ai}+v_{Bi}}{v_{Af}+v_{Bf}}<1</math> 

A perfectly inelastic collision (for example when two objects collide and then stick together) has a coefficient of restitution of 0.

===A Computational Model===
Here is a link to a glowscript simulation
http://www.glowscript.org/#/user/mmoreno181/folder/phys/program/cor 
Here is the code for the simulation: 
<code>
ball1=sphere(pos=vec(0,0,0), color=color.red, radius=.5)
ball2=sphere(pos=vec(10,0,0), color=color.blue, radius=.5)

#initialize velocity
ball1=sphere(pos=vec(0,0,0), color=color.red, radius=.5)
ball2=sphere(pos=vec(10,0,0), color=color.blue, radius=.5)

#initialize velocity
v1i=vec(.02,0,0)
v2i=vec(-.1,0,0)
v1f=vec(-.07,0,0)

#set coefficient of resitution
e=.5

#find v2f
v2f=v1f-(e*(v2i-v1i))

#find the time that the balls collide
tmeet= (ball2.pos.x-ball1.pos.x)/(v1i.x-v2i.x)

t=0

#animate
while t<150:
rate(100)
if t < tmeet: #if it's before the collision
ball1.pos=ball1.pos+v1i
ball2.pos=ball2.pos+v2i
else:
ball1.pos=ball1.pos+v1f
ball2.pos=ball2.pos+v2f

t=t+1
</code>

==Examples==
===Simple===
A ball is traveling at <math>4 \frac{m}{s}</math> and collides with a ball. After the collision the ball travels at <math>2.4\frac{m}{s}</math> in the opposite direction. Find the coefficient of restitution of the collision. 
Start with the definition of the coefficient of restitution: 
<math>e=\frac{v_{wall_f}-v_{ball_f}}{v_{ball_i}-v_{wall_i}}</math> 
Then plug in values and solve: 
<math>e=\frac{0-v_{ball_f}}{v_{ball_i}-0}</math> 
<math>e=\frac{-(-2.4)}{4}</math> 
<math>e=.6</math>

===Middling===
A hockey puck travels at a constant velocity <math>5\frac{m}{s}</math> and is hit by another puck with a velocity of <math>8\frac{m}{s}</math>. After the collision, it travels in the opposite direction at 4m/s. This collision has a coefficient of restitution of .55. Find the velocity of the second puck after the collision. 
Start with the definition of the coefficient of restitution. We are calling the velocity of the first puck <math>v_1</math> and the velocity of the second puck <math>v_2</math>. 
<math>e=\frac{v_{1f}-v_{2f}}{v_{2i}-v_{1i}}</math> 
Solve for <math>v_{2f}</math> 

<math>v_{2f}=v_{1f}-e(v_{2i}-v_{1i})</math> 
Plug in the values stated in the problem. 
<math>v_{2f}=4-.55(5-8)</math>
<math>v_{2f}=2.35\frac{m}{s}</math>

===Difficult===
[[File:Cor1.jpg|right]]
Bouncing Ball Example:

Consider a ball dropped from initial height <math>h_0</math>. When it bounces it reaches a height <math>h_f</math>. Find the coefficient of restitution of this collision. 
<math>e=\frac{v_{floor_f}-v_{ball_f}}{v_{ball_i}-v_{floor_i}}</math> 
The floor's velocity is always 0. 
<math>e=\frac{v_{ball_f}}{v_{ball_i}}</math> 
Recall the law of conservation of energy: 
<math> \Delta KE = \Delta PE</math> 
For the ball from h0 to h=0
<math> \frac{1}{2}mv_{ball_i}^2=mgh_0</math> 
<math> \frac{1}{2}v_{ball_i}^2=gh_0</math> 
<math> v_{ball_i}=2\sqrt{gh_0} </math> 
The same concept can be applied to the second bounce. 
<math>v_{ball_f}=2\sqrt{gh_f}</math> 
Plugging this in to the coefficient of restitution expression we get: 
<math>e=\frac{2\sqrt{mgh_f}}{2\sqrt{mgh_0}}</math> 
<math>e=\frac{\sqrt{h_f}}{\sqrt{h_0}}</math> 

==Connectedness==
'''How is this topic connected to something that you are interested in?''' 
This topic has major connections to sports. As a former high school volleyball player, it is very interesting to think about the collisions that occurred in the game and how the coefficient of restitution is related to those collisions. 
[[File:sportsgraph.png|right|thumb|This graph shows the initial and final velocities of many different balls from different sports.]]
'''How is it connected to your major (Mechanical Engineering)?''' 
The coefficient of restitution is a key concept in the study of Dynamics which is the foundation of many aspects of mechanical engineering. It also has to do with applications of mechanical engineering regarding materials. 

[[File:sportschart.png|right|thumb|This chart shows the relationship between different coefficients of restitution and the initial and final velocities of sports balls.]]

'''Is there an interesting industrial application?'''
In sports the coefficient of restitution is an important concept especially in sports like golf or tennis where collisions are a frequent and essential part of the game. In these sports sometimes limitations are put on how high the coefficient of restitution can be for certain clubs or bats. The maximum coefficient of restitution allowed in golf for clubs is .82. There have been studies done on the coefficients for different sports balls. These studies usually follow a procedure where different balls are dropped and left to bounce. Then data regarding their initial and final velocities or how high they bounced is recorded. The images below show some of the data from one of these experiments.

== See also ==

http://physicsbook.gatech.edu/Elastic_Collisions 
http://physicsbook.gatech.edu/Inelastic_Collisions 
http://physicsbook.gatech.edu/Collisions 
http://physicsbook.gatech.edu/Conservation_of_Momentum 
http://physicsbook.gatech.edu/Conservation_of_Energy 

===External links===
http://www.quintic.com/education/case_studies/coefficient_restitution.htm

==References==

McGill, David J., and Wilton W. King. Engineering mechanics : an introduction to dynamics. Bloomington, IN: Tichenor Pub, 2003. Print. 
"Quintic 4 Education Sports Video Analysis Software Education Package - The Coefficient of Restitution." Quintic 4 Education Sports Video Analysis Software Education Package - The Coefficient of Restitution. Quintic 4 Education, n.d. Web. 05 Dec. 2015. 
"Coefficient of Restitution (COR)." Yale. Web. 5 Dec. 2015. 

[[Category:Collisions]]

Coefficient of Restitution

2015-12-06T03:15:28Z

Mmoreno30: /* A Computational Model */

Claimed by Maria Moreno

The coefficient of restitution measures the amount of Kinetic Energy remains and how much energy is lost after a collision. It also shows the degree of elasticity of a collision. It is defined as the ratio of the difference in initial velocities divided by the difference in final velocities. The coefficient of restitution of a collision is usually between 0 and 1 except in rare cases when energy is released by a collision.
[[File:coe2.png|right]]

==The Main Idea==

The coefficient of restitution is a ratio that describes the degree of elasticity of a collision. It is used to solve problems dealing with collisions that are not perfectly elastic or inelastic.
The equation that describes the coefficient of restitution involved dividing the difference in the final velocities by the difference in the initial velocity.

Consider objects A and B with initial velocities vAi and vBi and final velocities vAf and vBf.
The coefficient of restitution, e is determined with the following formula:

<math> e = \frac{v_{Bf}-v_{Af}}{v_{Ai}-v_{Bi}} </math> 
The coefficient is a unitless form of measurement as it is the ratio of two velocities and the units cancel.

===A Mathematical Model===
Consider a perfectly elastic collision between objects A and B where vAi and vBi refer to the initial velocities of A and B and vAf and vBf refer to the final velocities of A and B.

Start with the conservation of momentum principle which states that 
<math>\vec{p_i} = \vec{p_f}</math> and 
<math>m_A\vec{v_{Ai}}+m_A\vec{v_{Bi}}=m_A\vec{v_{Af}}+m_A\vec{v_{Bf}}</math> (1) 

Remember that for a perfectly elastic collision kinetic energy is also conserved meaning: 
<math>KE_i = KE_f</math> and 
<math>\frac{1}{2}m_A v^2_{Ai}+\frac{1}{2}m_A v^2_Bi=\frac{1}{2}m_A v^2_{Af}+\frac{1}{2}m_A v^2_{Bf}</math> (2) 

Dividing (2) by (1) yields: 

<math>\vec{v_{Ai}}+\vec{v_{Bi}}=\vec{v_{Af}}+\vec{v_{Bf}}</math> 
<math>1=\frac{v_{Ai}+v_{Bi}}{v_{Af}+v_{Bf}}</math> 

<math>e=\frac{v_{Ai}+v_{Bi}}{v_{Af}+v_{Bf}}</math> 

For perfectly elastic collision e=1.

===Inelastic Collisions===

An inelastic collision is a type of collision where Kinetic Energy is not conserved. In this case the difference in initial velocities will not equal the difference in final velocities. This means that the coefficient of restitution will not equal 1.

Consider the same collision as previously described, except this collision is inelastic and Kinetic Energy is not conserved. Since some of the kinetic energy is dissipated: 
<math>\frac{1}{2}m_A v^2_{Ai}+\frac{1}{2}m_A v^2_Bi<\frac{1}{2}m_A v^2_{Af}+\frac{1}{2}m_A v^2_{Bf}</math> 
<math>\frac{v_{Ai}+v_{Bi}}{v_{Af}+v_{Bf}}<1</math> 

<math>e=\frac{v_{Ai}+v_{Bi}}{v_{Af}+v_{Bf}}<1</math> 

A perfectly inelastic collision (for example when two objects collide and then stick together) has a coefficient of restitution of 0.

===A Computational Model===
Here is a link to a glowscript simulation
http://www.glowscript.org/#/user/mmoreno181/folder/phys/program/cor 
Here is the code for the simulation: 
<code>
ball1=sphere(pos=vec(0,0,0), color=color.red, radius=.5)
ball2=sphere(pos=vec(10,0,0), color=color.blue, radius=.5)

#initialize velocity
ball1=sphere(pos=vec(0,0,0), color=color.red, radius=.5)
ball2=sphere(pos=vec(10,0,0), color=color.blue, radius=.5)

#initialize velocity
v1i=vec(.02,0,0)
v2i=vec(-.1,0,0)
v1f=vec(-.07,0,0)

#set coefficient of resitution
e=.5

#find v2f
v2f=v1f-(e*(v2i-v1i))

#find the time that the balls collide
tmeet= (ball2.pos.x-ball1.pos.x)/(v1i.x-v2i.x)

t=0

#animate
while t<150:
rate(100)
if t < tmeet: #if it's before the collision
ball1.pos=ball1.pos+v1i
ball2.pos=ball2.pos+v2i
else:
ball1.pos=ball1.pos+v1f
ball2.pos=ball2.pos+v2f

t=t+1
</code>

==Examples==
===Simple===
A ball is traveling at <math>4 \frac{m}{s}</math> and collides with a ball. After the collision the ball travels at <math>2.4\frac{m}{s}</math> in the opposite direction. Find the coefficient of restitution of the collision. 
Start with the definition of the coefficient of restitution: 
<math>e=\frac{v_{wall_f}-v_{ball_f}}{v_{ball_i}-v_{wall_i}}</math> 
Then plug in values and solve: 
<math>e=\frac{0-v_{ball_f}}{v_{ball_i}-0}</math> 
<math>e=\frac{-(-2.4)}{4}</math> 
<math>e=.6</math>

===Middling===
A hockey puck travels at a constant velocity <math>5\frac{m}{s}</math> and is hit by another puck with a velocity of <math>8\frac{m}{s}</math>. After the collision, it travels in the opposite direction at 4m/s. This collision has a coefficient of restitution of .55. Find the velocity of the second puck after the collision. 
Start with the definition of the coefficient of restitution. We are calling the velocity of the first puck <math>v_1</math> and the velocity of the second puck <math>v_2</math>. 
<math>e=\frac{v_{1f}-v_{2f}}{v_{2i}-v_{1i}}</math> 
Solve for <math>v_{2f}</math> 

<math>v_{2f}=v_{1f}-e(v_{2i}-v_{1i})</math> 
Plug in the values stated in the problem. 
<math>v_{2f}=4-.55(5-8)</math>
<math>v_{2f}=2.35\frac{m}{s}</math>

===Difficult===
[[File:Cor1.jpg|right]]
Bouncing Ball Example:

Consider a ball dropped from initial height <math>h_0</math>. When it bounces it reaches a height <math>h_f</math>. Find the coefficient of restitution of this collision. 
<math>e=\frac{v_{floor_f}-v_{ball_f}}{v_{ball_i}-v_{floor_i}}</math> 
The floor's velocity is always 0. 
<math>e=\frac{v_{ball_f}}{v_{ball_i}}</math> 
Recall the law of conservation of energy: 
<math> \Delta KE = \Delta PE</math> 
For the ball from h0 to h=0
<math> \frac{1}{2}mv_{ball_i}^2=mgh_0</math> 
<math> \frac{1}{2}v_{ball_i}^2=gh_0</math> 
<math> v_{ball_i}=2\sqrt{gh_0} </math> 
The same concept can be applied to the second bounce. 
<math>v_{ball_f}=2\sqrt{gh_f}</math> 
Plugging this in to the coefficient of restitution expression we get: 
<math>e=\frac{2\sqrt{mgh_f}}{2\sqrt{mgh_0}}</math> 
<math>e=\frac{\sqrt{h_f}}{\sqrt{h_0}}</math> 

==Connectedness==
'''How is this topic connected to something that you are interested in?''' 
This topic has major connections to sports. As a former high school volleyball player, it is very interesting to think about the collisions that occurred in the game and how the coefficient of restitution is related to those collisions. 
[[File:sportsgraph.png|right|thumb|This graph shows the initial and final velocities of many different balls from different sports.]]
'''How is it connected to your major (Mechanical Engineering)?''' 
The coefficient of restitution is a key concept in the study of Dynamics which is the foundation of many aspects of mechanical engineering. It also has to do with applications of mechanical engineering regarding materials. 

[[File:sportschart.png|right|thumb|This chart shows the relationship between different coefficients of restitution and the initial and final velocities of sports balls.]]

'''Is there an interesting industrial application?'''
In sports the coefficient of restitution is an important concept especially in sports like golf or tennis where collisions are a frequent and essential part of the game. In these sports sometimes limitations are put on how high the coefficient of restitution can be for certain clubs or bats. The maximum coefficient of restitution allowed in golf for clubs is .82. There have been studies done on the coefficients for different sports balls. These studies usually follow a procedure where different balls are dropped and left to bounce. Then data regarding their initial and final velocities or how high they bounced is recorded. The images below show some of the data from one of these experiments.

== See also ==

http://physicsbook.gatech.edu/Elastic_Collisions 
http://physicsbook.gatech.edu/Inelastic_Collisions 
http://physicsbook.gatech.edu/Collisions 
http://physicsbook.gatech.edu/Conservation_of_Momentum 
http://physicsbook.gatech.edu/Conservation_of_Energy 

===External links===
http://www.quintic.com/education/case_studies/coefficient_restitution.htm

==References==

McGill, David J., and Wilton W. King. Engineering mechanics : an introduction to dynamics. Bloomington, IN: Tichenor Pub, 2003. Print. 
"Quintic 4 Education Sports Video Analysis Software Education Package - The Coefficient of Restitution." Quintic 4 Education Sports Video Analysis Software Education Package - The Coefficient of Restitution. Quintic 4 Education, n.d. Web. 05 Dec. 2015. 
"Coefficient of Restitution (COR)." Yale. Web. 5 Dec. 2015. 

[[Category:Collisions]]

Coefficient of Restitution

2015-12-06T03:14:15Z

Mmoreno30: /* A Mathematical Model */

Claimed by Maria Moreno

The coefficient of restitution measures the amount of Kinetic Energy remains and how much energy is lost after a collision. It also shows the degree of elasticity of a collision. It is defined as the ratio of the difference in initial velocities divided by the difference in final velocities. The coefficient of restitution of a collision is usually between 0 and 1 except in rare cases when energy is released by a collision.
[[File:coe2.png|right]]

==The Main Idea==

The coefficient of restitution is a ratio that describes the degree of elasticity of a collision. It is used to solve problems dealing with collisions that are not perfectly elastic or inelastic.
The equation that describes the coefficient of restitution involved dividing the difference in the final velocities by the difference in the initial velocity.

Consider objects A and B with initial velocities vAi and vBi and final velocities vAf and vBf.
The coefficient of restitution, e is determined with the following formula:

<math> e = \frac{v_{Bf}-v_{Af}}{v_{Ai}-v_{Bi}} </math> 
The coefficient is a unitless form of measurement as it is the ratio of two velocities and the units cancel.

===A Mathematical Model===
Consider a perfectly elastic collision between objects A and B where vAi and vBi refer to the initial velocities of A and B and vAf and vBf refer to the final velocities of A and B.

Start with the conservation of momentum principle which states that 
<math>\vec{p_i} = \vec{p_f}</math> and 
<math>m_A\vec{v_{Ai}}+m_A\vec{v_{Bi}}=m_A\vec{v_{Af}}+m_A\vec{v_{Bf}}</math> (1) 

Remember that for a perfectly elastic collision kinetic energy is also conserved meaning: 
<math>KE_i = KE_f</math> and 
<math>\frac{1}{2}m_A v^2_{Ai}+\frac{1}{2}m_A v^2_Bi=\frac{1}{2}m_A v^2_{Af}+\frac{1}{2}m_A v^2_{Bf}</math> (2) 

Dividing (2) by (1) yields: 

<math>\vec{v_{Ai}}+\vec{v_{Bi}}=\vec{v_{Af}}+\vec{v_{Bf}}</math> 
<math>1=\frac{v_{Ai}+v_{Bi}}{v_{Af}+v_{Bf}}</math> 

<math>e=\frac{v_{Ai}+v_{Bi}}{v_{Af}+v_{Bf}}</math> 

For perfectly elastic collision e=1.

===Inelastic Collisions===

An inelastic collision is a type of collision where Kinetic Energy is not conserved. In this case the difference in initial velocities will not equal the difference in final velocities. This means that the coefficient of restitution will not equal 1.

Consider the same collision as previously described, except this collision is inelastic and Kinetic Energy is not conserved. Since some of the kinetic energy is dissipated: 
<math>\frac{1}{2}m_A v^2_{Ai}+\frac{1}{2}m_A v^2_Bi<\frac{1}{2}m_A v^2_{Af}+\frac{1}{2}m_A v^2_{Bf}</math> 
<math>\frac{v_{Ai}+v_{Bi}}{v_{Af}+v_{Bf}}<1</math> 

<math>e=\frac{v_{Ai}+v_{Bi}}{v_{Af}+v_{Bf}}<1</math> 

A perfectly inelastic collision (for example when two objects collide and then stick together) has a coefficient of restitution of 0.

===A Computational Model===
Here is a link to a glowscript simulation
http://www.glowscript.org/#/user/mmoreno181/folder/phys/program/cor 
Here is the code for the simulation: 
<code>
ball1=sphere(pos=vec(0,0,0), color=color.red, radius=.5)
ball2=sphere(pos=vec(10,0,0), color=color.blue, radius=.5)

#initialize velocity
v1i=vec(.02,0,0)
v2i=vec(-.1,0,0)
v1f=vec(-.07,0,0)

#set coefficient of resitution
e=.5

#find v2f
v2f=v1f-(e*(v2i-v1i))

#find the time that the balls collide
tmeet= (ball2.pos.x-ball1.pos.x)/(v1i.x-v2i.x)

t=0

#animate
while t<150:
rate(100)
if t < tmeet: #if it's before the collision
ball1.pos=ball1.pos+v1i
ball2.pos=ball2.pos+v2i
else:
ball1.pos=ball1.pos+v1f
ball2.pos=ball2.pos+v2f

t=t+1
</code>

==Examples==
===Simple===
A ball is traveling at <math>4 \frac{m}{s}</math> and collides with a ball. After the collision the ball travels at <math>2.4\frac{m}{s}</math> in the opposite direction. Find the coefficient of restitution of the collision. 
Start with the definition of the coefficient of restitution: 
<math>e=\frac{v_{wall_f}-v_{ball_f}}{v_{ball_i}-v_{wall_i}}</math> 
Then plug in values and solve: 
<math>e=\frac{0-v_{ball_f}}{v_{ball_i}-0}</math> 
<math>e=\frac{-(-2.4)}{4}</math> 
<math>e=.6</math>

===Middling===
A hockey puck travels at a constant velocity <math>5\frac{m}{s}</math> and is hit by another puck with a velocity of <math>8\frac{m}{s}</math>. After the collision, it travels in the opposite direction at 4m/s. This collision has a coefficient of restitution of .55. Find the velocity of the second puck after the collision. 
Start with the definition of the coefficient of restitution. We are calling the velocity of the first puck <math>v_1</math> and the velocity of the second puck <math>v_2</math>. 
<math>e=\frac{v_{1f}-v_{2f}}{v_{2i}-v_{1i}}</math> 
Solve for <math>v_{2f}</math> 

<math>v_{2f}=v_{1f}-e(v_{2i}-v_{1i})</math> 
Plug in the values stated in the problem. 
<math>v_{2f}=4-.55(5-8)</math>
<math>v_{2f}=2.35\frac{m}{s}</math>

===Difficult===
[[File:Cor1.jpg|right]]
Bouncing Ball Example:

Consider a ball dropped from initial height <math>h_0</math>. When it bounces it reaches a height <math>h_f</math>. Find the coefficient of restitution of this collision. 
<math>e=\frac{v_{floor_f}-v_{ball_f}}{v_{ball_i}-v_{floor_i}}</math> 
The floor's velocity is always 0. 
<math>e=\frac{v_{ball_f}}{v_{ball_i}}</math> 
Recall the law of conservation of energy: 
<math> \Delta KE = \Delta PE</math> 
For the ball from h0 to h=0
<math> \frac{1}{2}mv_{ball_i}^2=mgh_0</math> 
<math> \frac{1}{2}v_{ball_i}^2=gh_0</math> 
<math> v_{ball_i}=2\sqrt{gh_0} </math> 
The same concept can be applied to the second bounce. 
<math>v_{ball_f}=2\sqrt{gh_f}</math> 
Plugging this in to the coefficient of restitution expression we get: 
<math>e=\frac{2\sqrt{mgh_f}}{2\sqrt{mgh_0}}</math> 
<math>e=\frac{\sqrt{h_f}}{\sqrt{h_0}}</math> 

==Connectedness==
'''How is this topic connected to something that you are interested in?''' 
This topic has major connections to sports. As a former high school volleyball player, it is very interesting to think about the collisions that occurred in the game and how the coefficient of restitution is related to those collisions. 
[[File:sportsgraph.png|right|thumb|This graph shows the initial and final velocities of many different balls from different sports.]]
'''How is it connected to your major (Mechanical Engineering)?''' 
The coefficient of restitution is a key concept in the study of Dynamics which is the foundation of many aspects of mechanical engineering. It also has to do with applications of mechanical engineering regarding materials. 

[[File:sportschart.png|right|thumb|This chart shows the relationship between different coefficients of restitution and the initial and final velocities of sports balls.]]

'''Is there an interesting industrial application?'''
In sports the coefficient of restitution is an important concept especially in sports like golf or tennis where collisions are a frequent and essential part of the game. In these sports sometimes limitations are put on how high the coefficient of restitution can be for certain clubs or bats. The maximum coefficient of restitution allowed in golf for clubs is .82. There have been studies done on the coefficients for different sports balls. These studies usually follow a procedure where different balls are dropped and left to bounce. Then data regarding their initial and final velocities or how high they bounced is recorded. The images below show some of the data from one of these experiments.

== See also ==

http://physicsbook.gatech.edu/Elastic_Collisions 
http://physicsbook.gatech.edu/Inelastic_Collisions 
http://physicsbook.gatech.edu/Collisions 
http://physicsbook.gatech.edu/Conservation_of_Momentum 
http://physicsbook.gatech.edu/Conservation_of_Energy 

===External links===
http://www.quintic.com/education/case_studies/coefficient_restitution.htm

==References==

McGill, David J., and Wilton W. King. Engineering mechanics : an introduction to dynamics. Bloomington, IN: Tichenor Pub, 2003. Print. 
"Quintic 4 Education Sports Video Analysis Software Education Package - The Coefficient of Restitution." Quintic 4 Education Sports Video Analysis Software Education Package - The Coefficient of Restitution. Quintic 4 Education, n.d. Web. 05 Dec. 2015. 
"Coefficient of Restitution (COR)." Yale. Web. 5 Dec. 2015. 

[[Category:Collisions]]

Coefficient of Restitution

2015-12-06T03:13:59Z

Mmoreno30: /* A Mathematical Model */

Claimed by Maria Moreno

The coefficient of restitution measures the amount of Kinetic Energy remains and how much energy is lost after a collision. It also shows the degree of elasticity of a collision. It is defined as the ratio of the difference in initial velocities divided by the difference in final velocities. The coefficient of restitution of a collision is usually between 0 and 1 except in rare cases when energy is released by a collision.
[[File:coe2.png|right]]

==The Main Idea==

The coefficient of restitution is a ratio that describes the degree of elasticity of a collision. It is used to solve problems dealing with collisions that are not perfectly elastic or inelastic.
The equation that describes the coefficient of restitution involved dividing the difference in the final velocities by the difference in the initial velocity.

Consider objects A and B with initial velocities vAi and vBi and final velocities vAf and vBf.
The coefficient of restitution, e is determined with the following formula:

<math> e = \frac{v_{Bf}-v_{Af}}{v_{Ai}-v_{Bi}} </math> 
The coefficient is a unitless form of measurement as it is the ratio of two velocities and the units cancel.

===A Mathematical Model===
Consider a perfectly elastic collision between objects A and B where vAi and vBi refer to the initial velocities of A and B and vAf and vBf refer to the final velocities of A and B.

Start with the conservation of momentum principle which states that 
<math>\vec{p_i} = \vec{p_f}</math> and 
<math>m_A\vec{v_{Ai}}+m_A\vec{v_{Bi}}=m_A\vec{v_{Af}}+m_A\vec{v_{Bf}}</math> (1) 

Remember that for a perfectly elastic collision kinetic energy is also conserved meaning: 
<math>KE_i = KE_f</math> and 
<math>\frac{1}{2}m_A v^2_{Ai}+\frac{1}{2}m_A v^2_Bi=\frac{1}{2}m_A v^2_{Af}+\frac{1}{2}m_A v^2_{Bf}</math> (2) 

Dividing (2) by (1) yields: 

<math>\vec{v_{Ai}}+\vec{v_{Bi}}=\vec{v_{Af}}+\vec{v_{Bf}}</math> 
<math>1=\frac{v_{Ai}+v_{Bi}}{v_{Af}+v_{Bf}}</math> 

<math>e=\frac{v_{Ai}+v_{Bi}}{v_{Af}+v_{Bf}}</math> 

For perfectly elastic collision e=1.

===Inelastic Collisions===

An inelastic collision is a type of collision where Kinetic Energy is not conserved. In this case the difference in initial velocities will not equal the difference in final velocities. This means that the coefficient of restitution will not equal 1.

Consider the same collision as previously described, except this collision is inelastic and Kinetic Energy is not conserved. Since some of the kinetic energy is dissipated: 
<math>\frac{1}{2}m_A v^2_{Ai}+\frac{1}{2}m_A v^2_Bi<\frac{1}{2}m_A v^2_{Af}+\frac{1}{2}m_A v^2_{Bf}</math> 
<math>\frac{v_{Ai}+v_{Bi}}{v_{Af}+v_{Bf}}<1</math> 

<math>e=\frac{v_{Ai}+v_{Bi}}{v_{Af}+v_{Bf}}<1</math> 

A perfectly inelastic collision (for example when two objects collide and then stick together) has a coefficient of restitution of 0.

===A Computational Model===
Here is a link to a glowscript simulation
http://www.glowscript.org/#/user/mmoreno181/folder/phys/program/cor 
Here is the code for the simulation: 
<code>
ball1=sphere(pos=vec(0,0,0), color=color.red, radius=.5)
ball2=sphere(pos=vec(10,0,0), color=color.blue, radius=.5)

#initialize velocity
v1i=vec(.02,0,0)
v2i=vec(-.1,0,0)
v1f=vec(-.07,0,0)

#set coefficient of resitution
e=.5

#find v2f
v2f=v1f-(e*(v2i-v1i))

#find the time that the balls collide
tmeet= (ball2.pos.x-ball1.pos.x)/(v1i.x-v2i.x)

t=0

#animate
while t<150:
rate(100)
if t < tmeet: #if it's before the collision
ball1.pos=ball1.pos+v1i
ball2.pos=ball2.pos+v2i
else:
ball1.pos=ball1.pos+v1f
ball2.pos=ball2.pos+v2f

t=t+1
</code>

==Examples==
===Simple===
A ball is traveling at <math>4 \frac{m}{s}</math> and collides with a ball. After the collision the ball travels at <math>2.4\frac{m}{s}</math> in the opposite direction. Find the coefficient of restitution of the collision. 
Start with the definition of the coefficient of restitution: 
<math>e=\frac{v_{wall_f}-v_{ball_f}}{v_{ball_i}-v_{wall_i}}</math> 
Then plug in values and solve: 
<math>e=\frac{0-v_{ball_f}}{v_{ball_i}-0}</math> 
<math>e=\frac{-(-2.4)}{4}</math> 
<math>e=.6</math>

===Middling===
A hockey puck travels at a constant velocity <math>5\frac{m}{s}</math> and is hit by another puck with a velocity of <math>8\frac{m}{s}</math>. After the collision, it travels in the opposite direction at 4m/s. This collision has a coefficient of restitution of .55. Find the velocity of the second puck after the collision. 
Start with the definition of the coefficient of restitution. We are calling the velocity of the first puck <math>v_1</math> and the velocity of the second puck <math>v_2</math>. 
<math>e=\frac{v_{1f}-v_{2f}}{v_{2i}-v_{1i}}</math> 
Solve for <math>v_{2f}</math> 

<math>v_{2f}=v_{1f}-e(v_{2i}-v_{1i})</math> 
Plug in the values stated in the problem. 
<math>v_{2f}=4-.55(5-8)</math>
<math>v_{2f}=2.35\frac{m}{s}</math>

===Difficult===
[[File:Cor1.jpg|right]]
Bouncing Ball Example:

Consider a ball dropped from initial height <math>h_0</math>. When it bounces it reaches a height <math>h_f</math>. Find the coefficient of restitution of this collision. 
<math>e=\frac{v_{floor_f}-v_{ball_f}}{v_{ball_i}-v_{floor_i}}</math> 
The floor's velocity is always 0. 
<math>e=\frac{v_{ball_f}}{v_{ball_i}}</math> 
Recall the law of conservation of energy: 
<math> \Delta KE = \Delta PE</math> 
For the ball from h0 to h=0
<math> \frac{1}{2}mv_{ball_i}^2=mgh_0</math> 
<math> \frac{1}{2}v_{ball_i}^2=gh_0</math> 
<math> v_{ball_i}=2\sqrt{gh_0} </math> 
The same concept can be applied to the second bounce. 
<math>v_{ball_f}=2\sqrt{gh_f}</math> 
Plugging this in to the coefficient of restitution expression we get: 
<math>e=\frac{2\sqrt{mgh_f}}{2\sqrt{mgh_0}}</math> 
<math>e=\frac{\sqrt{h_f}}{\sqrt{h_0}}</math> 

==Connectedness==
'''How is this topic connected to something that you are interested in?''' 
This topic has major connections to sports. As a former high school volleyball player, it is very interesting to think about the collisions that occurred in the game and how the coefficient of restitution is related to those collisions. 
[[File:sportsgraph.png|right|thumb|This graph shows the initial and final velocities of many different balls from different sports.]]
'''How is it connected to your major (Mechanical Engineering)?''' 
The coefficient of restitution is a key concept in the study of Dynamics which is the foundation of many aspects of mechanical engineering. It also has to do with applications of mechanical engineering regarding materials. 

[[File:sportschart.png|right|thumb|This chart shows the relationship between different coefficients of restitution and the initial and final velocities of sports balls.]]

'''Is there an interesting industrial application?'''
In sports the coefficient of restitution is an important concept especially in sports like golf or tennis where collisions are a frequent and essential part of the game. In these sports sometimes limitations are put on how high the coefficient of restitution can be for certain clubs or bats. The maximum coefficient of restitution allowed in golf for clubs is .82. There have been studies done on the coefficients for different sports balls. These studies usually follow a procedure where different balls are dropped and left to bounce. Then data regarding their initial and final velocities or how high they bounced is recorded. The images below show some of the data from one of these experiments.

== See also ==

http://physicsbook.gatech.edu/Elastic_Collisions 
http://physicsbook.gatech.edu/Inelastic_Collisions 
http://physicsbook.gatech.edu/Collisions 
http://physicsbook.gatech.edu/Conservation_of_Momentum 
http://physicsbook.gatech.edu/Conservation_of_Energy 

===External links===
http://www.quintic.com/education/case_studies/coefficient_restitution.htm

==References==

McGill, David J., and Wilton W. King. Engineering mechanics : an introduction to dynamics. Bloomington, IN: Tichenor Pub, 2003. Print. 
"Quintic 4 Education Sports Video Analysis Software Education Package - The Coefficient of Restitution." Quintic 4 Education Sports Video Analysis Software Education Package - The Coefficient of Restitution. Quintic 4 Education, n.d. Web. 05 Dec. 2015. 
"Coefficient of Restitution (COR)." Yale. Web. 5 Dec. 2015. 

[[Category:Collisions]]

Coefficient of Restitution

2015-12-06T03:13:16Z

Mmoreno30: /* References */

Claimed by Maria Moreno

The coefficient of restitution measures the amount of Kinetic Energy remains and how much energy is lost after a collision. It also shows the degree of elasticity of a collision. It is defined as the ratio of the difference in initial velocities divided by the difference in final velocities. The coefficient of restitution of a collision is usually between 0 and 1 except in rare cases when energy is released by a collision.
[[File:coe2.png|right]]

==The Main Idea==

The coefficient of restitution is a ratio that describes the degree of elasticity of a collision. It is used to solve problems dealing with collisions that are not perfectly elastic or inelastic.
The equation that describes the coefficient of restitution involved dividing the difference in the final velocities by the difference in the initial velocity.

Consider objects A and B with initial velocities vAi and vBi and final velocities vAf and vBf.
The coefficient of restitution, e is determined with the following formula:

<math> e = \frac{v_{Bf}-v_{Af}}{v_{Ai}-v_{Bi}} </math> 
The coefficient is a unitless form of measurement as it is the ratio of two velocities and the units cancel.

===A Mathematical Model===
Consider a perfectly elastic collision between objects A and B where vAi and vBi refer to the initial velocities of A and B and vAf and vBf refer to the final velocities of A and B.

Start with the conservation of momentum principle which states that 
<math>\vec{p_i} = \vec{p_f}</math> and 
<math>m_A\vec{v_{Ai}}+m_A\vec{v_{Bi}}=m_A\vec{v_{Af}}+m_A\vec{v_{Bf}}</math> (1) 

Remember that for a perfectly elastic collision kinetic energy is also conserved meaning: 
<math>KE_i = KE_f</math> and 
<math>\frac{1}{2}m_A v^2_{Ai}+\frac{1}{2}m_A v^2_Bi=\frac{1}{2}m_A v^2_{Af}+\frac{1}{2}m_A v^2_{Bf}</math> (2) 

Dividing (2) by (1) yields: 

<math>\vec{v_{Ai}}+\vec{v_{Bi}}=\vec{v_{Af}}+\vec{v_{Bf}}</math> 
<math>1=\frac{v_{Ai}+v_{Bi}}{v_{Af}+v_{Bf}}</math> 

<math>e=\frac{v_{Ai}+v_{Bi}}{v_{Af}+v_{Bf}}</math> 

For perfectly elastic collision e=1.

===Inelastic Collisions===

An inelastic collision is a type of collision where Kinetic Energy is not conserved. In this case the difference in initial velocities will not equal the difference in final velocities. This means that the coefficient of restitution will not equal 1.

Consider the same collision as previously described, except this collision is inelastic and Kinetic Energy is not conserved. Since some of the kinetic energy is dissipated: 
<math>\frac{1}{2}m_A v^2_{Ai}+\frac{1}{2}m_A v^2_Bi<\frac{1}{2}m_A v^2_{Af}+\frac{1}{2}m_A v^2_{Bf}</math> 
<math>\frac{v_{Ai}+v_{Bi}}{v_{Af}+v_{Bf}}<1</math> 

<math>e=\frac{v_{Ai}+v_{Bi}}{v_{Af}+v_{Bf}}<1</math> 

A perfectly inelastic collision (for example when two objects collide and then stick together) has a coefficient of restitution of 0.

===A Computational Model===
Here is a link to a glowscript simulation
http://www.glowscript.org/#/user/mmoreno181/folder/phys/program/cor 
Here is the code for the simulation: 
<code>
ball1=sphere(pos=vec(0,0,0), color=color.red, radius=.5)
ball2=sphere(pos=vec(10,0,0), color=color.blue, radius=.5)

#initialize velocity
v1i=vec(.02,0,0)
v2i=vec(-.1,0,0)
v1f=vec(-.07,0,0)

#set coefficient of resitution
e=.5

#find v2f
v2f=v1f-(e*(v2i-v1i))

#find the time that the balls collide
tmeet= (ball2.pos.x-ball1.pos.x)/(v1i.x-v2i.x)

t=0

#animate
while t<150:
rate(100)
if t < tmeet: #if it's before the collision
ball1.pos=ball1.pos+v1i
ball2.pos=ball2.pos+v2i
else:
ball1.pos=ball1.pos+v1f
ball2.pos=ball2.pos+v2f

t=t+1
</code>

==Examples==
===Simple===
A ball is traveling at <math>4 \frac{m}{s}</math> and collides with a ball. After the collision the ball travels at <math>2.4\frac{m}{s}</math> in the opposite direction. Find the coefficient of restitution of the collision. 
Start with the definition of the coefficient of restitution: 
<math>e=\frac{v_{wall_f}-v_{ball_f}}{v_{ball_i}-v_{wall_i}}</math> 
Then plug in values and solve: 
<math>e=\frac{0-v_{ball_f}}{v_{ball_i}-0}</math> 
<math>e=\frac{-(-2.4)}{4}</math> 
<math>e=.6</math>

===Middling===
A hockey puck travels at a constant velocity <math>5\frac{m}{s}</math> and is hit by another puck with a velocity of <math>8\frac{m}{s}</math>. After the collision, it travels in the opposite direction at 4m/s. This collision has a coefficient of restitution of .55. Find the velocity of the second puck after the collision. 
Start with the definition of the coefficient of restitution. We are calling the velocity of the first puck <math>v_1</math> and the velocity of the second puck <math>v_2</math>. 
<math>e=\frac{v_{1f}-v_{2f}}{v_{2i}-v_{1i}}</math> 
Solve for <math>v_{2f}</math> 

<math>v_{2f}=v_{1f}-e(v_{2i}-v_{1i})</math> 
Plug in the values stated in the problem. 
<math>v_{2f}=4-.55(5-8)</math>
<math>v_{2f}=2.35\frac{m}{s}</math>

===Difficult===
[[File:Cor1.jpg|right]]
Bouncing Ball Example:

Consider a ball dropped from initial height <math>h_0</math>. When it bounces it reaches a height <math>h_f</math>. Find the coefficient of restitution of this collision. 
<math>e=\frac{v_{floor_f}-v_{ball_f}}{v_{ball_i}-v_{floor_i}}</math> 
The floor's velocity is always 0. 
<math>e=\frac{v_{ball_f}}{v_{ball_i}}</math> 
Recall the law of conservation of energy: 
<math> \Delta KE = \Delta PE</math> 
For the ball from h0 to h=0
<math> \frac{1}{2}mv_{ball_i}^2=mgh_0</math> 
<math> \frac{1}{2}v_{ball_i}^2=gh_0</math> 
<math> v_{ball_i}=2\sqrt{gh_0} </math> 
The same concept can be applied to the second bounce. 
<math>v_{ball_f}=2\sqrt{gh_f}</math> 
Plugging this in to the coefficient of restitution expression we get: 
<math>e=\frac{2\sqrt{mgh_f}}{2\sqrt{mgh_0}}</math> 
<math>e=\frac{\sqrt{h_f}}{\sqrt{h_0}}</math> 

==Connectedness==
'''How is this topic connected to something that you are interested in?''' 
This topic has major connections to sports. As a former high school volleyball player, it is very interesting to think about the collisions that occurred in the game and how the coefficient of restitution is related to those collisions. 
[[File:sportsgraph.png|right|thumb|This graph shows the initial and final velocities of many different balls from different sports.]]
'''How is it connected to your major (Mechanical Engineering)?''' 
The coefficient of restitution is a key concept in the study of Dynamics which is the foundation of many aspects of mechanical engineering. It also has to do with applications of mechanical engineering regarding materials. 

[[File:sportschart.png|right|thumb|This chart shows the relationship between different coefficients of restitution and the initial and final velocities of sports balls.]]

'''Is there an interesting industrial application?'''
In sports the coefficient of restitution is an important concept especially in sports like golf or tennis where collisions are a frequent and essential part of the game. In these sports sometimes limitations are put on how high the coefficient of restitution can be for certain clubs or bats. The maximum coefficient of restitution allowed in golf for clubs is .82. There have been studies done on the coefficients for different sports balls. These studies usually follow a procedure where different balls are dropped and left to bounce. Then data regarding their initial and final velocities or how high they bounced is recorded. The images below show some of the data from one of these experiments.

== See also ==

http://physicsbook.gatech.edu/Elastic_Collisions 
http://physicsbook.gatech.edu/Inelastic_Collisions 
http://physicsbook.gatech.edu/Collisions 
http://physicsbook.gatech.edu/Conservation_of_Momentum 
http://physicsbook.gatech.edu/Conservation_of_Energy 

===External links===
http://www.quintic.com/education/case_studies/coefficient_restitution.htm

==References==

McGill, David J., and Wilton W. King. Engineering mechanics : an introduction to dynamics. Bloomington, IN: Tichenor Pub, 2003. Print. 
"Quintic 4 Education Sports Video Analysis Software Education Package - The Coefficient of Restitution." Quintic 4 Education Sports Video Analysis Software Education Package - The Coefficient of Restitution. Quintic 4 Education, n.d. Web. 05 Dec. 2015. 
"Coefficient of Restitution (COR)." Yale. Web. 5 Dec. 2015. 

[[Category:Collisions]]

Coefficient of Restitution

2015-12-06T03:12:28Z

Mmoreno30: /* See also */

Claimed by Maria Moreno

The coefficient of restitution measures the amount of Kinetic Energy remains and how much energy is lost after a collision. It also shows the degree of elasticity of a collision. It is defined as the ratio of the difference in initial velocities divided by the difference in final velocities. The coefficient of restitution of a collision is usually between 0 and 1 except in rare cases when energy is released by a collision.
[[File:coe2.png|right]]

==The Main Idea==

The coefficient of restitution is a ratio that describes the degree of elasticity of a collision. It is used to solve problems dealing with collisions that are not perfectly elastic or inelastic.
The equation that describes the coefficient of restitution involved dividing the difference in the final velocities by the difference in the initial velocity.

Consider objects A and B with initial velocities vAi and vBi and final velocities vAf and vBf.
The coefficient of restitution, e is determined with the following formula:

<math> e = \frac{v_{Bf}-v_{Af}}{v_{Ai}-v_{Bi}} </math> 
The coefficient is a unitless form of measurement as it is the ratio of two velocities and the units cancel.

===A Mathematical Model===
Consider a perfectly elastic collision between objects A and B where vAi and vBi refer to the initial velocities of A and B and vAf and vBf refer to the final velocities of A and B.

Start with the conservation of momentum principle which states that 
<math>\vec{p_i} = \vec{p_f}</math> and 
<math>m_A\vec{v_{Ai}}+m_A\vec{v_{Bi}}=m_A\vec{v_{Af}}+m_A\vec{v_{Bf}}</math> (1) 

Remember that for a perfectly elastic collision kinetic energy is also conserved meaning: 
<math>KE_i = KE_f</math> and 
<math>\frac{1}{2}m_A v^2_{Ai}+\frac{1}{2}m_A v^2_Bi=\frac{1}{2}m_A v^2_{Af}+\frac{1}{2}m_A v^2_{Bf}</math> (2) 

Dividing (2) by (1) yields: 

<math>\vec{v_{Ai}}+\vec{v_{Bi}}=\vec{v_{Af}}+\vec{v_{Bf}}</math> 
<math>1=\frac{v_{Ai}+v_{Bi}}{v_{Af}+v_{Bf}}</math> 

<math>e=\frac{v_{Ai}+v_{Bi}}{v_{Af}+v_{Bf}}</math> 

For perfectly elastic collision e=1.

===Inelastic Collisions===

An inelastic collision is a type of collision where Kinetic Energy is not conserved. In this case the difference in initial velocities will not equal the difference in final velocities. This means that the coefficient of restitution will not equal 1.

Consider the same collision as previously described, except this collision is inelastic and Kinetic Energy is not conserved. Since some of the kinetic energy is dissipated: 
<math>\frac{1}{2}m_A v^2_{Ai}+\frac{1}{2}m_A v^2_Bi<\frac{1}{2}m_A v^2_{Af}+\frac{1}{2}m_A v^2_{Bf}</math> 
<math>\frac{v_{Ai}+v_{Bi}}{v_{Af}+v_{Bf}}<1</math> 

<math>e=\frac{v_{Ai}+v_{Bi}}{v_{Af}+v_{Bf}}<1</math> 

A perfectly inelastic collision (for example when two objects collide and then stick together) has a coefficient of restitution of 0.

===A Computational Model===
Here is a link to a glowscript simulation
http://www.glowscript.org/#/user/mmoreno181/folder/phys/program/cor 
Here is the code for the simulation: 
<code>
ball1=sphere(pos=vec(0,0,0), color=color.red, radius=.5)
ball2=sphere(pos=vec(10,0,0), color=color.blue, radius=.5)

#initialize velocity
v1i=vec(.02,0,0)
v2i=vec(-.1,0,0)
v1f=vec(-.07,0,0)

#set coefficient of resitution
e=.5

#find v2f
v2f=v1f-(e*(v2i-v1i))

#find the time that the balls collide
tmeet= (ball2.pos.x-ball1.pos.x)/(v1i.x-v2i.x)

t=0

#animate
while t<150:
rate(100)
if t < tmeet: #if it's before the collision
ball1.pos=ball1.pos+v1i
ball2.pos=ball2.pos+v2i
else:
ball1.pos=ball1.pos+v1f
ball2.pos=ball2.pos+v2f

t=t+1
</code>

==Examples==
===Simple===
A ball is traveling at <math>4 \frac{m}{s}</math> and collides with a ball. After the collision the ball travels at <math>2.4\frac{m}{s}</math> in the opposite direction. Find the coefficient of restitution of the collision. 
Start with the definition of the coefficient of restitution: 
<math>e=\frac{v_{wall_f}-v_{ball_f}}{v_{ball_i}-v_{wall_i}}</math> 
Then plug in values and solve: 
<math>e=\frac{0-v_{ball_f}}{v_{ball_i}-0}</math> 
<math>e=\frac{-(-2.4)}{4}</math> 
<math>e=.6</math>

===Middling===
A hockey puck travels at a constant velocity <math>5\frac{m}{s}</math> and is hit by another puck with a velocity of <math>8\frac{m}{s}</math>. After the collision, it travels in the opposite direction at 4m/s. This collision has a coefficient of restitution of .55. Find the velocity of the second puck after the collision. 
Start with the definition of the coefficient of restitution. We are calling the velocity of the first puck <math>v_1</math> and the velocity of the second puck <math>v_2</math>. 
<math>e=\frac{v_{1f}-v_{2f}}{v_{2i}-v_{1i}}</math> 
Solve for <math>v_{2f}</math> 

<math>v_{2f}=v_{1f}-e(v_{2i}-v_{1i})</math> 
Plug in the values stated in the problem. 
<math>v_{2f}=4-.55(5-8)</math>
<math>v_{2f}=2.35\frac{m}{s}</math>

===Difficult===
[[File:Cor1.jpg|right]]
Bouncing Ball Example:

Consider a ball dropped from initial height <math>h_0</math>. When it bounces it reaches a height <math>h_f</math>. Find the coefficient of restitution of this collision. 
<math>e=\frac{v_{floor_f}-v_{ball_f}}{v_{ball_i}-v_{floor_i}}</math> 
The floor's velocity is always 0. 
<math>e=\frac{v_{ball_f}}{v_{ball_i}}</math> 
Recall the law of conservation of energy: 
<math> \Delta KE = \Delta PE</math> 
For the ball from h0 to h=0
<math> \frac{1}{2}mv_{ball_i}^2=mgh_0</math> 
<math> \frac{1}{2}v_{ball_i}^2=gh_0</math> 
<math> v_{ball_i}=2\sqrt{gh_0} </math> 
The same concept can be applied to the second bounce. 
<math>v_{ball_f}=2\sqrt{gh_f}</math> 
Plugging this in to the coefficient of restitution expression we get: 
<math>e=\frac{2\sqrt{mgh_f}}{2\sqrt{mgh_0}}</math> 
<math>e=\frac{\sqrt{h_f}}{\sqrt{h_0}}</math> 

==Connectedness==
'''How is this topic connected to something that you are interested in?''' 
This topic has major connections to sports. As a former high school volleyball player, it is very interesting to think about the collisions that occurred in the game and how the coefficient of restitution is related to those collisions. 
[[File:sportsgraph.png|right|thumb|This graph shows the initial and final velocities of many different balls from different sports.]]
'''How is it connected to your major (Mechanical Engineering)?''' 
The coefficient of restitution is a key concept in the study of Dynamics which is the foundation of many aspects of mechanical engineering. It also has to do with applications of mechanical engineering regarding materials. 

[[File:sportschart.png|right|thumb|This chart shows the relationship between different coefficients of restitution and the initial and final velocities of sports balls.]]

'''Is there an interesting industrial application?'''
In sports the coefficient of restitution is an important concept especially in sports like golf or tennis where collisions are a frequent and essential part of the game. In these sports sometimes limitations are put on how high the coefficient of restitution can be for certain clubs or bats. The maximum coefficient of restitution allowed in golf for clubs is .82. There have been studies done on the coefficients for different sports balls. These studies usually follow a procedure where different balls are dropped and left to bounce. Then data regarding their initial and final velocities or how high they bounced is recorded. The images below show some of the data from one of these experiments.

== See also ==

http://physicsbook.gatech.edu/Elastic_Collisions 
http://physicsbook.gatech.edu/Inelastic_Collisions 
http://physicsbook.gatech.edu/Collisions 
http://physicsbook.gatech.edu/Conservation_of_Momentum 
http://physicsbook.gatech.edu/Conservation_of_Energy 

===External links===
http://www.quintic.com/education/case_studies/coefficient_restitution.htm

==References==

McGill, David J., and Wilton W. King. Engineering mechanics : an introduction to dynamics. Bloomington, IN: Tichenor Pub, 2003. Print. 
"Quintic 4 Education Sports Video Analysis Software Education Package - The Coefficient of Restitution." Quintic 4 Education Sports Video Analysis Software Education Package - The Coefficient of Restitution. Quintic 4 Education, n.d. Web. 05 Dec. 2015. 
"Coefficient of Restitution (COR)." ( ) (n.d.): n. pag. Yale. Web. 5 Dec. 2015. 

[[Category:Collisions]]

Coefficient of Restitution

2015-12-06T03:11:41Z

Mmoreno30: /* See also */

Claimed by Maria Moreno

The coefficient of restitution measures the amount of Kinetic Energy remains and how much energy is lost after a collision. It also shows the degree of elasticity of a collision. It is defined as the ratio of the difference in initial velocities divided by the difference in final velocities. The coefficient of restitution of a collision is usually between 0 and 1 except in rare cases when energy is released by a collision.
[[File:coe2.png|right]]

==The Main Idea==

The coefficient of restitution is a ratio that describes the degree of elasticity of a collision. It is used to solve problems dealing with collisions that are not perfectly elastic or inelastic.
The equation that describes the coefficient of restitution involved dividing the difference in the final velocities by the difference in the initial velocity.

Consider objects A and B with initial velocities vAi and vBi and final velocities vAf and vBf.
The coefficient of restitution, e is determined with the following formula:

<math> e = \frac{v_{Bf}-v_{Af}}{v_{Ai}-v_{Bi}} </math> 
The coefficient is a unitless form of measurement as it is the ratio of two velocities and the units cancel.

===A Mathematical Model===
Consider a perfectly elastic collision between objects A and B where vAi and vBi refer to the initial velocities of A and B and vAf and vBf refer to the final velocities of A and B.

Start with the conservation of momentum principle which states that 
<math>\vec{p_i} = \vec{p_f}</math> and 
<math>m_A\vec{v_{Ai}}+m_A\vec{v_{Bi}}=m_A\vec{v_{Af}}+m_A\vec{v_{Bf}}</math> (1) 

Remember that for a perfectly elastic collision kinetic energy is also conserved meaning: 
<math>KE_i = KE_f</math> and 
<math>\frac{1}{2}m_A v^2_{Ai}+\frac{1}{2}m_A v^2_Bi=\frac{1}{2}m_A v^2_{Af}+\frac{1}{2}m_A v^2_{Bf}</math> (2) 

Dividing (2) by (1) yields: 

<math>\vec{v_{Ai}}+\vec{v_{Bi}}=\vec{v_{Af}}+\vec{v_{Bf}}</math> 
<math>1=\frac{v_{Ai}+v_{Bi}}{v_{Af}+v_{Bf}}</math> 

<math>e=\frac{v_{Ai}+v_{Bi}}{v_{Af}+v_{Bf}}</math> 

For perfectly elastic collision e=1.

===Inelastic Collisions===

An inelastic collision is a type of collision where Kinetic Energy is not conserved. In this case the difference in initial velocities will not equal the difference in final velocities. This means that the coefficient of restitution will not equal 1.

Consider the same collision as previously described, except this collision is inelastic and Kinetic Energy is not conserved. Since some of the kinetic energy is dissipated: 
<math>\frac{1}{2}m_A v^2_{Ai}+\frac{1}{2}m_A v^2_Bi<\frac{1}{2}m_A v^2_{Af}+\frac{1}{2}m_A v^2_{Bf}</math> 
<math>\frac{v_{Ai}+v_{Bi}}{v_{Af}+v_{Bf}}<1</math> 

<math>e=\frac{v_{Ai}+v_{Bi}}{v_{Af}+v_{Bf}}<1</math> 

A perfectly inelastic collision (for example when two objects collide and then stick together) has a coefficient of restitution of 0.

===A Computational Model===
Here is a link to a glowscript simulation
http://www.glowscript.org/#/user/mmoreno181/folder/phys/program/cor 
Here is the code for the simulation: 
<code>
ball1=sphere(pos=vec(0,0,0), color=color.red, radius=.5)
ball2=sphere(pos=vec(10,0,0), color=color.blue, radius=.5)

#initialize velocity
v1i=vec(.02,0,0)
v2i=vec(-.1,0,0)
v1f=vec(-.07,0,0)

#set coefficient of resitution
e=.5

#find v2f
v2f=v1f-(e*(v2i-v1i))

#find the time that the balls collide
tmeet= (ball2.pos.x-ball1.pos.x)/(v1i.x-v2i.x)

t=0

#animate
while t<150:
rate(100)
if t < tmeet: #if it's before the collision
ball1.pos=ball1.pos+v1i
ball2.pos=ball2.pos+v2i
else:
ball1.pos=ball1.pos+v1f
ball2.pos=ball2.pos+v2f

t=t+1
</code>

==Examples==
===Simple===
A ball is traveling at <math>4 \frac{m}{s}</math> and collides with a ball. After the collision the ball travels at <math>2.4\frac{m}{s}</math> in the opposite direction. Find the coefficient of restitution of the collision. 
Start with the definition of the coefficient of restitution: 
<math>e=\frac{v_{wall_f}-v_{ball_f}}{v_{ball_i}-v_{wall_i}}</math> 
Then plug in values and solve: 
<math>e=\frac{0-v_{ball_f}}{v_{ball_i}-0}</math> 
<math>e=\frac{-(-2.4)}{4}</math> 
<math>e=.6</math>

===Middling===
A hockey puck travels at a constant velocity <math>5\frac{m}{s}</math> and is hit by another puck with a velocity of <math>8\frac{m}{s}</math>. After the collision, it travels in the opposite direction at 4m/s. This collision has a coefficient of restitution of .55. Find the velocity of the second puck after the collision. 
Start with the definition of the coefficient of restitution. We are calling the velocity of the first puck <math>v_1</math> and the velocity of the second puck <math>v_2</math>. 
<math>e=\frac{v_{1f}-v_{2f}}{v_{2i}-v_{1i}}</math> 
Solve for <math>v_{2f}</math> 

<math>v_{2f}=v_{1f}-e(v_{2i}-v_{1i})</math> 
Plug in the values stated in the problem. 
<math>v_{2f}=4-.55(5-8)</math>
<math>v_{2f}=2.35\frac{m}{s}</math>

===Difficult===
[[File:Cor1.jpg|right]]
Bouncing Ball Example:

Consider a ball dropped from initial height <math>h_0</math>. When it bounces it reaches a height <math>h_f</math>. Find the coefficient of restitution of this collision. 
<math>e=\frac{v_{floor_f}-v_{ball_f}}{v_{ball_i}-v_{floor_i}}</math> 
The floor's velocity is always 0. 
<math>e=\frac{v_{ball_f}}{v_{ball_i}}</math> 
Recall the law of conservation of energy: 
<math> \Delta KE = \Delta PE</math> 
For the ball from h0 to h=0
<math> \frac{1}{2}mv_{ball_i}^2=mgh_0</math> 
<math> \frac{1}{2}v_{ball_i}^2=gh_0</math> 
<math> v_{ball_i}=2\sqrt{gh_0} </math> 
The same concept can be applied to the second bounce. 
<math>v_{ball_f}=2\sqrt{gh_f}</math> 
Plugging this in to the coefficient of restitution expression we get: 
<math>e=\frac{2\sqrt{mgh_f}}{2\sqrt{mgh_0}}</math> 
<math>e=\frac{\sqrt{h_f}}{\sqrt{h_0}}</math> 

==Connectedness==
'''How is this topic connected to something that you are interested in?''' 
This topic has major connections to sports. As a former high school volleyball player, it is very interesting to think about the collisions that occurred in the game and how the coefficient of restitution is related to those collisions. 
[[File:sportsgraph.png|right|thumb|This graph shows the initial and final velocities of many different balls from different sports.]]
'''How is it connected to your major (Mechanical Engineering)?''' 
The coefficient of restitution is a key concept in the study of Dynamics which is the foundation of many aspects of mechanical engineering. It also has to do with applications of mechanical engineering regarding materials. 

[[File:sportschart.png|right|thumb|This chart shows the relationship between different coefficients of restitution and the initial and final velocities of sports balls.]]

'''Is there an interesting industrial application?'''
In sports the coefficient of restitution is an important concept especially in sports like golf or tennis where collisions are a frequent and essential part of the game. In these sports sometimes limitations are put on how high the coefficient of restitution can be for certain clubs or bats. The maximum coefficient of restitution allowed in golf for clubs is .82. There have been studies done on the coefficients for different sports balls. These studies usually follow a procedure where different balls are dropped and left to bounce. Then data regarding their initial and final velocities or how high they bounced is recorded. The images below show some of the data from one of these experiments.

== See also ==

http://physicsbook.gatech.edu/Elastic_Collisions 
http://physicsbook.gatech.edu/Inelastic_Collisions 
http://physicsbook.gatech.edu/Collisions 
http://physicsbook.gatech.edu/Momentum 

===External links===
http://www.quintic.com/education/case_studies/coefficient_restitution.htm

==References==

McGill, David J., and Wilton W. King. Engineering mechanics : an introduction to dynamics. Bloomington, IN: Tichenor Pub, 2003. Print. 
"Quintic 4 Education Sports Video Analysis Software Education Package - The Coefficient of Restitution." Quintic 4 Education Sports Video Analysis Software Education Package - The Coefficient of Restitution. Quintic 4 Education, n.d. Web. 05 Dec. 2015. 
"Coefficient of Restitution (COR)." ( ) (n.d.): n. pag. Yale. Web. 5 Dec. 2015. 

[[Category:Collisions]]

Coefficient of Restitution

2015-12-06T03:08:52Z

Mmoreno30: /* References */

Claimed by Maria Moreno

The coefficient of restitution measures the amount of Kinetic Energy remains and how much energy is lost after a collision. It also shows the degree of elasticity of a collision. It is defined as the ratio of the difference in initial velocities divided by the difference in final velocities. The coefficient of restitution of a collision is usually between 0 and 1 except in rare cases when energy is released by a collision.
[[File:coe2.png|right]]

==The Main Idea==

The coefficient of restitution is a ratio that describes the degree of elasticity of a collision. It is used to solve problems dealing with collisions that are not perfectly elastic or inelastic.
The equation that describes the coefficient of restitution involved dividing the difference in the final velocities by the difference in the initial velocity.

Consider objects A and B with initial velocities vAi and vBi and final velocities vAf and vBf.
The coefficient of restitution, e is determined with the following formula:

<math> e = \frac{v_{Bf}-v_{Af}}{v_{Ai}-v_{Bi}} </math> 
The coefficient is a unitless form of measurement as it is the ratio of two velocities and the units cancel.

===A Mathematical Model===
Consider a perfectly elastic collision between objects A and B where vAi and vBi refer to the initial velocities of A and B and vAf and vBf refer to the final velocities of A and B.

Start with the conservation of momentum principle which states that 
<math>\vec{p_i} = \vec{p_f}</math> and 
<math>m_A\vec{v_{Ai}}+m_A\vec{v_{Bi}}=m_A\vec{v_{Af}}+m_A\vec{v_{Bf}}</math> (1) 

Remember that for a perfectly elastic collision kinetic energy is also conserved meaning: 
<math>KE_i = KE_f</math> and 
<math>\frac{1}{2}m_A v^2_{Ai}+\frac{1}{2}m_A v^2_Bi=\frac{1}{2}m_A v^2_{Af}+\frac{1}{2}m_A v^2_{Bf}</math> (2) 

Dividing (2) by (1) yields: 

<math>\vec{v_{Ai}}+\vec{v_{Bi}}=\vec{v_{Af}}+\vec{v_{Bf}}</math> 
<math>1=\frac{v_{Ai}+v_{Bi}}{v_{Af}+v_{Bf}}</math> 

<math>e=\frac{v_{Ai}+v_{Bi}}{v_{Af}+v_{Bf}}</math> 

For perfectly elastic collision e=1.

===Inelastic Collisions===

An inelastic collision is a type of collision where Kinetic Energy is not conserved. In this case the difference in initial velocities will not equal the difference in final velocities. This means that the coefficient of restitution will not equal 1.

Consider the same collision as previously described, except this collision is inelastic and Kinetic Energy is not conserved. Since some of the kinetic energy is dissipated: 
<math>\frac{1}{2}m_A v^2_{Ai}+\frac{1}{2}m_A v^2_Bi<\frac{1}{2}m_A v^2_{Af}+\frac{1}{2}m_A v^2_{Bf}</math> 
<math>\frac{v_{Ai}+v_{Bi}}{v_{Af}+v_{Bf}}<1</math> 

<math>e=\frac{v_{Ai}+v_{Bi}}{v_{Af}+v_{Bf}}<1</math> 

A perfectly inelastic collision (for example when two objects collide and then stick together) has a coefficient of restitution of 0.

===A Computational Model===
Here is a link to a glowscript simulation
http://www.glowscript.org/#/user/mmoreno181/folder/phys/program/cor 
Here is the code for the simulation: 
<code>
ball1=sphere(pos=vec(0,0,0), color=color.red, radius=.5)
ball2=sphere(pos=vec(10,0,0), color=color.blue, radius=.5)

#initialize velocity
v1i=vec(.02,0,0)
v2i=vec(-.1,0,0)
v1f=vec(-.07,0,0)

#set coefficient of resitution
e=.5

#find v2f
v2f=v1f-(e*(v2i-v1i))

#find the time that the balls collide
tmeet= (ball2.pos.x-ball1.pos.x)/(v1i.x-v2i.x)

t=0

#animate
while t<150:
rate(100)
if t < tmeet: #if it's before the collision
ball1.pos=ball1.pos+v1i
ball2.pos=ball2.pos+v2i
else:
ball1.pos=ball1.pos+v1f
ball2.pos=ball2.pos+v2f

t=t+1
</code>

==Examples==
===Simple===
A ball is traveling at <math>4 \frac{m}{s}</math> and collides with a ball. After the collision the ball travels at <math>2.4\frac{m}{s}</math> in the opposite direction. Find the coefficient of restitution of the collision. 
Start with the definition of the coefficient of restitution: 
<math>e=\frac{v_{wall_f}-v_{ball_f}}{v_{ball_i}-v_{wall_i}}</math> 
Then plug in values and solve: 
<math>e=\frac{0-v_{ball_f}}{v_{ball_i}-0}</math> 
<math>e=\frac{-(-2.4)}{4}</math> 
<math>e=.6</math>

===Middling===
A hockey puck travels at a constant velocity <math>5\frac{m}{s}</math> and is hit by another puck with a velocity of <math>8\frac{m}{s}</math>. After the collision, it travels in the opposite direction at 4m/s. This collision has a coefficient of restitution of .55. Find the velocity of the second puck after the collision. 
Start with the definition of the coefficient of restitution. We are calling the velocity of the first puck <math>v_1</math> and the velocity of the second puck <math>v_2</math>. 
<math>e=\frac{v_{1f}-v_{2f}}{v_{2i}-v_{1i}}</math> 
Solve for <math>v_{2f}</math> 

<math>v_{2f}=v_{1f}-e(v_{2i}-v_{1i})</math> 
Plug in the values stated in the problem. 
<math>v_{2f}=4-.55(5-8)</math>
<math>v_{2f}=2.35\frac{m}{s}</math>

===Difficult===
[[File:Cor1.jpg|right]]
Bouncing Ball Example:

Consider a ball dropped from initial height <math>h_0</math>. When it bounces it reaches a height <math>h_f</math>. Find the coefficient of restitution of this collision. 
<math>e=\frac{v_{floor_f}-v_{ball_f}}{v_{ball_i}-v_{floor_i}}</math> 
The floor's velocity is always 0. 
<math>e=\frac{v_{ball_f}}{v_{ball_i}}</math> 
Recall the law of conservation of energy: 
<math> \Delta KE = \Delta PE</math> 
For the ball from h0 to h=0
<math> \frac{1}{2}mv_{ball_i}^2=mgh_0</math> 
<math> \frac{1}{2}v_{ball_i}^2=gh_0</math> 
<math> v_{ball_i}=2\sqrt{gh_0} </math> 
The same concept can be applied to the second bounce. 
<math>v_{ball_f}=2\sqrt{gh_f}</math> 
Plugging this in to the coefficient of restitution expression we get: 
<math>e=\frac{2\sqrt{mgh_f}}{2\sqrt{mgh_0}}</math> 
<math>e=\frac{\sqrt{h_f}}{\sqrt{h_0}}</math> 

==Connectedness==
'''How is this topic connected to something that you are interested in?''' 
This topic has major connections to sports. As a former high school volleyball player, it is very interesting to think about the collisions that occurred in the game and how the coefficient of restitution is related to those collisions. 
[[File:sportsgraph.png|right|thumb|This graph shows the initial and final velocities of many different balls from different sports.]]
'''How is it connected to your major (Mechanical Engineering)?''' 
The coefficient of restitution is a key concept in the study of Dynamics which is the foundation of many aspects of mechanical engineering. It also has to do with applications of mechanical engineering regarding materials. 

[[File:sportschart.png|right|thumb|This chart shows the relationship between different coefficients of restitution and the initial and final velocities of sports balls.]]

'''Is there an interesting industrial application?'''
In sports the coefficient of restitution is an important concept especially in sports like golf or tennis where collisions are a frequent and essential part of the game. In these sports sometimes limitations are put on how high the coefficient of restitution can be for certain clubs or bats. The maximum coefficient of restitution allowed in golf for clubs is .82. There have been studies done on the coefficients for different sports balls. These studies usually follow a procedure where different balls are dropped and left to bounce. Then data regarding their initial and final velocities or how high they bounced is recorded. The images below show some of the data from one of these experiments.

== See also ==

Inelastic Collision
Elastic Collision
Momentum
Conservation of Energy

===External links===
http://www.quintic.com/education/case_studies/coefficient_restitution.htm

==References==

McGill, David J., and Wilton W. King. Engineering mechanics : an introduction to dynamics. Bloomington, IN: Tichenor Pub, 2003. Print. 
"Quintic 4 Education Sports Video Analysis Software Education Package - The Coefficient of Restitution." Quintic 4 Education Sports Video Analysis Software Education Package - The Coefficient of Restitution. Quintic 4 Education, n.d. Web. 05 Dec. 2015. 
"Coefficient of Restitution (COR)." ( ) (n.d.): n. pag. Yale. Web. 5 Dec. 2015. 

[[Category:Collisions]]

Coefficient of Restitution

2015-12-06T03:06:33Z

Mmoreno30: /* References */

Claimed by Maria Moreno

The coefficient of restitution measures the amount of Kinetic Energy remains and how much energy is lost after a collision. It also shows the degree of elasticity of a collision. It is defined as the ratio of the difference in initial velocities divided by the difference in final velocities. The coefficient of restitution of a collision is usually between 0 and 1 except in rare cases when energy is released by a collision.
[[File:coe2.png|right]]

==The Main Idea==

The coefficient of restitution is a ratio that describes the degree of elasticity of a collision. It is used to solve problems dealing with collisions that are not perfectly elastic or inelastic.
The equation that describes the coefficient of restitution involved dividing the difference in the final velocities by the difference in the initial velocity.

Consider objects A and B with initial velocities vAi and vBi and final velocities vAf and vBf.
The coefficient of restitution, e is determined with the following formula:

<math> e = \frac{v_{Bf}-v_{Af}}{v_{Ai}-v_{Bi}} </math> 
The coefficient is a unitless form of measurement as it is the ratio of two velocities and the units cancel.

===A Mathematical Model===
Consider a perfectly elastic collision between objects A and B where vAi and vBi refer to the initial velocities of A and B and vAf and vBf refer to the final velocities of A and B.

Start with the conservation of momentum principle which states that 
<math>\vec{p_i} = \vec{p_f}</math> and 
<math>m_A\vec{v_{Ai}}+m_A\vec{v_{Bi}}=m_A\vec{v_{Af}}+m_A\vec{v_{Bf}}</math> (1) 

Remember that for a perfectly elastic collision kinetic energy is also conserved meaning: 
<math>KE_i = KE_f</math> and 
<math>\frac{1}{2}m_A v^2_{Ai}+\frac{1}{2}m_A v^2_Bi=\frac{1}{2}m_A v^2_{Af}+\frac{1}{2}m_A v^2_{Bf}</math> (2) 

Dividing (2) by (1) yields: 

<math>\vec{v_{Ai}}+\vec{v_{Bi}}=\vec{v_{Af}}+\vec{v_{Bf}}</math> 
<math>1=\frac{v_{Ai}+v_{Bi}}{v_{Af}+v_{Bf}}</math> 

<math>e=\frac{v_{Ai}+v_{Bi}}{v_{Af}+v_{Bf}}</math> 

For perfectly elastic collision e=1.

===Inelastic Collisions===

An inelastic collision is a type of collision where Kinetic Energy is not conserved. In this case the difference in initial velocities will not equal the difference in final velocities. This means that the coefficient of restitution will not equal 1.

Consider the same collision as previously described, except this collision is inelastic and Kinetic Energy is not conserved. Since some of the kinetic energy is dissipated: 
<math>\frac{1}{2}m_A v^2_{Ai}+\frac{1}{2}m_A v^2_Bi<\frac{1}{2}m_A v^2_{Af}+\frac{1}{2}m_A v^2_{Bf}</math> 
<math>\frac{v_{Ai}+v_{Bi}}{v_{Af}+v_{Bf}}<1</math> 

<math>e=\frac{v_{Ai}+v_{Bi}}{v_{Af}+v_{Bf}}<1</math> 

A perfectly inelastic collision (for example when two objects collide and then stick together) has a coefficient of restitution of 0.

===A Computational Model===
Here is a link to a glowscript simulation
http://www.glowscript.org/#/user/mmoreno181/folder/phys/program/cor 
Here is the code for the simulation: 
<code>
ball1=sphere(pos=vec(0,0,0), color=color.red, radius=.5)
ball2=sphere(pos=vec(10,0,0), color=color.blue, radius=.5)

#initialize velocity
v1i=vec(.02,0,0)
v2i=vec(-.1,0,0)
v1f=vec(-.07,0,0)

#set coefficient of resitution
e=.5

#find v2f
v2f=v1f-(e*(v2i-v1i))

#find the time that the balls collide
tmeet= (ball2.pos.x-ball1.pos.x)/(v1i.x-v2i.x)

t=0

#animate
while t<150:
rate(100)
if t < tmeet: #if it's before the collision
ball1.pos=ball1.pos+v1i
ball2.pos=ball2.pos+v2i
else:
ball1.pos=ball1.pos+v1f
ball2.pos=ball2.pos+v2f

t=t+1
</code>

==Examples==
===Simple===
A ball is traveling at <math>4 \frac{m}{s}</math> and collides with a ball. After the collision the ball travels at <math>2.4\frac{m}{s}</math> in the opposite direction. Find the coefficient of restitution of the collision. 
Start with the definition of the coefficient of restitution: 
<math>e=\frac{v_{wall_f}-v_{ball_f}}{v_{ball_i}-v_{wall_i}}</math> 
Then plug in values and solve: 
<math>e=\frac{0-v_{ball_f}}{v_{ball_i}-0}</math> 
<math>e=\frac{-(-2.4)}{4}</math> 
<math>e=.6</math>

===Middling===
A hockey puck travels at a constant velocity <math>5\frac{m}{s}</math> and is hit by another puck with a velocity of <math>8\frac{m}{s}</math>. After the collision, it travels in the opposite direction at 4m/s. This collision has a coefficient of restitution of .55. Find the velocity of the second puck after the collision. 
Start with the definition of the coefficient of restitution. We are calling the velocity of the first puck <math>v_1</math> and the velocity of the second puck <math>v_2</math>. 
<math>e=\frac{v_{1f}-v_{2f}}{v_{2i}-v_{1i}}</math> 
Solve for <math>v_{2f}</math> 

<math>v_{2f}=v_{1f}-e(v_{2i}-v_{1i})</math> 
Plug in the values stated in the problem. 
<math>v_{2f}=4-.55(5-8)</math>
<math>v_{2f}=2.35\frac{m}{s}</math>

===Difficult===
[[File:Cor1.jpg|right]]
Bouncing Ball Example:

Consider a ball dropped from initial height <math>h_0</math>. When it bounces it reaches a height <math>h_f</math>. Find the coefficient of restitution of this collision. 
<math>e=\frac{v_{floor_f}-v_{ball_f}}{v_{ball_i}-v_{floor_i}}</math> 
The floor's velocity is always 0. 
<math>e=\frac{v_{ball_f}}{v_{ball_i}}</math> 
Recall the law of conservation of energy: 
<math> \Delta KE = \Delta PE</math> 
For the ball from h0 to h=0
<math> \frac{1}{2}mv_{ball_i}^2=mgh_0</math> 
<math> \frac{1}{2}v_{ball_i}^2=gh_0</math> 
<math> v_{ball_i}=2\sqrt{gh_0} </math> 
The same concept can be applied to the second bounce. 
<math>v_{ball_f}=2\sqrt{gh_f}</math> 
Plugging this in to the coefficient of restitution expression we get: 
<math>e=\frac{2\sqrt{mgh_f}}{2\sqrt{mgh_0}}</math> 
<math>e=\frac{\sqrt{h_f}}{\sqrt{h_0}}</math> 

==Connectedness==
'''How is this topic connected to something that you are interested in?''' 
This topic has major connections to sports. As a former high school volleyball player, it is very interesting to think about the collisions that occurred in the game and how the coefficient of restitution is related to those collisions. 
[[File:sportsgraph.png|right|thumb|This graph shows the initial and final velocities of many different balls from different sports.]]
'''How is it connected to your major (Mechanical Engineering)?''' 
The coefficient of restitution is a key concept in the study of Dynamics which is the foundation of many aspects of mechanical engineering. It also has to do with applications of mechanical engineering regarding materials. 

[[File:sportschart.png|right|thumb|This chart shows the relationship between different coefficients of restitution and the initial and final velocities of sports balls.]]

'''Is there an interesting industrial application?'''
In sports the coefficient of restitution is an important concept especially in sports like golf or tennis where collisions are a frequent and essential part of the game. In these sports sometimes limitations are put on how high the coefficient of restitution can be for certain clubs or bats. The maximum coefficient of restitution allowed in golf for clubs is .82. There have been studies done on the coefficients for different sports balls. These studies usually follow a procedure where different balls are dropped and left to bounce. Then data regarding their initial and final velocities or how high they bounced is recorded. The images below show some of the data from one of these experiments.

== See also ==

Inelastic Collision
Elastic Collision
Momentum
Conservation of Energy

===External links===
http://www.quintic.com/education/case_studies/coefficient_restitution.htm

==References==

McGill, David J., and Wilton W. King. Engineering mechanics : an introduction to dynamics. Bloomington, IN: Tichenor Pub, 2003. Print. 
"Quintic 4 Education Sports Video Analysis Software Education Package - The Coefficient of Restitution." Quintic 4 Education Sports Video Analysis Software Education Package - The Coefficient of Restitution. Quintic 4 Education, n.d. Web. 05 Dec. 2015. 

[[Category:Collisions]]

Coefficient of Restitution

2015-12-06T03:02:24Z

Mmoreno30: /* See also */

Claimed by Maria Moreno

The coefficient of restitution measures the amount of Kinetic Energy remains and how much energy is lost after a collision. It also shows the degree of elasticity of a collision. It is defined as the ratio of the difference in initial velocities divided by the difference in final velocities. The coefficient of restitution of a collision is usually between 0 and 1 except in rare cases when energy is released by a collision.
[[File:coe2.png|right]]

==The Main Idea==

The coefficient of restitution is a ratio that describes the degree of elasticity of a collision. It is used to solve problems dealing with collisions that are not perfectly elastic or inelastic.
The equation that describes the coefficient of restitution involved dividing the difference in the final velocities by the difference in the initial velocity.

Consider objects A and B with initial velocities vAi and vBi and final velocities vAf and vBf.
The coefficient of restitution, e is determined with the following formula:

<math> e = \frac{v_{Bf}-v_{Af}}{v_{Ai}-v_{Bi}} </math> 
The coefficient is a unitless form of measurement as it is the ratio of two velocities and the units cancel.

===A Mathematical Model===
Consider a perfectly elastic collision between objects A and B where vAi and vBi refer to the initial velocities of A and B and vAf and vBf refer to the final velocities of A and B.

Start with the conservation of momentum principle which states that 
<math>\vec{p_i} = \vec{p_f}</math> and 
<math>m_A\vec{v_{Ai}}+m_A\vec{v_{Bi}}=m_A\vec{v_{Af}}+m_A\vec{v_{Bf}}</math> (1) 

Remember that for a perfectly elastic collision kinetic energy is also conserved meaning: 
<math>KE_i = KE_f</math> and 
<math>\frac{1}{2}m_A v^2_{Ai}+\frac{1}{2}m_A v^2_Bi=\frac{1}{2}m_A v^2_{Af}+\frac{1}{2}m_A v^2_{Bf}</math> (2) 

Dividing (2) by (1) yields: 

<math>\vec{v_{Ai}}+\vec{v_{Bi}}=\vec{v_{Af}}+\vec{v_{Bf}}</math> 
<math>1=\frac{v_{Ai}+v_{Bi}}{v_{Af}+v_{Bf}}</math> 

<math>e=\frac{v_{Ai}+v_{Bi}}{v_{Af}+v_{Bf}}</math> 

For perfectly elastic collision e=1.

===Inelastic Collisions===

An inelastic collision is a type of collision where Kinetic Energy is not conserved. In this case the difference in initial velocities will not equal the difference in final velocities. This means that the coefficient of restitution will not equal 1.

Consider the same collision as previously described, except this collision is inelastic and Kinetic Energy is not conserved. Since some of the kinetic energy is dissipated: 
<math>\frac{1}{2}m_A v^2_{Ai}+\frac{1}{2}m_A v^2_Bi<\frac{1}{2}m_A v^2_{Af}+\frac{1}{2}m_A v^2_{Bf}</math> 
<math>\frac{v_{Ai}+v_{Bi}}{v_{Af}+v_{Bf}}<1</math> 

<math>e=\frac{v_{Ai}+v_{Bi}}{v_{Af}+v_{Bf}}<1</math> 

A perfectly inelastic collision (for example when two objects collide and then stick together) has a coefficient of restitution of 0.

===A Computational Model===
Here is a link to a glowscript simulation
http://www.glowscript.org/#/user/mmoreno181/folder/phys/program/cor 
Here is the code for the simulation: 
<code>
ball1=sphere(pos=vec(0,0,0), color=color.red, radius=.5)
ball2=sphere(pos=vec(10,0,0), color=color.blue, radius=.5)

#initialize velocity
v1i=vec(.02,0,0)
v2i=vec(-.1,0,0)
v1f=vec(-.07,0,0)

#set coefficient of resitution
e=.5

#find v2f
v2f=v1f-(e*(v2i-v1i))

#find the time that the balls collide
tmeet= (ball2.pos.x-ball1.pos.x)/(v1i.x-v2i.x)

t=0

#animate
while t<150:
rate(100)
if t < tmeet: #if it's before the collision
ball1.pos=ball1.pos+v1i
ball2.pos=ball2.pos+v2i
else:
ball1.pos=ball1.pos+v1f
ball2.pos=ball2.pos+v2f

t=t+1
</code>

==Examples==
===Simple===
A ball is traveling at <math>4 \frac{m}{s}</math> and collides with a ball. After the collision the ball travels at <math>2.4\frac{m}{s}</math> in the opposite direction. Find the coefficient of restitution of the collision. 
Start with the definition of the coefficient of restitution: 
<math>e=\frac{v_{wall_f}-v_{ball_f}}{v_{ball_i}-v_{wall_i}}</math> 
Then plug in values and solve: 
<math>e=\frac{0-v_{ball_f}}{v_{ball_i}-0}</math> 
<math>e=\frac{-(-2.4)}{4}</math> 
<math>e=.6</math>

===Middling===
A hockey puck travels at a constant velocity <math>5\frac{m}{s}</math> and is hit by another puck with a velocity of <math>8\frac{m}{s}</math>. After the collision, it travels in the opposite direction at 4m/s. This collision has a coefficient of restitution of .55. Find the velocity of the second puck after the collision. 
Start with the definition of the coefficient of restitution. We are calling the velocity of the first puck <math>v_1</math> and the velocity of the second puck <math>v_2</math>. 
<math>e=\frac{v_{1f}-v_{2f}}{v_{2i}-v_{1i}}</math> 
Solve for <math>v_{2f}</math> 

<math>v_{2f}=v_{1f}-e(v_{2i}-v_{1i})</math> 
Plug in the values stated in the problem. 
<math>v_{2f}=4-.55(5-8)</math>
<math>v_{2f}=2.35\frac{m}{s}</math>

===Difficult===
[[File:Cor1.jpg|right]]
Bouncing Ball Example:

Consider a ball dropped from initial height <math>h_0</math>. When it bounces it reaches a height <math>h_f</math>. Find the coefficient of restitution of this collision. 
<math>e=\frac{v_{floor_f}-v_{ball_f}}{v_{ball_i}-v_{floor_i}}</math> 
The floor's velocity is always 0. 
<math>e=\frac{v_{ball_f}}{v_{ball_i}}</math> 
Recall the law of conservation of energy: 
<math> \Delta KE = \Delta PE</math> 
For the ball from h0 to h=0
<math> \frac{1}{2}mv_{ball_i}^2=mgh_0</math> 
<math> \frac{1}{2}v_{ball_i}^2=gh_0</math> 
<math> v_{ball_i}=2\sqrt{gh_0} </math> 
The same concept can be applied to the second bounce. 
<math>v_{ball_f}=2\sqrt{gh_f}</math> 
Plugging this in to the coefficient of restitution expression we get: 
<math>e=\frac{2\sqrt{mgh_f}}{2\sqrt{mgh_0}}</math> 
<math>e=\frac{\sqrt{h_f}}{\sqrt{h_0}}</math> 

==Connectedness==
'''How is this topic connected to something that you are interested in?''' 
This topic has major connections to sports. As a former high school volleyball player, it is very interesting to think about the collisions that occurred in the game and how the coefficient of restitution is related to those collisions. 
[[File:sportsgraph.png|right|thumb|This graph shows the initial and final velocities of many different balls from different sports.]]
'''How is it connected to your major (Mechanical Engineering)?''' 
The coefficient of restitution is a key concept in the study of Dynamics which is the foundation of many aspects of mechanical engineering. It also has to do with applications of mechanical engineering regarding materials. 

[[File:sportschart.png|right|thumb|This chart shows the relationship between different coefficients of restitution and the initial and final velocities of sports balls.]]

'''Is there an interesting industrial application?'''
In sports the coefficient of restitution is an important concept especially in sports like golf or tennis where collisions are a frequent and essential part of the game. In these sports sometimes limitations are put on how high the coefficient of restitution can be for certain clubs or bats. The maximum coefficient of restitution allowed in golf for clubs is .82. There have been studies done on the coefficients for different sports balls. These studies usually follow a procedure where different balls are dropped and left to bounce. Then data regarding their initial and final velocities or how high they bounced is recorded. The images below show some of the data from one of these experiments.

== See also ==

Inelastic Collision
Elastic Collision
Momentum
Conservation of Energy

===External links===
http://www.quintic.com/education/case_studies/coefficient_restitution.htm

==References==

[[Category:Collisions]]

Coefficient of Restitution

2015-12-06T03:00:08Z

Mmoreno30: /* Connectedness */

Claimed by Maria Moreno

The coefficient of restitution measures the amount of Kinetic Energy remains and how much energy is lost after a collision. It also shows the degree of elasticity of a collision. It is defined as the ratio of the difference in initial velocities divided by the difference in final velocities. The coefficient of restitution of a collision is usually between 0 and 1 except in rare cases when energy is released by a collision.
[[File:coe2.png|right]]

==The Main Idea==

The coefficient of restitution is a ratio that describes the degree of elasticity of a collision. It is used to solve problems dealing with collisions that are not perfectly elastic or inelastic.
The equation that describes the coefficient of restitution involved dividing the difference in the final velocities by the difference in the initial velocity.

Consider objects A and B with initial velocities vAi and vBi and final velocities vAf and vBf.
The coefficient of restitution, e is determined with the following formula:

<math> e = \frac{v_{Bf}-v_{Af}}{v_{Ai}-v_{Bi}} </math> 
The coefficient is a unitless form of measurement as it is the ratio of two velocities and the units cancel.

===A Mathematical Model===
Consider a perfectly elastic collision between objects A and B where vAi and vBi refer to the initial velocities of A and B and vAf and vBf refer to the final velocities of A and B.

Start with the conservation of momentum principle which states that 
<math>\vec{p_i} = \vec{p_f}</math> and 
<math>m_A\vec{v_{Ai}}+m_A\vec{v_{Bi}}=m_A\vec{v_{Af}}+m_A\vec{v_{Bf}}</math> (1) 

Remember that for a perfectly elastic collision kinetic energy is also conserved meaning: 
<math>KE_i = KE_f</math> and 
<math>\frac{1}{2}m_A v^2_{Ai}+\frac{1}{2}m_A v^2_Bi=\frac{1}{2}m_A v^2_{Af}+\frac{1}{2}m_A v^2_{Bf}</math> (2) 

Dividing (2) by (1) yields: 

<math>\vec{v_{Ai}}+\vec{v_{Bi}}=\vec{v_{Af}}+\vec{v_{Bf}}</math> 
<math>1=\frac{v_{Ai}+v_{Bi}}{v_{Af}+v_{Bf}}</math> 

<math>e=\frac{v_{Ai}+v_{Bi}}{v_{Af}+v_{Bf}}</math> 

For perfectly elastic collision e=1.

===Inelastic Collisions===

An inelastic collision is a type of collision where Kinetic Energy is not conserved. In this case the difference in initial velocities will not equal the difference in final velocities. This means that the coefficient of restitution will not equal 1.

Consider the same collision as previously described, except this collision is inelastic and Kinetic Energy is not conserved. Since some of the kinetic energy is dissipated: 
<math>\frac{1}{2}m_A v^2_{Ai}+\frac{1}{2}m_A v^2_Bi<\frac{1}{2}m_A v^2_{Af}+\frac{1}{2}m_A v^2_{Bf}</math> 
<math>\frac{v_{Ai}+v_{Bi}}{v_{Af}+v_{Bf}}<1</math> 

<math>e=\frac{v_{Ai}+v_{Bi}}{v_{Af}+v_{Bf}}<1</math> 

A perfectly inelastic collision (for example when two objects collide and then stick together) has a coefficient of restitution of 0.

===A Computational Model===
Here is a link to a glowscript simulation
http://www.glowscript.org/#/user/mmoreno181/folder/phys/program/cor 
Here is the code for the simulation: 
<code>
ball1=sphere(pos=vec(0,0,0), color=color.red, radius=.5)
ball2=sphere(pos=vec(10,0,0), color=color.blue, radius=.5)

#initialize velocity
v1i=vec(.02,0,0)
v2i=vec(-.1,0,0)
v1f=vec(-.07,0,0)

#set coefficient of resitution
e=.5

#find v2f
v2f=v1f-(e*(v2i-v1i))

#find the time that the balls collide
tmeet= (ball2.pos.x-ball1.pos.x)/(v1i.x-v2i.x)

t=0

#animate
while t<150:
rate(100)
if t < tmeet: #if it's before the collision
ball1.pos=ball1.pos+v1i
ball2.pos=ball2.pos+v2i
else:
ball1.pos=ball1.pos+v1f
ball2.pos=ball2.pos+v2f

t=t+1
</code>

==Examples==
===Simple===
A ball is traveling at <math>4 \frac{m}{s}</math> and collides with a ball. After the collision the ball travels at <math>2.4\frac{m}{s}</math> in the opposite direction. Find the coefficient of restitution of the collision. 
Start with the definition of the coefficient of restitution: 
<math>e=\frac{v_{wall_f}-v_{ball_f}}{v_{ball_i}-v_{wall_i}}</math> 
Then plug in values and solve: 
<math>e=\frac{0-v_{ball_f}}{v_{ball_i}-0}</math> 
<math>e=\frac{-(-2.4)}{4}</math> 
<math>e=.6</math>

===Middling===
A hockey puck travels at a constant velocity <math>5\frac{m}{s}</math> and is hit by another puck with a velocity of <math>8\frac{m}{s}</math>. After the collision, it travels in the opposite direction at 4m/s. This collision has a coefficient of restitution of .55. Find the velocity of the second puck after the collision. 
Start with the definition of the coefficient of restitution. We are calling the velocity of the first puck <math>v_1</math> and the velocity of the second puck <math>v_2</math>. 
<math>e=\frac{v_{1f}-v_{2f}}{v_{2i}-v_{1i}}</math> 
Solve for <math>v_{2f}</math> 

<math>v_{2f}=v_{1f}-e(v_{2i}-v_{1i})</math> 
Plug in the values stated in the problem. 
<math>v_{2f}=4-.55(5-8)</math>
<math>v_{2f}=2.35\frac{m}{s}</math>

===Difficult===
[[File:Cor1.jpg|right]]
Bouncing Ball Example:

Consider a ball dropped from initial height <math>h_0</math>. When it bounces it reaches a height <math>h_f</math>. Find the coefficient of restitution of this collision. 
<math>e=\frac{v_{floor_f}-v_{ball_f}}{v_{ball_i}-v_{floor_i}}</math> 
The floor's velocity is always 0. 
<math>e=\frac{v_{ball_f}}{v_{ball_i}}</math> 
Recall the law of conservation of energy: 
<math> \Delta KE = \Delta PE</math> 
For the ball from h0 to h=0
<math> \frac{1}{2}mv_{ball_i}^2=mgh_0</math> 
<math> \frac{1}{2}v_{ball_i}^2=gh_0</math> 
<math> v_{ball_i}=2\sqrt{gh_0} </math> 
The same concept can be applied to the second bounce. 
<math>v_{ball_f}=2\sqrt{gh_f}</math> 
Plugging this in to the coefficient of restitution expression we get: 
<math>e=\frac{2\sqrt{mgh_f}}{2\sqrt{mgh_0}}</math> 
<math>e=\frac{\sqrt{h_f}}{\sqrt{h_0}}</math> 

==Connectedness==
'''How is this topic connected to something that you are interested in?''' 
This topic has major connections to sports. As a former high school volleyball player, it is very interesting to think about the collisions that occurred in the game and how the coefficient of restitution is related to those collisions. 
[[File:sportsgraph.png|right|thumb|This graph shows the initial and final velocities of many different balls from different sports.]]
'''How is it connected to your major (Mechanical Engineering)?''' 
The coefficient of restitution is a key concept in the study of Dynamics which is the foundation of many aspects of mechanical engineering. It also has to do with applications of mechanical engineering regarding materials. 

[[File:sportschart.png|right|thumb|This chart shows the relationship between different coefficients of restitution and the initial and final velocities of sports balls.]]

'''Is there an interesting industrial application?'''
In sports the coefficient of restitution is an important concept especially in sports like golf or tennis where collisions are a frequent and essential part of the game. In these sports sometimes limitations are put on how high the coefficient of restitution can be for certain clubs or bats. The maximum coefficient of restitution allowed in golf for clubs is .82. There have been studies done on the coefficients for different sports balls. These studies usually follow a procedure where different balls are dropped and left to bounce. Then data regarding their initial and final velocities or how high they bounced is recorded. The images below show some of the data from one of these experiments.

== See also ==

Are there related topics or categories in this wiki resource for the curious reader to explore? How does this topic fit into that context?

===Further reading===

Books, Articles or other print media on this topic

===External links===

==References==

[[Category:Collisions]]

Coefficient of Restitution

2015-12-06T02:59:46Z

Mmoreno30: /* Connectedness */

Claimed by Maria Moreno

The coefficient of restitution measures the amount of Kinetic Energy remains and how much energy is lost after a collision. It also shows the degree of elasticity of a collision. It is defined as the ratio of the difference in initial velocities divided by the difference in final velocities. The coefficient of restitution of a collision is usually between 0 and 1 except in rare cases when energy is released by a collision.
[[File:coe2.png|right]]

==The Main Idea==

The coefficient of restitution is a ratio that describes the degree of elasticity of a collision. It is used to solve problems dealing with collisions that are not perfectly elastic or inelastic.
The equation that describes the coefficient of restitution involved dividing the difference in the final velocities by the difference in the initial velocity.

Consider objects A and B with initial velocities vAi and vBi and final velocities vAf and vBf.
The coefficient of restitution, e is determined with the following formula:

<math> e = \frac{v_{Bf}-v_{Af}}{v_{Ai}-v_{Bi}} </math> 
The coefficient is a unitless form of measurement as it is the ratio of two velocities and the units cancel.

===A Mathematical Model===
Consider a perfectly elastic collision between objects A and B where vAi and vBi refer to the initial velocities of A and B and vAf and vBf refer to the final velocities of A and B.

Start with the conservation of momentum principle which states that 
<math>\vec{p_i} = \vec{p_f}</math> and 
<math>m_A\vec{v_{Ai}}+m_A\vec{v_{Bi}}=m_A\vec{v_{Af}}+m_A\vec{v_{Bf}}</math> (1) 

Remember that for a perfectly elastic collision kinetic energy is also conserved meaning: 
<math>KE_i = KE_f</math> and 
<math>\frac{1}{2}m_A v^2_{Ai}+\frac{1}{2}m_A v^2_Bi=\frac{1}{2}m_A v^2_{Af}+\frac{1}{2}m_A v^2_{Bf}</math> (2) 

Dividing (2) by (1) yields: 

<math>\vec{v_{Ai}}+\vec{v_{Bi}}=\vec{v_{Af}}+\vec{v_{Bf}}</math> 
<math>1=\frac{v_{Ai}+v_{Bi}}{v_{Af}+v_{Bf}}</math> 

<math>e=\frac{v_{Ai}+v_{Bi}}{v_{Af}+v_{Bf}}</math> 

For perfectly elastic collision e=1.

===Inelastic Collisions===

An inelastic collision is a type of collision where Kinetic Energy is not conserved. In this case the difference in initial velocities will not equal the difference in final velocities. This means that the coefficient of restitution will not equal 1.

Consider the same collision as previously described, except this collision is inelastic and Kinetic Energy is not conserved. Since some of the kinetic energy is dissipated: 
<math>\frac{1}{2}m_A v^2_{Ai}+\frac{1}{2}m_A v^2_Bi<\frac{1}{2}m_A v^2_{Af}+\frac{1}{2}m_A v^2_{Bf}</math> 
<math>\frac{v_{Ai}+v_{Bi}}{v_{Af}+v_{Bf}}<1</math> 

<math>e=\frac{v_{Ai}+v_{Bi}}{v_{Af}+v_{Bf}}<1</math> 

A perfectly inelastic collision (for example when two objects collide and then stick together) has a coefficient of restitution of 0.

===A Computational Model===
Here is a link to a glowscript simulation
http://www.glowscript.org/#/user/mmoreno181/folder/phys/program/cor 
Here is the code for the simulation: 
<code>
ball1=sphere(pos=vec(0,0,0), color=color.red, radius=.5)
ball2=sphere(pos=vec(10,0,0), color=color.blue, radius=.5)

#initialize velocity
v1i=vec(.02,0,0)
v2i=vec(-.1,0,0)
v1f=vec(-.07,0,0)

#set coefficient of resitution
e=.5

#find v2f
v2f=v1f-(e*(v2i-v1i))

#find the time that the balls collide
tmeet= (ball2.pos.x-ball1.pos.x)/(v1i.x-v2i.x)

t=0

#animate
while t<150:
rate(100)
if t < tmeet: #if it's before the collision
ball1.pos=ball1.pos+v1i
ball2.pos=ball2.pos+v2i
else:
ball1.pos=ball1.pos+v1f
ball2.pos=ball2.pos+v2f

t=t+1
</code>

==Examples==
===Simple===
A ball is traveling at <math>4 \frac{m}{s}</math> and collides with a ball. After the collision the ball travels at <math>2.4\frac{m}{s}</math> in the opposite direction. Find the coefficient of restitution of the collision. 
Start with the definition of the coefficient of restitution: 
<math>e=\frac{v_{wall_f}-v_{ball_f}}{v_{ball_i}-v_{wall_i}}</math> 
Then plug in values and solve: 
<math>e=\frac{0-v_{ball_f}}{v_{ball_i}-0}</math> 
<math>e=\frac{-(-2.4)}{4}</math> 
<math>e=.6</math>

===Middling===
A hockey puck travels at a constant velocity <math>5\frac{m}{s}</math> and is hit by another puck with a velocity of <math>8\frac{m}{s}</math>. After the collision, it travels in the opposite direction at 4m/s. This collision has a coefficient of restitution of .55. Find the velocity of the second puck after the collision. 
Start with the definition of the coefficient of restitution. We are calling the velocity of the first puck <math>v_1</math> and the velocity of the second puck <math>v_2</math>. 
<math>e=\frac{v_{1f}-v_{2f}}{v_{2i}-v_{1i}}</math> 
Solve for <math>v_{2f}</math> 

<math>v_{2f}=v_{1f}-e(v_{2i}-v_{1i})</math> 
Plug in the values stated in the problem. 
<math>v_{2f}=4-.55(5-8)</math>
<math>v_{2f}=2.35\frac{m}{s}</math>

===Difficult===
[[File:Cor1.jpg|right]]
Bouncing Ball Example:

Consider a ball dropped from initial height <math>h_0</math>. When it bounces it reaches a height <math>h_f</math>. Find the coefficient of restitution of this collision. 
<math>e=\frac{v_{floor_f}-v_{ball_f}}{v_{ball_i}-v_{floor_i}}</math> 
The floor's velocity is always 0. 
<math>e=\frac{v_{ball_f}}{v_{ball_i}}</math> 
Recall the law of conservation of energy: 
<math> \Delta KE = \Delta PE</math> 
For the ball from h0 to h=0
<math> \frac{1}{2}mv_{ball_i}^2=mgh_0</math> 
<math> \frac{1}{2}v_{ball_i}^2=gh_0</math> 
<math> v_{ball_i}=2\sqrt{gh_0} </math> 
The same concept can be applied to the second bounce. 
<math>v_{ball_f}=2\sqrt{gh_f}</math> 
Plugging this in to the coefficient of restitution expression we get: 
<math>e=\frac{2\sqrt{mgh_f}}{2\sqrt{mgh_0}}</math> 
<math>e=\frac{\sqrt{h_f}}{\sqrt{h_0}}</math> 

==Connectedness==
'''How is this topic connected to something that you are interested in?''' 
This topic has major connections to sports. As a former high school volleyball player, it is very interesting to think about the collisions that occurred in the game and how the coefficient of restitution is related to those collisions. 
'''How is it connected to your major (Mechanical Engineering)?''' 
The coefficient of restitution is a key concept in the study of Dynamics which is the foundation of many aspects of mechanical engineering. It also has to do with applications of mechanical engineering regarding materials. 

[[File:sportsgraph.png|right|thumb|This graph shows the initial and final velocities of many different balls from different sports.]]
[[File:sportschart.png|right|thumb|This chart shows the relationship between different coefficients of restitution and the initial and final velocities of sports balls.]]

'''Is there an interesting industrial application?'''
In sports the coefficient of restitution is an important concept especially in sports like golf or tennis where collisions are a frequent and essential part of the game. In these sports sometimes limitations are put on how high the coefficient of restitution can be for certain clubs or bats. The maximum coefficient of restitution allowed in golf for clubs is .82. There have been studies done on the coefficients for different sports balls. These studies usually follow a procedure where different balls are dropped and left to bounce. Then data regarding their initial and final velocities or how high they bounced is recorded. The images below show some of the data from one of these experiments.

== See also ==

Are there related topics or categories in this wiki resource for the curious reader to explore? How does this topic fit into that context?

===Further reading===

Books, Articles or other print media on this topic

===External links===

==References==

[[Category:Collisions]]

Coefficient of Restitution

2015-12-06T02:59:08Z

Mmoreno30: /* Connectedness */

Claimed by Maria Moreno

The coefficient of restitution measures the amount of Kinetic Energy remains and how much energy is lost after a collision. It also shows the degree of elasticity of a collision. It is defined as the ratio of the difference in initial velocities divided by the difference in final velocities. The coefficient of restitution of a collision is usually between 0 and 1 except in rare cases when energy is released by a collision.
[[File:coe2.png|right]]

==The Main Idea==

The coefficient of restitution is a ratio that describes the degree of elasticity of a collision. It is used to solve problems dealing with collisions that are not perfectly elastic or inelastic.
The equation that describes the coefficient of restitution involved dividing the difference in the final velocities by the difference in the initial velocity.

Consider objects A and B with initial velocities vAi and vBi and final velocities vAf and vBf.
The coefficient of restitution, e is determined with the following formula:

<math> e = \frac{v_{Bf}-v_{Af}}{v_{Ai}-v_{Bi}} </math> 
The coefficient is a unitless form of measurement as it is the ratio of two velocities and the units cancel.

===A Mathematical Model===
Consider a perfectly elastic collision between objects A and B where vAi and vBi refer to the initial velocities of A and B and vAf and vBf refer to the final velocities of A and B.

Start with the conservation of momentum principle which states that 
<math>\vec{p_i} = \vec{p_f}</math> and 
<math>m_A\vec{v_{Ai}}+m_A\vec{v_{Bi}}=m_A\vec{v_{Af}}+m_A\vec{v_{Bf}}</math> (1) 

Remember that for a perfectly elastic collision kinetic energy is also conserved meaning: 
<math>KE_i = KE_f</math> and 
<math>\frac{1}{2}m_A v^2_{Ai}+\frac{1}{2}m_A v^2_Bi=\frac{1}{2}m_A v^2_{Af}+\frac{1}{2}m_A v^2_{Bf}</math> (2) 

Dividing (2) by (1) yields: 

<math>\vec{v_{Ai}}+\vec{v_{Bi}}=\vec{v_{Af}}+\vec{v_{Bf}}</math> 
<math>1=\frac{v_{Ai}+v_{Bi}}{v_{Af}+v_{Bf}}</math> 

<math>e=\frac{v_{Ai}+v_{Bi}}{v_{Af}+v_{Bf}}</math> 

For perfectly elastic collision e=1.

===Inelastic Collisions===

An inelastic collision is a type of collision where Kinetic Energy is not conserved. In this case the difference in initial velocities will not equal the difference in final velocities. This means that the coefficient of restitution will not equal 1.

Consider the same collision as previously described, except this collision is inelastic and Kinetic Energy is not conserved. Since some of the kinetic energy is dissipated: 
<math>\frac{1}{2}m_A v^2_{Ai}+\frac{1}{2}m_A v^2_Bi<\frac{1}{2}m_A v^2_{Af}+\frac{1}{2}m_A v^2_{Bf}</math> 
<math>\frac{v_{Ai}+v_{Bi}}{v_{Af}+v_{Bf}}<1</math> 

<math>e=\frac{v_{Ai}+v_{Bi}}{v_{Af}+v_{Bf}}<1</math> 

A perfectly inelastic collision (for example when two objects collide and then stick together) has a coefficient of restitution of 0.

===A Computational Model===
Here is a link to a glowscript simulation
http://www.glowscript.org/#/user/mmoreno181/folder/phys/program/cor 
Here is the code for the simulation: 
<code>
ball1=sphere(pos=vec(0,0,0), color=color.red, radius=.5)
ball2=sphere(pos=vec(10,0,0), color=color.blue, radius=.5)

#initialize velocity
v1i=vec(.02,0,0)
v2i=vec(-.1,0,0)
v1f=vec(-.07,0,0)

#set coefficient of resitution
e=.5

#find v2f
v2f=v1f-(e*(v2i-v1i))

#find the time that the balls collide
tmeet= (ball2.pos.x-ball1.pos.x)/(v1i.x-v2i.x)

t=0

#animate
while t<150:
rate(100)
if t < tmeet: #if it's before the collision
ball1.pos=ball1.pos+v1i
ball2.pos=ball2.pos+v2i
else:
ball1.pos=ball1.pos+v1f
ball2.pos=ball2.pos+v2f

t=t+1
</code>

==Examples==
===Simple===
A ball is traveling at <math>4 \frac{m}{s}</math> and collides with a ball. After the collision the ball travels at <math>2.4\frac{m}{s}</math> in the opposite direction. Find the coefficient of restitution of the collision. 
Start with the definition of the coefficient of restitution: 
<math>e=\frac{v_{wall_f}-v_{ball_f}}{v_{ball_i}-v_{wall_i}}</math> 
Then plug in values and solve: 
<math>e=\frac{0-v_{ball_f}}{v_{ball_i}-0}</math> 
<math>e=\frac{-(-2.4)}{4}</math> 
<math>e=.6</math>

===Middling===
A hockey puck travels at a constant velocity <math>5\frac{m}{s}</math> and is hit by another puck with a velocity of <math>8\frac{m}{s}</math>. After the collision, it travels in the opposite direction at 4m/s. This collision has a coefficient of restitution of .55. Find the velocity of the second puck after the collision. 
Start with the definition of the coefficient of restitution. We are calling the velocity of the first puck <math>v_1</math> and the velocity of the second puck <math>v_2</math>. 
<math>e=\frac{v_{1f}-v_{2f}}{v_{2i}-v_{1i}}</math> 
Solve for <math>v_{2f}</math> 

<math>v_{2f}=v_{1f}-e(v_{2i}-v_{1i})</math> 
Plug in the values stated in the problem. 
<math>v_{2f}=4-.55(5-8)</math>
<math>v_{2f}=2.35\frac{m}{s}</math>

===Difficult===
[[File:Cor1.jpg|right]]
Bouncing Ball Example:

Consider a ball dropped from initial height <math>h_0</math>. When it bounces it reaches a height <math>h_f</math>. Find the coefficient of restitution of this collision. 
<math>e=\frac{v_{floor_f}-v_{ball_f}}{v_{ball_i}-v_{floor_i}}</math> 
The floor's velocity is always 0. 
<math>e=\frac{v_{ball_f}}{v_{ball_i}}</math> 
Recall the law of conservation of energy: 
<math> \Delta KE = \Delta PE</math> 
For the ball from h0 to h=0
<math> \frac{1}{2}mv_{ball_i}^2=mgh_0</math> 
<math> \frac{1}{2}v_{ball_i}^2=gh_0</math> 
<math> v_{ball_i}=2\sqrt{gh_0} </math> 
The same concept can be applied to the second bounce. 
<math>v_{ball_f}=2\sqrt{gh_f}</math> 
Plugging this in to the coefficient of restitution expression we get: 
<math>e=\frac{2\sqrt{mgh_f}}{2\sqrt{mgh_0}}</math> 
<math>e=\frac{\sqrt{h_f}}{\sqrt{h_0}}</math> 

==Connectedness==
'''How is this topic connected to something that you are interested in?''' 
This topic has major connections to sports. As a former high school volleyball player, it is very interesting to think about the collisions that occurred in the game and how the coefficient of restitution is related to those collisions. 
'''How is it connected to your major (Mechanical Engineering)?''' 
The coefficient of restitution is a key concept in the study of Dynamics which is the foundation of many aspects of mechanical engineering. It also has to do with applications of mechanical engineering regarding materials. 

'''Is there an interesting industrial application?'''
In sports the coefficient of restitution is an important concept especially in sports like golf or tennis where collisions are a frequent and essential part of the game. In these sports sometimes limitations are put on how high the coefficient of restitution can be for certain clubs or bats. The maximum coefficient of restitution allowed in golf for clubs is .82. There have been studies done on the coefficients for different sports balls. These studies usually follow a procedure where different balls are dropped and left to bounce. Then data regarding their initial and final velocities or how high they bounced is recorded. The images below show some of the data from one of these experiments.

[[File:sportsgraph.png|right|This graph shows the initial and final velocities of many different balls from different sports.]]
[[File:sportschart.png|right|This chart shows the relationship between different coefficients of restitution and the initial and final velocities of sports balls.]]

== See also ==

Are there related topics or categories in this wiki resource for the curious reader to explore? How does this topic fit into that context?

===Further reading===

Books, Articles or other print media on this topic

===External links===

==References==

[[Category:Collisions]]

Coefficient of Restitution

2015-12-06T02:58:08Z

Mmoreno30: /* A Computational Model */

Claimed by Maria Moreno

The coefficient of restitution measures the amount of Kinetic Energy remains and how much energy is lost after a collision. It also shows the degree of elasticity of a collision. It is defined as the ratio of the difference in initial velocities divided by the difference in final velocities. The coefficient of restitution of a collision is usually between 0 and 1 except in rare cases when energy is released by a collision.
[[File:coe2.png|right]]

==The Main Idea==

The coefficient of restitution is a ratio that describes the degree of elasticity of a collision. It is used to solve problems dealing with collisions that are not perfectly elastic or inelastic.
The equation that describes the coefficient of restitution involved dividing the difference in the final velocities by the difference in the initial velocity.

Consider objects A and B with initial velocities vAi and vBi and final velocities vAf and vBf.
The coefficient of restitution, e is determined with the following formula:

<math> e = \frac{v_{Bf}-v_{Af}}{v_{Ai}-v_{Bi}} </math> 
The coefficient is a unitless form of measurement as it is the ratio of two velocities and the units cancel.

===A Mathematical Model===
Consider a perfectly elastic collision between objects A and B where vAi and vBi refer to the initial velocities of A and B and vAf and vBf refer to the final velocities of A and B.

Start with the conservation of momentum principle which states that 
<math>\vec{p_i} = \vec{p_f}</math> and 
<math>m_A\vec{v_{Ai}}+m_A\vec{v_{Bi}}=m_A\vec{v_{Af}}+m_A\vec{v_{Bf}}</math> (1) 

Remember that for a perfectly elastic collision kinetic energy is also conserved meaning: 
<math>KE_i = KE_f</math> and 
<math>\frac{1}{2}m_A v^2_{Ai}+\frac{1}{2}m_A v^2_Bi=\frac{1}{2}m_A v^2_{Af}+\frac{1}{2}m_A v^2_{Bf}</math> (2) 

Dividing (2) by (1) yields: 

<math>\vec{v_{Ai}}+\vec{v_{Bi}}=\vec{v_{Af}}+\vec{v_{Bf}}</math> 
<math>1=\frac{v_{Ai}+v_{Bi}}{v_{Af}+v_{Bf}}</math> 

<math>e=\frac{v_{Ai}+v_{Bi}}{v_{Af}+v_{Bf}}</math> 

For perfectly elastic collision e=1.

===Inelastic Collisions===

An inelastic collision is a type of collision where Kinetic Energy is not conserved. In this case the difference in initial velocities will not equal the difference in final velocities. This means that the coefficient of restitution will not equal 1.

Consider the same collision as previously described, except this collision is inelastic and Kinetic Energy is not conserved. Since some of the kinetic energy is dissipated: 
<math>\frac{1}{2}m_A v^2_{Ai}+\frac{1}{2}m_A v^2_Bi<\frac{1}{2}m_A v^2_{Af}+\frac{1}{2}m_A v^2_{Bf}</math> 
<math>\frac{v_{Ai}+v_{Bi}}{v_{Af}+v_{Bf}}<1</math> 

<math>e=\frac{v_{Ai}+v_{Bi}}{v_{Af}+v_{Bf}}<1</math> 

A perfectly inelastic collision (for example when two objects collide and then stick together) has a coefficient of restitution of 0.

===A Computational Model===
Here is a link to a glowscript simulation
http://www.glowscript.org/#/user/mmoreno181/folder/phys/program/cor 
Here is the code for the simulation: 
<code>
ball1=sphere(pos=vec(0,0,0), color=color.red, radius=.5)
ball2=sphere(pos=vec(10,0,0), color=color.blue, radius=.5)

#initialize velocity
v1i=vec(.02,0,0)
v2i=vec(-.1,0,0)
v1f=vec(-.07,0,0)

#set coefficient of resitution
e=.5

#find v2f
v2f=v1f-(e*(v2i-v1i))

#find the time that the balls collide
tmeet= (ball2.pos.x-ball1.pos.x)/(v1i.x-v2i.x)

t=0

#animate
while t<150:
rate(100)
if t < tmeet: #if it's before the collision
ball1.pos=ball1.pos+v1i
ball2.pos=ball2.pos+v2i
else:
ball1.pos=ball1.pos+v1f
ball2.pos=ball2.pos+v2f

t=t+1
</code>

==Examples==
===Simple===
A ball is traveling at <math>4 \frac{m}{s}</math> and collides with a ball. After the collision the ball travels at <math>2.4\frac{m}{s}</math> in the opposite direction. Find the coefficient of restitution of the collision. 
Start with the definition of the coefficient of restitution: 
<math>e=\frac{v_{wall_f}-v_{ball_f}}{v_{ball_i}-v_{wall_i}}</math> 
Then plug in values and solve: 
<math>e=\frac{0-v_{ball_f}}{v_{ball_i}-0}</math> 
<math>e=\frac{-(-2.4)}{4}</math> 
<math>e=.6</math>

===Middling===
A hockey puck travels at a constant velocity <math>5\frac{m}{s}</math> and is hit by another puck with a velocity of <math>8\frac{m}{s}</math>. After the collision, it travels in the opposite direction at 4m/s. This collision has a coefficient of restitution of .55. Find the velocity of the second puck after the collision. 
Start with the definition of the coefficient of restitution. We are calling the velocity of the first puck <math>v_1</math> and the velocity of the second puck <math>v_2</math>. 
<math>e=\frac{v_{1f}-v_{2f}}{v_{2i}-v_{1i}}</math> 
Solve for <math>v_{2f}</math> 

<math>v_{2f}=v_{1f}-e(v_{2i}-v_{1i})</math> 
Plug in the values stated in the problem. 
<math>v_{2f}=4-.55(5-8)</math>
<math>v_{2f}=2.35\frac{m}{s}</math>

===Difficult===
[[File:Cor1.jpg|right]]
Bouncing Ball Example:

Consider a ball dropped from initial height <math>h_0</math>. When it bounces it reaches a height <math>h_f</math>. Find the coefficient of restitution of this collision. 
<math>e=\frac{v_{floor_f}-v_{ball_f}}{v_{ball_i}-v_{floor_i}}</math> 
The floor's velocity is always 0. 
<math>e=\frac{v_{ball_f}}{v_{ball_i}}</math> 
Recall the law of conservation of energy: 
<math> \Delta KE = \Delta PE</math> 
For the ball from h0 to h=0
<math> \frac{1}{2}mv_{ball_i}^2=mgh_0</math> 
<math> \frac{1}{2}v_{ball_i}^2=gh_0</math> 
<math> v_{ball_i}=2\sqrt{gh_0} </math> 
The same concept can be applied to the second bounce. 
<math>v_{ball_f}=2\sqrt{gh_f}</math> 
Plugging this in to the coefficient of restitution expression we get: 
<math>e=\frac{2\sqrt{mgh_f}}{2\sqrt{mgh_0}}</math> 
<math>e=\frac{\sqrt{h_f}}{\sqrt{h_0}}</math> 

==Connectedness==
'''How is this topic connected to something that you are interested in?''' 
This topic has major connections to sports. As a former high school volleyball player, it is very interesting to think about the collisions that occurred in the game and how the coefficient of restitution is related to those collisions. 
'''How is it connected to your major (Mechanical Engineering)?''' 
The coefficient of restitution is a key concept in the study of Dynamics which is the foundation of many aspects of mechanical engineering. It also has to do with applications of mechanical engineering regarding materials. 

'''Is there an interesting industrial application?'''
In sports the coefficient of restitution is an important concept especially in sports like golf or tennis where collisions are a frequent and essential part of the game. In these sports sometimes limitations are put on how high the coefficient of restitution can be for certain clubs or bats. The maximum coefficient of restitution allowed in golf for clubs is .82. There have been studies done on the coefficients for different sports balls. These studies usually follow a procedure where different balls are dropped and left to bounce. Then data regarding their initial and final velocities or how high they bounced is recorded. The images below show some of the data from one of these experiments.

[[File:sportsgraph.png|This graph shows the initial and final velocities of many different balls from different sports.]]
[[File:sportschart.png|This chart shows the relationship between different coefficients of restitution and the initial and final velocities of sports balls.]]

== See also ==

Are there related topics or categories in this wiki resource for the curious reader to explore? How does this topic fit into that context?

===Further reading===

Books, Articles or other print media on this topic

===External links===

==References==

[[Category:Collisions]]

Coefficient of Restitution

2015-12-06T02:56:07Z

Mmoreno30: /* A Computational Model */

Claimed by Maria Moreno

The coefficient of restitution measures the amount of Kinetic Energy remains and how much energy is lost after a collision. It also shows the degree of elasticity of a collision. It is defined as the ratio of the difference in initial velocities divided by the difference in final velocities. The coefficient of restitution of a collision is usually between 0 and 1 except in rare cases when energy is released by a collision.
[[File:coe2.png|right]]

==The Main Idea==

The coefficient of restitution is a ratio that describes the degree of elasticity of a collision. It is used to solve problems dealing with collisions that are not perfectly elastic or inelastic.
The equation that describes the coefficient of restitution involved dividing the difference in the final velocities by the difference in the initial velocity.

Consider objects A and B with initial velocities vAi and vBi and final velocities vAf and vBf.
The coefficient of restitution, e is determined with the following formula:

<math> e = \frac{v_{Bf}-v_{Af}}{v_{Ai}-v_{Bi}} </math> 
The coefficient is a unitless form of measurement as it is the ratio of two velocities and the units cancel.

===A Mathematical Model===
Consider a perfectly elastic collision between objects A and B where vAi and vBi refer to the initial velocities of A and B and vAf and vBf refer to the final velocities of A and B.

Start with the conservation of momentum principle which states that 
<math>\vec{p_i} = \vec{p_f}</math> and 
<math>m_A\vec{v_{Ai}}+m_A\vec{v_{Bi}}=m_A\vec{v_{Af}}+m_A\vec{v_{Bf}}</math> (1) 

Remember that for a perfectly elastic collision kinetic energy is also conserved meaning: 
<math>KE_i = KE_f</math> and 
<math>\frac{1}{2}m_A v^2_{Ai}+\frac{1}{2}m_A v^2_Bi=\frac{1}{2}m_A v^2_{Af}+\frac{1}{2}m_A v^2_{Bf}</math> (2) 

Dividing (2) by (1) yields: 

<math>\vec{v_{Ai}}+\vec{v_{Bi}}=\vec{v_{Af}}+\vec{v_{Bf}}</math> 
<math>1=\frac{v_{Ai}+v_{Bi}}{v_{Af}+v_{Bf}}</math> 

<math>e=\frac{v_{Ai}+v_{Bi}}{v_{Af}+v_{Bf}}</math> 

For perfectly elastic collision e=1.

===Inelastic Collisions===

An inelastic collision is a type of collision where Kinetic Energy is not conserved. In this case the difference in initial velocities will not equal the difference in final velocities. This means that the coefficient of restitution will not equal 1.

Consider the same collision as previously described, except this collision is inelastic and Kinetic Energy is not conserved. Since some of the kinetic energy is dissipated: 
<math>\frac{1}{2}m_A v^2_{Ai}+\frac{1}{2}m_A v^2_Bi<\frac{1}{2}m_A v^2_{Af}+\frac{1}{2}m_A v^2_{Bf}</math> 
<math>\frac{v_{Ai}+v_{Bi}}{v_{Af}+v_{Bf}}<1</math> 

<math>e=\frac{v_{Ai}+v_{Bi}}{v_{Af}+v_{Bf}}<1</math> 

A perfectly inelastic collision (for example when two objects collide and then stick together) has a coefficient of restitution of 0.

===A Computational Model===
Here is a link to a glowscript simulation
[http://www.glowscript.org/#/user/mmoreno181/folder/phys/program/cor] 

ball1=sphere(pos=vec(0,0,0), color=color.red, radius=.5)
ball2=sphere(pos=vec(10,0,0), color=color.blue, radius=.5)

#initialize velocity
v1i=vec(.02,0,0)
v2i=vec(-.1,0,0)
v1f=vec(-.07,0,0)

#set coefficient of resitution
e=.5

#find v2f
v2f=v1f-(e*(v2i-v1i))

#find the time that the balls collide
tmeet= (ball2.pos.x-ball1.pos.x)/(v1i.x-v2i.x)

t=0

#animate
while t<150:
rate(100)
if t < tmeet: #if it's before the collision
ball1.pos=ball1.pos+v1i
ball2.pos=ball2.pos+v2i
else:
ball1.pos=ball1.pos+v1f
ball2.pos=ball2.pos+v2f

t=t+1

==Examples==
===Simple===
A ball is traveling at <math>4 \frac{m}{s}</math> and collides with a ball. After the collision the ball travels at <math>2.4\frac{m}{s}</math> in the opposite direction. Find the coefficient of restitution of the collision. 
Start with the definition of the coefficient of restitution: 
<math>e=\frac{v_{wall_f}-v_{ball_f}}{v_{ball_i}-v_{wall_i}}</math> 
Then plug in values and solve: 
<math>e=\frac{0-v_{ball_f}}{v_{ball_i}-0}</math> 
<math>e=\frac{-(-2.4)}{4}</math> 
<math>e=.6</math>

===Middling===
A hockey puck travels at a constant velocity <math>5\frac{m}{s}</math> and is hit by another puck with a velocity of <math>8\frac{m}{s}</math>. After the collision, it travels in the opposite direction at 4m/s. This collision has a coefficient of restitution of .55. Find the velocity of the second puck after the collision. 
Start with the definition of the coefficient of restitution. We are calling the velocity of the first puck <math>v_1</math> and the velocity of the second puck <math>v_2</math>. 
<math>e=\frac{v_{1f}-v_{2f}}{v_{2i}-v_{1i}}</math> 
Solve for <math>v_{2f}</math> 

<math>v_{2f}=v_{1f}-e(v_{2i}-v_{1i})</math> 
Plug in the values stated in the problem. 
<math>v_{2f}=4-.55(5-8)</math>
<math>v_{2f}=2.35\frac{m}{s}</math>

===Difficult===
[[File:Cor1.jpg|right]]
Bouncing Ball Example:

Consider a ball dropped from initial height <math>h_0</math>. When it bounces it reaches a height <math>h_f</math>. Find the coefficient of restitution of this collision. 
<math>e=\frac{v_{floor_f}-v_{ball_f}}{v_{ball_i}-v_{floor_i}}</math> 
The floor's velocity is always 0. 
<math>e=\frac{v_{ball_f}}{v_{ball_i}}</math> 
Recall the law of conservation of energy: 
<math> \Delta KE = \Delta PE</math> 
For the ball from h0 to h=0
<math> \frac{1}{2}mv_{ball_i}^2=mgh_0</math> 
<math> \frac{1}{2}v_{ball_i}^2=gh_0</math> 
<math> v_{ball_i}=2\sqrt{gh_0} </math> 
The same concept can be applied to the second bounce. 
<math>v_{ball_f}=2\sqrt{gh_f}</math> 
Plugging this in to the coefficient of restitution expression we get: 
<math>e=\frac{2\sqrt{mgh_f}}{2\sqrt{mgh_0}}</math> 
<math>e=\frac{\sqrt{h_f}}{\sqrt{h_0}}</math> 

==Connectedness==
'''How is this topic connected to something that you are interested in?''' 
This topic has major connections to sports. As a former high school volleyball player, it is very interesting to think about the collisions that occurred in the game and how the coefficient of restitution is related to those collisions. 
'''How is it connected to your major (Mechanical Engineering)?''' 
The coefficient of restitution is a key concept in the study of Dynamics which is the foundation of many aspects of mechanical engineering. It also has to do with applications of mechanical engineering regarding materials. 

'''Is there an interesting industrial application?'''
In sports the coefficient of restitution is an important concept especially in sports like golf or tennis where collisions are a frequent and essential part of the game. In these sports sometimes limitations are put on how high the coefficient of restitution can be for certain clubs or bats. The maximum coefficient of restitution allowed in golf for clubs is .82. There have been studies done on the coefficients for different sports balls. These studies usually follow a procedure where different balls are dropped and left to bounce. Then data regarding their initial and final velocities or how high they bounced is recorded. The images below show some of the data from one of these experiments.

[[File:sportsgraph.png|This graph shows the initial and final velocities of many different balls from different sports.]]
[[File:sportschart.png|This chart shows the relationship between different coefficients of restitution and the initial and final velocities of sports balls.]]

== See also ==

Are there related topics or categories in this wiki resource for the curious reader to explore? How does this topic fit into that context?

===Further reading===

Books, Articles or other print media on this topic

===External links===

==References==

[[Category:Collisions]]

Coefficient of Restitution

2015-12-06T02:55:39Z

Mmoreno30: /* References */

Claimed by Maria Moreno

The coefficient of restitution measures the amount of Kinetic Energy remains and how much energy is lost after a collision. It also shows the degree of elasticity of a collision. It is defined as the ratio of the difference in initial velocities divided by the difference in final velocities. The coefficient of restitution of a collision is usually between 0 and 1 except in rare cases when energy is released by a collision.
[[File:coe2.png|right]]

==The Main Idea==

The coefficient of restitution is a ratio that describes the degree of elasticity of a collision. It is used to solve problems dealing with collisions that are not perfectly elastic or inelastic.
The equation that describes the coefficient of restitution involved dividing the difference in the final velocities by the difference in the initial velocity.

Consider objects A and B with initial velocities vAi and vBi and final velocities vAf and vBf.
The coefficient of restitution, e is determined with the following formula:

<math> e = \frac{v_{Bf}-v_{Af}}{v_{Ai}-v_{Bi}} </math> 
The coefficient is a unitless form of measurement as it is the ratio of two velocities and the units cancel.

===A Mathematical Model===
Consider a perfectly elastic collision between objects A and B where vAi and vBi refer to the initial velocities of A and B and vAf and vBf refer to the final velocities of A and B.

Start with the conservation of momentum principle which states that 
<math>\vec{p_i} = \vec{p_f}</math> and 
<math>m_A\vec{v_{Ai}}+m_A\vec{v_{Bi}}=m_A\vec{v_{Af}}+m_A\vec{v_{Bf}}</math> (1) 

Remember that for a perfectly elastic collision kinetic energy is also conserved meaning: 
<math>KE_i = KE_f</math> and 
<math>\frac{1}{2}m_A v^2_{Ai}+\frac{1}{2}m_A v^2_Bi=\frac{1}{2}m_A v^2_{Af}+\frac{1}{2}m_A v^2_{Bf}</math> (2) 

Dividing (2) by (1) yields: 

<math>\vec{v_{Ai}}+\vec{v_{Bi}}=\vec{v_{Af}}+\vec{v_{Bf}}</math> 
<math>1=\frac{v_{Ai}+v_{Bi}}{v_{Af}+v_{Bf}}</math> 

<math>e=\frac{v_{Ai}+v_{Bi}}{v_{Af}+v_{Bf}}</math> 

For perfectly elastic collision e=1.

===Inelastic Collisions===

An inelastic collision is a type of collision where Kinetic Energy is not conserved. In this case the difference in initial velocities will not equal the difference in final velocities. This means that the coefficient of restitution will not equal 1.

Consider the same collision as previously described, except this collision is inelastic and Kinetic Energy is not conserved. Since some of the kinetic energy is dissipated: 
<math>\frac{1}{2}m_A v^2_{Ai}+\frac{1}{2}m_A v^2_Bi<\frac{1}{2}m_A v^2_{Af}+\frac{1}{2}m_A v^2_{Bf}</math> 
<math>\frac{v_{Ai}+v_{Bi}}{v_{Af}+v_{Bf}}<1</math> 

<math>e=\frac{v_{Ai}+v_{Bi}}{v_{Af}+v_{Bf}}<1</math> 

A perfectly inelastic collision (for example when two objects collide and then stick together) has a coefficient of restitution of 0.

===A Computational Model===

==Examples==
===Simple===
A ball is traveling at <math>4 \frac{m}{s}</math> and collides with a ball. After the collision the ball travels at <math>2.4\frac{m}{s}</math> in the opposite direction. Find the coefficient of restitution of the collision. 
Start with the definition of the coefficient of restitution: 
<math>e=\frac{v_{wall_f}-v_{ball_f}}{v_{ball_i}-v_{wall_i}}</math> 
Then plug in values and solve: 
<math>e=\frac{0-v_{ball_f}}{v_{ball_i}-0}</math> 
<math>e=\frac{-(-2.4)}{4}</math> 
<math>e=.6</math>

===Middling===
A hockey puck travels at a constant velocity <math>5\frac{m}{s}</math> and is hit by another puck with a velocity of <math>8\frac{m}{s}</math>. After the collision, it travels in the opposite direction at 4m/s. This collision has a coefficient of restitution of .55. Find the velocity of the second puck after the collision. 
Start with the definition of the coefficient of restitution. We are calling the velocity of the first puck <math>v_1</math> and the velocity of the second puck <math>v_2</math>. 
<math>e=\frac{v_{1f}-v_{2f}}{v_{2i}-v_{1i}}</math> 
Solve for <math>v_{2f}</math> 

<math>v_{2f}=v_{1f}-e(v_{2i}-v_{1i})</math> 
Plug in the values stated in the problem. 
<math>v_{2f}=4-.55(5-8)</math>
<math>v_{2f}=2.35\frac{m}{s}</math>

===Difficult===
[[File:Cor1.jpg|right]]
Bouncing Ball Example:

Consider a ball dropped from initial height <math>h_0</math>. When it bounces it reaches a height <math>h_f</math>. Find the coefficient of restitution of this collision. 
<math>e=\frac{v_{floor_f}-v_{ball_f}}{v_{ball_i}-v_{floor_i}}</math> 
The floor's velocity is always 0. 
<math>e=\frac{v_{ball_f}}{v_{ball_i}}</math> 
Recall the law of conservation of energy: 
<math> \Delta KE = \Delta PE</math> 
For the ball from h0 to h=0
<math> \frac{1}{2}mv_{ball_i}^2=mgh_0</math> 
<math> \frac{1}{2}v_{ball_i}^2=gh_0</math> 
<math> v_{ball_i}=2\sqrt{gh_0} </math> 
The same concept can be applied to the second bounce. 
<math>v_{ball_f}=2\sqrt{gh_f}</math> 
Plugging this in to the coefficient of restitution expression we get: 
<math>e=\frac{2\sqrt{mgh_f}}{2\sqrt{mgh_0}}</math> 
<math>e=\frac{\sqrt{h_f}}{\sqrt{h_0}}</math> 

==Connectedness==
'''How is this topic connected to something that you are interested in?''' 
This topic has major connections to sports. As a former high school volleyball player, it is very interesting to think about the collisions that occurred in the game and how the coefficient of restitution is related to those collisions. 
'''How is it connected to your major (Mechanical Engineering)?''' 
The coefficient of restitution is a key concept in the study of Dynamics which is the foundation of many aspects of mechanical engineering. It also has to do with applications of mechanical engineering regarding materials. 

'''Is there an interesting industrial application?'''
In sports the coefficient of restitution is an important concept especially in sports like golf or tennis where collisions are a frequent and essential part of the game. In these sports sometimes limitations are put on how high the coefficient of restitution can be for certain clubs or bats. The maximum coefficient of restitution allowed in golf for clubs is .82. There have been studies done on the coefficients for different sports balls. These studies usually follow a procedure where different balls are dropped and left to bounce. Then data regarding their initial and final velocities or how high they bounced is recorded. The images below show some of the data from one of these experiments.

[[File:sportsgraph.png|This graph shows the initial and final velocities of many different balls from different sports.]]
[[File:sportschart.png|This chart shows the relationship between different coefficients of restitution and the initial and final velocities of sports balls.]]

== See also ==

Are there related topics or categories in this wiki resource for the curious reader to explore? How does this topic fit into that context?

===Further reading===

Books, Articles or other print media on this topic

===External links===

==References==

[[Category:Collisions]]

Coefficient of Restitution

2015-12-06T02:52:15Z

Mmoreno30: /* References */

Claimed by Maria Moreno

The coefficient of restitution measures the amount of Kinetic Energy remains and how much energy is lost after a collision. It also shows the degree of elasticity of a collision. It is defined as the ratio of the difference in initial velocities divided by the difference in final velocities. The coefficient of restitution of a collision is usually between 0 and 1 except in rare cases when energy is released by a collision.
[[File:coe2.png|right]]

==The Main Idea==

The coefficient of restitution is a ratio that describes the degree of elasticity of a collision. It is used to solve problems dealing with collisions that are not perfectly elastic or inelastic.
The equation that describes the coefficient of restitution involved dividing the difference in the final velocities by the difference in the initial velocity.

Consider objects A and B with initial velocities vAi and vBi and final velocities vAf and vBf.
The coefficient of restitution, e is determined with the following formula:

<math> e = \frac{v_{Bf}-v_{Af}}{v_{Ai}-v_{Bi}} </math> 
The coefficient is a unitless form of measurement as it is the ratio of two velocities and the units cancel.

===A Mathematical Model===
Consider a perfectly elastic collision between objects A and B where vAi and vBi refer to the initial velocities of A and B and vAf and vBf refer to the final velocities of A and B.

Start with the conservation of momentum principle which states that 
<math>\vec{p_i} = \vec{p_f}</math> and 
<math>m_A\vec{v_{Ai}}+m_A\vec{v_{Bi}}=m_A\vec{v_{Af}}+m_A\vec{v_{Bf}}</math> (1) 

Remember that for a perfectly elastic collision kinetic energy is also conserved meaning: 
<math>KE_i = KE_f</math> and 
<math>\frac{1}{2}m_A v^2_{Ai}+\frac{1}{2}m_A v^2_Bi=\frac{1}{2}m_A v^2_{Af}+\frac{1}{2}m_A v^2_{Bf}</math> (2) 

Dividing (2) by (1) yields: 

<math>\vec{v_{Ai}}+\vec{v_{Bi}}=\vec{v_{Af}}+\vec{v_{Bf}}</math> 
<math>1=\frac{v_{Ai}+v_{Bi}}{v_{Af}+v_{Bf}}</math> 

<math>e=\frac{v_{Ai}+v_{Bi}}{v_{Af}+v_{Bf}}</math> 

For perfectly elastic collision e=1.

===Inelastic Collisions===

An inelastic collision is a type of collision where Kinetic Energy is not conserved. In this case the difference in initial velocities will not equal the difference in final velocities. This means that the coefficient of restitution will not equal 1.

Consider the same collision as previously described, except this collision is inelastic and Kinetic Energy is not conserved. Since some of the kinetic energy is dissipated: 
<math>\frac{1}{2}m_A v^2_{Ai}+\frac{1}{2}m_A v^2_Bi<\frac{1}{2}m_A v^2_{Af}+\frac{1}{2}m_A v^2_{Bf}</math> 
<math>\frac{v_{Ai}+v_{Bi}}{v_{Af}+v_{Bf}}<1</math> 

<math>e=\frac{v_{Ai}+v_{Bi}}{v_{Af}+v_{Bf}}<1</math> 

A perfectly inelastic collision (for example when two objects collide and then stick together) has a coefficient of restitution of 0.

===A Computational Model===

==Examples==
===Simple===
A ball is traveling at <math>4 \frac{m}{s}</math> and collides with a ball. After the collision the ball travels at <math>2.4\frac{m}{s}</math> in the opposite direction. Find the coefficient of restitution of the collision. 
Start with the definition of the coefficient of restitution: 
<math>e=\frac{v_{wall_f}-v_{ball_f}}{v_{ball_i}-v_{wall_i}}</math> 
Then plug in values and solve: 
<math>e=\frac{0-v_{ball_f}}{v_{ball_i}-0}</math> 
<math>e=\frac{-(-2.4)}{4}</math> 
<math>e=.6</math>

===Middling===
A hockey puck travels at a constant velocity <math>5\frac{m}{s}</math> and is hit by another puck with a velocity of <math>8\frac{m}{s}</math>. After the collision, it travels in the opposite direction at 4m/s. This collision has a coefficient of restitution of .55. Find the velocity of the second puck after the collision. 
Start with the definition of the coefficient of restitution. We are calling the velocity of the first puck <math>v_1</math> and the velocity of the second puck <math>v_2</math>. 
<math>e=\frac{v_{1f}-v_{2f}}{v_{2i}-v_{1i}}</math> 
Solve for <math>v_{2f}</math> 

<math>v_{2f}=v_{1f}-e(v_{2i}-v_{1i})</math> 
Plug in the values stated in the problem. 
<math>v_{2f}=4-.55(5-8)</math>
<math>v_{2f}=2.35\frac{m}{s}</math>

===Difficult===
[[File:Cor1.jpg|right]]
Bouncing Ball Example:

Consider a ball dropped from initial height <math>h_0</math>. When it bounces it reaches a height <math>h_f</math>. Find the coefficient of restitution of this collision. 
<math>e=\frac{v_{floor_f}-v_{ball_f}}{v_{ball_i}-v_{floor_i}}</math> 
The floor's velocity is always 0. 
<math>e=\frac{v_{ball_f}}{v_{ball_i}}</math> 
Recall the law of conservation of energy: 
<math> \Delta KE = \Delta PE</math> 
For the ball from h0 to h=0
<math> \frac{1}{2}mv_{ball_i}^2=mgh_0</math> 
<math> \frac{1}{2}v_{ball_i}^2=gh_0</math> 
<math> v_{ball_i}=2\sqrt{gh_0} </math> 
The same concept can be applied to the second bounce. 
<math>v_{ball_f}=2\sqrt{gh_f}</math> 
Plugging this in to the coefficient of restitution expression we get: 
<math>e=\frac{2\sqrt{mgh_f}}{2\sqrt{mgh_0}}</math> 
<math>e=\frac{\sqrt{h_f}}{\sqrt{h_0}}</math> 

==Connectedness==
'''How is this topic connected to something that you are interested in?''' 
This topic has major connections to sports. As a former high school volleyball player, it is very interesting to think about the collisions that occurred in the game and how the coefficient of restitution is related to those collisions. 
'''How is it connected to your major (Mechanical Engineering)?''' 
The coefficient of restitution is a key concept in the study of Dynamics which is the foundation of many aspects of mechanical engineering. It also has to do with applications of mechanical engineering regarding materials. 

'''Is there an interesting industrial application?'''
In sports the coefficient of restitution is an important concept especially in sports like golf or tennis where collisions are a frequent and essential part of the game. In these sports sometimes limitations are put on how high the coefficient of restitution can be for certain clubs or bats. The maximum coefficient of restitution allowed in golf for clubs is .82. There have been studies done on the coefficients for different sports balls. These studies usually follow a procedure where different balls are dropped and left to bounce. Then data regarding their initial and final velocities or how high they bounced is recorded. The images below show some of the data from one of these experiments.

[[File:sportsgraph.png|This graph shows the initial and final velocities of many different balls from different sports.]]
[[File:sportschart.png|This chart shows the relationship between different coefficients of restitution and the initial and final velocities of sports balls.]]

== See also ==

Are there related topics or categories in this wiki resource for the curious reader to explore? How does this topic fit into that context?

===Further reading===

Books, Articles or other print media on this topic

===External links===

==References==

Here is a link to a glowscript simulation
[http://www.glowscript.org/#/user/mmoreno181/folder/phys/program/cor]

<blockquote>ball1=sphere(pos=vec(0,0,0), color=color.red, radius=.5)
ball2=sphere(pos=vec(10,0,0), color=color.blue, radius=.5)

#initialize velocity
v1i=vec(.02,0,0)
v2i=vec(-.1,0,0)
v1f=vec(-.07,0,0)

#set coefficient of resitution
e=.5

#find v2f
v2f=v1f-(e*(v2i-v1i))

#find the time that the balls collide
tmeet= (ball2.pos.x-ball1.pos.x)/(v1i.x-v2i.x)

t=0

#animate
while t<150:
rate(100)
if t < tmeet: #if it's before the collision
ball1.pos=ball1.pos+v1i
ball2.pos=ball2.pos+v2i
else:
ball1.pos=ball1.pos+v1f
ball2.pos=ball2.pos+v2f

t=t+1</blockquote>

[[Category:Collisions]]

Coefficient of Restitution

2015-12-06T02:48:50Z

Mmoreno30: /* References */

Claimed by Maria Moreno

The coefficient of restitution measures the amount of Kinetic Energy remains and how much energy is lost after a collision. It also shows the degree of elasticity of a collision. It is defined as the ratio of the difference in initial velocities divided by the difference in final velocities. The coefficient of restitution of a collision is usually between 0 and 1 except in rare cases when energy is released by a collision.
[[File:coe2.png|right]]

==The Main Idea==

The coefficient of restitution is a ratio that describes the degree of elasticity of a collision. It is used to solve problems dealing with collisions that are not perfectly elastic or inelastic.
The equation that describes the coefficient of restitution involved dividing the difference in the final velocities by the difference in the initial velocity.

Consider objects A and B with initial velocities vAi and vBi and final velocities vAf and vBf.
The coefficient of restitution, e is determined with the following formula:

<math> e = \frac{v_{Bf}-v_{Af}}{v_{Ai}-v_{Bi}} </math> 
The coefficient is a unitless form of measurement as it is the ratio of two velocities and the units cancel.

===A Mathematical Model===
Consider a perfectly elastic collision between objects A and B where vAi and vBi refer to the initial velocities of A and B and vAf and vBf refer to the final velocities of A and B.

Start with the conservation of momentum principle which states that 
<math>\vec{p_i} = \vec{p_f}</math> and 
<math>m_A\vec{v_{Ai}}+m_A\vec{v_{Bi}}=m_A\vec{v_{Af}}+m_A\vec{v_{Bf}}</math> (1) 

Remember that for a perfectly elastic collision kinetic energy is also conserved meaning: 
<math>KE_i = KE_f</math> and 
<math>\frac{1}{2}m_A v^2_{Ai}+\frac{1}{2}m_A v^2_Bi=\frac{1}{2}m_A v^2_{Af}+\frac{1}{2}m_A v^2_{Bf}</math> (2) 

Dividing (2) by (1) yields: 

<math>\vec{v_{Ai}}+\vec{v_{Bi}}=\vec{v_{Af}}+\vec{v_{Bf}}</math> 
<math>1=\frac{v_{Ai}+v_{Bi}}{v_{Af}+v_{Bf}}</math> 

<math>e=\frac{v_{Ai}+v_{Bi}}{v_{Af}+v_{Bf}}</math> 

For perfectly elastic collision e=1.

===Inelastic Collisions===

An inelastic collision is a type of collision where Kinetic Energy is not conserved. In this case the difference in initial velocities will not equal the difference in final velocities. This means that the coefficient of restitution will not equal 1.

Consider the same collision as previously described, except this collision is inelastic and Kinetic Energy is not conserved. Since some of the kinetic energy is dissipated: 
<math>\frac{1}{2}m_A v^2_{Ai}+\frac{1}{2}m_A v^2_Bi<\frac{1}{2}m_A v^2_{Af}+\frac{1}{2}m_A v^2_{Bf}</math> 
<math>\frac{v_{Ai}+v_{Bi}}{v_{Af}+v_{Bf}}<1</math> 

<math>e=\frac{v_{Ai}+v_{Bi}}{v_{Af}+v_{Bf}}<1</math> 

A perfectly inelastic collision (for example when two objects collide and then stick together) has a coefficient of restitution of 0.

===A Computational Model===

==Examples==
===Simple===
A ball is traveling at <math>4 \frac{m}{s}</math> and collides with a ball. After the collision the ball travels at <math>2.4\frac{m}{s}</math> in the opposite direction. Find the coefficient of restitution of the collision. 
Start with the definition of the coefficient of restitution: 
<math>e=\frac{v_{wall_f}-v_{ball_f}}{v_{ball_i}-v_{wall_i}}</math> 
Then plug in values and solve: 
<math>e=\frac{0-v_{ball_f}}{v_{ball_i}-0}</math> 
<math>e=\frac{-(-2.4)}{4}</math> 
<math>e=.6</math>

===Middling===
A hockey puck travels at a constant velocity <math>5\frac{m}{s}</math> and is hit by another puck with a velocity of <math>8\frac{m}{s}</math>. After the collision, it travels in the opposite direction at 4m/s. This collision has a coefficient of restitution of .55. Find the velocity of the second puck after the collision. 
Start with the definition of the coefficient of restitution. We are calling the velocity of the first puck <math>v_1</math> and the velocity of the second puck <math>v_2</math>. 
<math>e=\frac{v_{1f}-v_{2f}}{v_{2i}-v_{1i}}</math> 
Solve for <math>v_{2f}</math> 

<math>v_{2f}=v_{1f}-e(v_{2i}-v_{1i})</math> 
Plug in the values stated in the problem. 
<math>v_{2f}=4-.55(5-8)</math>
<math>v_{2f}=2.35\frac{m}{s}</math>

===Difficult===
[[File:Cor1.jpg|right]]
Bouncing Ball Example:

Consider a ball dropped from initial height <math>h_0</math>. When it bounces it reaches a height <math>h_f</math>. Find the coefficient of restitution of this collision. 
<math>e=\frac{v_{floor_f}-v_{ball_f}}{v_{ball_i}-v_{floor_i}}</math> 
The floor's velocity is always 0. 
<math>e=\frac{v_{ball_f}}{v_{ball_i}}</math> 
Recall the law of conservation of energy: 
<math> \Delta KE = \Delta PE</math> 
For the ball from h0 to h=0
<math> \frac{1}{2}mv_{ball_i}^2=mgh_0</math> 
<math> \frac{1}{2}v_{ball_i}^2=gh_0</math> 
<math> v_{ball_i}=2\sqrt{gh_0} </math> 
The same concept can be applied to the second bounce. 
<math>v_{ball_f}=2\sqrt{gh_f}</math> 
Plugging this in to the coefficient of restitution expression we get: 
<math>e=\frac{2\sqrt{mgh_f}}{2\sqrt{mgh_0}}</math> 
<math>e=\frac{\sqrt{h_f}}{\sqrt{h_0}}</math> 

==Connectedness==
'''How is this topic connected to something that you are interested in?''' 
This topic has major connections to sports. As a former high school volleyball player, it is very interesting to think about the collisions that occurred in the game and how the coefficient of restitution is related to those collisions. 
'''How is it connected to your major (Mechanical Engineering)?''' 
The coefficient of restitution is a key concept in the study of Dynamics which is the foundation of many aspects of mechanical engineering. It also has to do with applications of mechanical engineering regarding materials. 

'''Is there an interesting industrial application?'''
In sports the coefficient of restitution is an important concept especially in sports like golf or tennis where collisions are a frequent and essential part of the game. In these sports sometimes limitations are put on how high the coefficient of restitution can be for certain clubs or bats. The maximum coefficient of restitution allowed in golf for clubs is .82. There have been studies done on the coefficients for different sports balls. These studies usually follow a procedure where different balls are dropped and left to bounce. Then data regarding their initial and final velocities or how high they bounced is recorded. The images below show some of the data from one of these experiments.

[[File:sportsgraph.png|This graph shows the initial and final velocities of many different balls from different sports.]]
[[File:sportschart.png|This chart shows the relationship between different coefficients of restitution and the initial and final velocities of sports balls.]]

== See also ==

Are there related topics or categories in this wiki resource for the curious reader to explore? How does this topic fit into that context?

===Further reading===

Books, Articles or other print media on this topic

===External links===

==References==

Here is a link to a glowscript simulation
[http://www.glowscript.org/#/user/mmoreno181/folder/phys/program/cor]

<nowiki>ball1=sphere(pos=vec(0,0,0), color=color.red, radius=.5)
ball2=sphere(pos=vec(10,0,0), color=color.blue, radius=.5)

#initialize velocity
v1i=vec(.02,0,0)
v2i=vec(-.1,0,0)
v1f=vec(-.07,0,0)

#set coefficient of resitution
e=.5

#find v2f
v2f=v1f-(e*(v2i-v1i))

#find the time that the balls collide
tmeet= (ball2.pos.x-ball1.pos.x)/(v1i.x-v2i.x)

t=0

#animate
while t<150:
rate(100)
if t < tmeet: #if it's before the collision
ball1.pos=ball1.pos+v1i
ball2.pos=ball2.pos+v2i
else:
ball1.pos=ball1.pos+v1f
ball2.pos=ball2.pos+v2f

t=t+1</nowiki>

[[Category:Collisions]]

Coefficient of Restitution

2015-12-06T02:20:02Z

Mmoreno30: /* References */

Claimed by Maria Moreno

The coefficient of restitution measures the amount of Kinetic Energy remains and how much energy is lost after a collision. It also shows the degree of elasticity of a collision. It is defined as the ratio of the difference in initial velocities divided by the difference in final velocities. The coefficient of restitution of a collision is usually between 0 and 1 except in rare cases when energy is released by a collision.
[[File:coe2.png|right]]

==The Main Idea==

The coefficient of restitution is a ratio that describes the degree of elasticity of a collision. It is used to solve problems dealing with collisions that are not perfectly elastic or inelastic.
The equation that describes the coefficient of restitution involved dividing the difference in the final velocities by the difference in the initial velocity.

Consider objects A and B with initial velocities vAi and vBi and final velocities vAf and vBf.
The coefficient of restitution, e is determined with the following formula:

<math> e = \frac{v_{Bf}-v_{Af}}{v_{Ai}-v_{Bi}} </math> 
The coefficient is a unitless form of measurement as it is the ratio of two velocities and the units cancel.

===A Mathematical Model===
Consider a perfectly elastic collision between objects A and B where vAi and vBi refer to the initial velocities of A and B and vAf and vBf refer to the final velocities of A and B.

Start with the conservation of momentum principle which states that 
<math>\vec{p_i} = \vec{p_f}</math> and 
<math>m_A\vec{v_{Ai}}+m_A\vec{v_{Bi}}=m_A\vec{v_{Af}}+m_A\vec{v_{Bf}}</math> (1) 

Remember that for a perfectly elastic collision kinetic energy is also conserved meaning: 
<math>KE_i = KE_f</math> and 
<math>\frac{1}{2}m_A v^2_{Ai}+\frac{1}{2}m_A v^2_Bi=\frac{1}{2}m_A v^2_{Af}+\frac{1}{2}m_A v^2_{Bf}</math> (2) 

Dividing (2) by (1) yields: 

<math>\vec{v_{Ai}}+\vec{v_{Bi}}=\vec{v_{Af}}+\vec{v_{Bf}}</math> 
<math>1=\frac{v_{Ai}+v_{Bi}}{v_{Af}+v_{Bf}}</math> 

<math>e=\frac{v_{Ai}+v_{Bi}}{v_{Af}+v_{Bf}}</math> 

For perfectly elastic collision e=1.

===Inelastic Collisions===

An inelastic collision is a type of collision where Kinetic Energy is not conserved. In this case the difference in initial velocities will not equal the difference in final velocities. This means that the coefficient of restitution will not equal 1.

Consider the same collision as previously described, except this collision is inelastic and Kinetic Energy is not conserved. Since some of the kinetic energy is dissipated: 
<math>\frac{1}{2}m_A v^2_{Ai}+\frac{1}{2}m_A v^2_Bi<\frac{1}{2}m_A v^2_{Af}+\frac{1}{2}m_A v^2_{Bf}</math> 
<math>\frac{v_{Ai}+v_{Bi}}{v_{Af}+v_{Bf}}<1</math> 

<math>e=\frac{v_{Ai}+v_{Bi}}{v_{Af}+v_{Bf}}<1</math> 

A perfectly inelastic collision (for example when two objects collide and then stick together) has a coefficient of restitution of 0.

===A Computational Model===

==Examples==
===Simple===
A ball is traveling at <math>4 \frac{m}{s}</math> and collides with a ball. After the collision the ball travels at <math>2.4\frac{m}{s}</math> in the opposite direction. Find the coefficient of restitution of the collision. 
Start with the definition of the coefficient of restitution: 
<math>e=\frac{v_{wall_f}-v_{ball_f}}{v_{ball_i}-v_{wall_i}}</math> 
Then plug in values and solve: 
<math>e=\frac{0-v_{ball_f}}{v_{ball_i}-0}</math> 
<math>e=\frac{-(-2.4)}{4}</math> 
<math>e=.6</math>

===Middling===
A hockey puck travels at a constant velocity <math>5\frac{m}{s}</math> and is hit by another puck with a velocity of <math>8\frac{m}{s}</math>. After the collision, it travels in the opposite direction at 4m/s. This collision has a coefficient of restitution of .55. Find the velocity of the second puck after the collision. 
Start with the definition of the coefficient of restitution. We are calling the velocity of the first puck <math>v_1</math> and the velocity of the second puck <math>v_2</math>. 
<math>e=\frac{v_{1f}-v_{2f}}{v_{2i}-v_{1i}}</math> 
Solve for <math>v_{2f}</math> 

<math>v_{2f}=v_{1f}-e(v_{2i}-v_{1i})</math> 
Plug in the values stated in the problem. 
<math>v_{2f}=4-.55(5-8)</math>
<math>v_{2f}=2.35\frac{m}{s}</math>

===Difficult===
[[File:Cor1.jpg|right]]
Bouncing Ball Example:

Consider a ball dropped from initial height <math>h_0</math>. When it bounces it reaches a height <math>h_f</math>. Find the coefficient of restitution of this collision. 
<math>e=\frac{v_{floor_f}-v_{ball_f}}{v_{ball_i}-v_{floor_i}}</math> 
The floor's velocity is always 0. 
<math>e=\frac{v_{ball_f}}{v_{ball_i}}</math> 
Recall the law of conservation of energy: 
<math> \Delta KE = \Delta PE</math> 
For the ball from h0 to h=0
<math> \frac{1}{2}mv_{ball_i}^2=mgh_0</math> 
<math> \frac{1}{2}v_{ball_i}^2=gh_0</math> 
<math> v_{ball_i}=2\sqrt{gh_0} </math> 
The same concept can be applied to the second bounce. 
<math>v_{ball_f}=2\sqrt{gh_f}</math> 
Plugging this in to the coefficient of restitution expression we get: 
<math>e=\frac{2\sqrt{mgh_f}}{2\sqrt{mgh_0}}</math> 
<math>e=\frac{\sqrt{h_f}}{\sqrt{h_0}}</math> 

==Connectedness==
'''How is this topic connected to something that you are interested in?''' 
This topic has major connections to sports. As a former high school volleyball player, it is very interesting to think about the collisions that occurred in the game and how the coefficient of restitution is related to those collisions. 
'''How is it connected to your major (Mechanical Engineering)?''' 
The coefficient of restitution is a key concept in the study of Dynamics which is the foundation of many aspects of mechanical engineering. It also has to do with applications of mechanical engineering regarding materials. 

'''Is there an interesting industrial application?'''
In sports the coefficient of restitution is an important concept especially in sports like golf or tennis where collisions are a frequent and essential part of the game. In these sports sometimes limitations are put on how high the coefficient of restitution can be for certain clubs or bats. The maximum coefficient of restitution allowed in golf for clubs is .82. There have been studies done on the coefficients for different sports balls. These studies usually follow a procedure where different balls are dropped and left to bounce. Then data regarding their initial and final velocities or how high they bounced is recorded. The images below show some of the data from one of these experiments.

[[File:sportsgraph.png|This graph shows the initial and final velocities of many different balls from different sports.]]
[[File:sportschart.png|This chart shows the relationship between different coefficients of restitution and the initial and final velocities of sports balls.]]

== See also ==

Are there related topics or categories in this wiki resource for the curious reader to explore? How does this topic fit into that context?

===Further reading===

Books, Articles or other print media on this topic

===External links===

==References==

This section contains the the references you used while writing this page

[[Category:Collisions]]

Coefficient of Restitution

2015-12-06T02:19:23Z

Mmoreno30: /* The Main Idea */

Claimed by Maria Moreno

The coefficient of restitution measures the amount of Kinetic Energy remains and how much energy is lost after a collision. It also shows the degree of elasticity of a collision. It is defined as the ratio of the difference in initial velocities divided by the difference in final velocities. The coefficient of restitution of a collision is usually between 0 and 1 except in rare cases when energy is released by a collision.
[[File:coe2.png|right]]

==The Main Idea==

The coefficient of restitution is a ratio that describes the degree of elasticity of a collision. It is used to solve problems dealing with collisions that are not perfectly elastic or inelastic.
The equation that describes the coefficient of restitution involved dividing the difference in the final velocities by the difference in the initial velocity.

Consider objects A and B with initial velocities vAi and vBi and final velocities vAf and vBf.
The coefficient of restitution, e is determined with the following formula:

<math> e = \frac{v_{Bf}-v_{Af}}{v_{Ai}-v_{Bi}} </math> 
The coefficient is a unitless form of measurement as it is the ratio of two velocities and the units cancel.

===A Mathematical Model===
Consider a perfectly elastic collision between objects A and B where vAi and vBi refer to the initial velocities of A and B and vAf and vBf refer to the final velocities of A and B.

Start with the conservation of momentum principle which states that 
<math>\vec{p_i} = \vec{p_f}</math> and 
<math>m_A\vec{v_{Ai}}+m_A\vec{v_{Bi}}=m_A\vec{v_{Af}}+m_A\vec{v_{Bf}}</math> (1) 

Remember that for a perfectly elastic collision kinetic energy is also conserved meaning: 
<math>KE_i = KE_f</math> and 
<math>\frac{1}{2}m_A v^2_{Ai}+\frac{1}{2}m_A v^2_Bi=\frac{1}{2}m_A v^2_{Af}+\frac{1}{2}m_A v^2_{Bf}</math> (2) 

Dividing (2) by (1) yields: 

<math>\vec{v_{Ai}}+\vec{v_{Bi}}=\vec{v_{Af}}+\vec{v_{Bf}}</math> 
<math>1=\frac{v_{Ai}+v_{Bi}}{v_{Af}+v_{Bf}}</math> 

<math>e=\frac{v_{Ai}+v_{Bi}}{v_{Af}+v_{Bf}}</math> 

For perfectly elastic collision e=1.

===Inelastic Collisions===

An inelastic collision is a type of collision where Kinetic Energy is not conserved. In this case the difference in initial velocities will not equal the difference in final velocities. This means that the coefficient of restitution will not equal 1.

Consider the same collision as previously described, except this collision is inelastic and Kinetic Energy is not conserved. Since some of the kinetic energy is dissipated: 
<math>\frac{1}{2}m_A v^2_{Ai}+\frac{1}{2}m_A v^2_Bi<\frac{1}{2}m_A v^2_{Af}+\frac{1}{2}m_A v^2_{Bf}</math> 
<math>\frac{v_{Ai}+v_{Bi}}{v_{Af}+v_{Bf}}<1</math> 

<math>e=\frac{v_{Ai}+v_{Bi}}{v_{Af}+v_{Bf}}<1</math> 

A perfectly inelastic collision (for example when two objects collide and then stick together) has a coefficient of restitution of 0.

===A Computational Model===

==Examples==
===Simple===
A ball is traveling at <math>4 \frac{m}{s}</math> and collides with a ball. After the collision the ball travels at <math>2.4\frac{m}{s}</math> in the opposite direction. Find the coefficient of restitution of the collision. 
Start with the definition of the coefficient of restitution: 
<math>e=\frac{v_{wall_f}-v_{ball_f}}{v_{ball_i}-v_{wall_i}}</math> 
Then plug in values and solve: 
<math>e=\frac{0-v_{ball_f}}{v_{ball_i}-0}</math> 
<math>e=\frac{-(-2.4)}{4}</math> 
<math>e=.6</math>

===Middling===
A hockey puck travels at a constant velocity <math>5\frac{m}{s}</math> and is hit by another puck with a velocity of <math>8\frac{m}{s}</math>. After the collision, it travels in the opposite direction at 4m/s. This collision has a coefficient of restitution of .55. Find the velocity of the second puck after the collision. 
Start with the definition of the coefficient of restitution. We are calling the velocity of the first puck <math>v_1</math> and the velocity of the second puck <math>v_2</math>. 
<math>e=\frac{v_{1f}-v_{2f}}{v_{2i}-v_{1i}}</math> 
Solve for <math>v_{2f}</math> 

<math>v_{2f}=v_{1f}-e(v_{2i}-v_{1i})</math> 
Plug in the values stated in the problem. 
<math>v_{2f}=4-.55(5-8)</math>
<math>v_{2f}=2.35\frac{m}{s}</math>

===Difficult===
[[File:Cor1.jpg|right]]
Bouncing Ball Example:

Consider a ball dropped from initial height <math>h_0</math>. When it bounces it reaches a height <math>h_f</math>. Find the coefficient of restitution of this collision. 
<math>e=\frac{v_{floor_f}-v_{ball_f}}{v_{ball_i}-v_{floor_i}}</math> 
The floor's velocity is always 0. 
<math>e=\frac{v_{ball_f}}{v_{ball_i}}</math> 
Recall the law of conservation of energy: 
<math> \Delta KE = \Delta PE</math> 
For the ball from h0 to h=0
<math> \frac{1}{2}mv_{ball_i}^2=mgh_0</math> 
<math> \frac{1}{2}v_{ball_i}^2=gh_0</math> 
<math> v_{ball_i}=2\sqrt{gh_0} </math> 
The same concept can be applied to the second bounce. 
<math>v_{ball_f}=2\sqrt{gh_f}</math> 
Plugging this in to the coefficient of restitution expression we get: 
<math>e=\frac{2\sqrt{mgh_f}}{2\sqrt{mgh_0}}</math> 
<math>e=\frac{\sqrt{h_f}}{\sqrt{h_0}}</math> 

==Connectedness==
'''How is this topic connected to something that you are interested in?''' 
This topic has major connections to sports. As a former high school volleyball player, it is very interesting to think about the collisions that occurred in the game and how the coefficient of restitution is related to those collisions. 
'''How is it connected to your major (Mechanical Engineering)?''' 
The coefficient of restitution is a key concept in the study of Dynamics which is the foundation of many aspects of mechanical engineering. It also has to do with applications of mechanical engineering regarding materials. 

'''Is there an interesting industrial application?'''
In sports the coefficient of restitution is an important concept especially in sports like golf or tennis where collisions are a frequent and essential part of the game. In these sports sometimes limitations are put on how high the coefficient of restitution can be for certain clubs or bats. The maximum coefficient of restitution allowed in golf for clubs is .82. There have been studies done on the coefficients for different sports balls. These studies usually follow a procedure where different balls are dropped and left to bounce. Then data regarding their initial and final velocities or how high they bounced is recorded. The images below show some of the data from one of these experiments.

[[File:sportsgraph.png|This graph shows the initial and final velocities of many different balls from different sports.]]
[[File:sportschart.png|This chart shows the relationship between different coefficients of restitution and the initial and final velocities of sports balls.]]

== See also ==

Are there related topics or categories in this wiki resource for the curious reader to explore? How does this topic fit into that context?

===Further reading===

Books, Articles or other print media on this topic

===External links===

==References==

This section contains the the references you used while writing this page

[[Category:Which Category did you place this in?]]

Coefficient of Restitution

2015-12-06T02:18:19Z

Mmoreno30: /* Middling */

Claimed by Maria Moreno

The coefficient of restitution measures the amount of Kinetic Energy remains and how much energy is lost after a collision. It also shows the degree of elasticity of a collision. It is defined as the ratio of the difference in initial velocities divided by the difference in final velocities. The coefficient of restitution of a collision is usually between 0 and 1 except in rare cases when energy is released by a collision.
[[File:coe2.png|right]]

==The Main Idea==

The coefficient of restitution is a ratio that describes the degree of elasticity of a collision. It is used to solve problems dealing with collisions that are not perfectly elastic or inelastic.
The equation that describes the coefficient of restitution involved dividing the difference in the final velocities by the difference in the initial velocity.

Consider objects A and B with initial velocities vAi and vBi and final velocities vAf and vBf.
The coefficient of restitution, e is determined with the following formula:

<math> e = \frac{v_{Bf}-v_{Af}}{v_{Ai}-v_{Bi}} </math>

===A Mathematical Model===
Consider a perfectly elastic collision between objects A and B where vAi and vBi refer to the initial velocities of A and B and vAf and vBf refer to the final velocities of A and B.

Start with the conservation of momentum principle which states that 
<math>\vec{p_i} = \vec{p_f}</math> and 
<math>m_A\vec{v_{Ai}}+m_A\vec{v_{Bi}}=m_A\vec{v_{Af}}+m_A\vec{v_{Bf}}</math> (1) 

Remember that for a perfectly elastic collision kinetic energy is also conserved meaning: 
<math>KE_i = KE_f</math> and 
<math>\frac{1}{2}m_A v^2_{Ai}+\frac{1}{2}m_A v^2_Bi=\frac{1}{2}m_A v^2_{Af}+\frac{1}{2}m_A v^2_{Bf}</math> (2) 

Dividing (2) by (1) yields: 

<math>\vec{v_{Ai}}+\vec{v_{Bi}}=\vec{v_{Af}}+\vec{v_{Bf}}</math> 
<math>1=\frac{v_{Ai}+v_{Bi}}{v_{Af}+v_{Bf}}</math> 

<math>e=\frac{v_{Ai}+v_{Bi}}{v_{Af}+v_{Bf}}</math> 

For perfectly elastic collision e=1.

===Inelastic Collisions===

An inelastic collision is a type of collision where Kinetic Energy is not conserved. In this case the difference in initial velocities will not equal the difference in final velocities. This means that the coefficient of restitution will not equal 1.

Consider the same collision as previously described, except this collision is inelastic and Kinetic Energy is not conserved. Since some of the kinetic energy is dissipated: 
<math>\frac{1}{2}m_A v^2_{Ai}+\frac{1}{2}m_A v^2_Bi<\frac{1}{2}m_A v^2_{Af}+\frac{1}{2}m_A v^2_{Bf}</math> 
<math>\frac{v_{Ai}+v_{Bi}}{v_{Af}+v_{Bf}}<1</math> 

<math>e=\frac{v_{Ai}+v_{Bi}}{v_{Af}+v_{Bf}}<1</math> 

A perfectly inelastic collision (for example when two objects collide and then stick together) has a coefficient of restitution of 0.

===A Computational Model===

==Examples==
===Simple===
A ball is traveling at <math>4 \frac{m}{s}</math> and collides with a ball. After the collision the ball travels at <math>2.4\frac{m}{s}</math> in the opposite direction. Find the coefficient of restitution of the collision. 
Start with the definition of the coefficient of restitution: 
<math>e=\frac{v_{wall_f}-v_{ball_f}}{v_{ball_i}-v_{wall_i}}</math> 
Then plug in values and solve: 
<math>e=\frac{0-v_{ball_f}}{v_{ball_i}-0}</math> 
<math>e=\frac{-(-2.4)}{4}</math> 
<math>e=.6</math>

===Middling===
A hockey puck travels at a constant velocity <math>5\frac{m}{s}</math> and is hit by another puck with a velocity of <math>8\frac{m}{s}</math>. After the collision, it travels in the opposite direction at 4m/s. This collision has a coefficient of restitution of .55. Find the velocity of the second puck after the collision. 
Start with the definition of the coefficient of restitution. We are calling the velocity of the first puck <math>v_1</math> and the velocity of the second puck <math>v_2</math>. 
<math>e=\frac{v_{1f}-v_{2f}}{v_{2i}-v_{1i}}</math> 
Solve for <math>v_{2f}</math> 

<math>v_{2f}=v_{1f}-e(v_{2i}-v_{1i})</math> 
Plug in the values stated in the problem. 
<math>v_{2f}=4-.55(5-8)</math>
<math>v_{2f}=2.35\frac{m}{s}</math>

===Difficult===
[[File:Cor1.jpg|right]]
Bouncing Ball Example:

Consider a ball dropped from initial height <math>h_0</math>. When it bounces it reaches a height <math>h_f</math>. Find the coefficient of restitution of this collision. 
<math>e=\frac{v_{floor_f}-v_{ball_f}}{v_{ball_i}-v_{floor_i}}</math> 
The floor's velocity is always 0. 
<math>e=\frac{v_{ball_f}}{v_{ball_i}}</math> 
Recall the law of conservation of energy: 
<math> \Delta KE = \Delta PE</math> 
For the ball from h0 to h=0
<math> \frac{1}{2}mv_{ball_i}^2=mgh_0</math> 
<math> \frac{1}{2}v_{ball_i}^2=gh_0</math> 
<math> v_{ball_i}=2\sqrt{gh_0} </math> 
The same concept can be applied to the second bounce. 
<math>v_{ball_f}=2\sqrt{gh_f}</math> 
Plugging this in to the coefficient of restitution expression we get: 
<math>e=\frac{2\sqrt{mgh_f}}{2\sqrt{mgh_0}}</math> 
<math>e=\frac{\sqrt{h_f}}{\sqrt{h_0}}</math> 

==Connectedness==
'''How is this topic connected to something that you are interested in?''' 
This topic has major connections to sports. As a former high school volleyball player, it is very interesting to think about the collisions that occurred in the game and how the coefficient of restitution is related to those collisions. 
'''How is it connected to your major (Mechanical Engineering)?''' 
The coefficient of restitution is a key concept in the study of Dynamics which is the foundation of many aspects of mechanical engineering. It also has to do with applications of mechanical engineering regarding materials. 

'''Is there an interesting industrial application?'''
In sports the coefficient of restitution is an important concept especially in sports like golf or tennis where collisions are a frequent and essential part of the game. In these sports sometimes limitations are put on how high the coefficient of restitution can be for certain clubs or bats. The maximum coefficient of restitution allowed in golf for clubs is .82. There have been studies done on the coefficients for different sports balls. These studies usually follow a procedure where different balls are dropped and left to bounce. Then data regarding their initial and final velocities or how high they bounced is recorded. The images below show some of the data from one of these experiments.

[[File:sportsgraph.png|This graph shows the initial and final velocities of many different balls from different sports.]]
[[File:sportschart.png|This chart shows the relationship between different coefficients of restitution and the initial and final velocities of sports balls.]]

== See also ==

Are there related topics or categories in this wiki resource for the curious reader to explore? How does this topic fit into that context?

===Further reading===

Books, Articles or other print media on this topic

===External links===

==References==

This section contains the the references you used while writing this page

[[Category:Which Category did you place this in?]]

Coefficient of Restitution

2015-12-06T02:03:35Z

Mmoreno30: /* Middling */

Claimed by Maria Moreno

The coefficient of restitution measures the amount of Kinetic Energy remains and how much energy is lost after a collision. It also shows the degree of elasticity of a collision. It is defined as the ratio of the difference in initial velocities divided by the difference in final velocities. The coefficient of restitution of a collision is usually between 0 and 1 except in rare cases when energy is released by a collision.
[[File:coe2.png|right]]

==The Main Idea==

The coefficient of restitution is a ratio that describes the degree of elasticity of a collision. It is used to solve problems dealing with collisions that are not perfectly elastic or inelastic.
The equation that describes the coefficient of restitution involved dividing the difference in the final velocities by the difference in the initial velocity.

Consider objects A and B with initial velocities vAi and vBi and final velocities vAf and vBf.
The coefficient of restitution, e is determined with the following formula:

<math> e = \frac{v_{Bf}-v_{Af}}{v_{Ai}-v_{Bi}} </math>

===A Mathematical Model===
Consider a perfectly elastic collision between objects A and B where vAi and vBi refer to the initial velocities of A and B and vAf and vBf refer to the final velocities of A and B.

Start with the conservation of momentum principle which states that 
<math>\vec{p_i} = \vec{p_f}</math> and 
<math>m_A\vec{v_{Ai}}+m_A\vec{v_{Bi}}=m_A\vec{v_{Af}}+m_A\vec{v_{Bf}}</math> (1) 

Remember that for a perfectly elastic collision kinetic energy is also conserved meaning: 
<math>KE_i = KE_f</math> and 
<math>\frac{1}{2}m_A v^2_{Ai}+\frac{1}{2}m_A v^2_Bi=\frac{1}{2}m_A v^2_{Af}+\frac{1}{2}m_A v^2_{Bf}</math> (2) 

Dividing (2) by (1) yields: 

<math>\vec{v_{Ai}}+\vec{v_{Bi}}=\vec{v_{Af}}+\vec{v_{Bf}}</math> 
<math>1=\frac{v_{Ai}+v_{Bi}}{v_{Af}+v_{Bf}}</math> 

<math>e=\frac{v_{Ai}+v_{Bi}}{v_{Af}+v_{Bf}}</math> 

For perfectly elastic collision e=1.

===Inelastic Collisions===

An inelastic collision is a type of collision where Kinetic Energy is not conserved. In this case the difference in initial velocities will not equal the difference in final velocities. This means that the coefficient of restitution will not equal 1.

Consider the same collision as previously described, except this collision is inelastic and Kinetic Energy is not conserved. Since some of the kinetic energy is dissipated: 
<math>\frac{1}{2}m_A v^2_{Ai}+\frac{1}{2}m_A v^2_Bi<\frac{1}{2}m_A v^2_{Af}+\frac{1}{2}m_A v^2_{Bf}</math> 
<math>\frac{v_{Ai}+v_{Bi}}{v_{Af}+v_{Bf}}<1</math> 

<math>e=\frac{v_{Ai}+v_{Bi}}{v_{Af}+v_{Bf}}<1</math> 

A perfectly inelastic collision (for example when two objects collide and then stick together) has a coefficient of restitution of 0.

===A Computational Model===

==Examples==
===Simple===
A ball is traveling at <math>4 \frac{m}{s}</math> and collides with a ball. After the collision the ball travels at <math>2.4\frac{m}{s}</math> in the opposite direction. Find the coefficient of restitution of the collision. 
Start with the definition of the coefficient of restitution: 
<math>e=\frac{v_{wall_f}-v_{ball_f}}{v_{ball_i}-v_{wall_i}}</math> 
Then plug in values and solve: 
<math>e=\frac{0-v_{ball_f}}{v_{ball_i}-0}</math> 
<math>e=\frac{-(-2.4)}{4}</math> 
<math>e=.6</math>

===Middling===
A hockey puck travels at a constant velocity <math>5\frac{m}{s}</math> and is hit by another puck with a velocity of <math>8\frac{m}{s}</math>. After the collision, it travels in the opposite direction at 4m/s. This collision has a coefficient of restitution of .55. Find the velocity of the second puck after the collision. 
Start with the definition of the coefficient of restitution. We are calling the velocity of the first puck <math>v_1</math> and the velocity of the second puck <math>v_2</math>. 
<math>e=\frac{v_{1f}-v_{2f}}{v_{2i}-v_{1i}}</math> 
Solve for <math>v_{2f}</math> 

<math>v_{2f}=v_{1f}-e(v_{2i}-v_{1i})</math> 
Plug in the values stated in the problem. 

===Difficult===
[[File:Cor1.jpg|right]]
Bouncing Ball Example:

Consider a ball dropped from initial height <math>h_0</math>. When it bounces it reaches a height <math>h_f</math>. Find the coefficient of restitution of this collision. 
<math>e=\frac{v_{floor_f}-v_{ball_f}}{v_{ball_i}-v_{floor_i}}</math> 
The floor's velocity is always 0. 
<math>e=\frac{v_{ball_f}}{v_{ball_i}}</math> 
Recall the law of conservation of energy: 
<math> \Delta KE = \Delta PE</math> 
For the ball from h0 to h=0
<math> \frac{1}{2}mv_{ball_i}^2=mgh_0</math> 
<math> \frac{1}{2}v_{ball_i}^2=gh_0</math> 
<math> v_{ball_i}=2\sqrt{gh_0} </math> 
The same concept can be applied to the second bounce. 
<math>v_{ball_f}=2\sqrt{gh_f}</math> 
Plugging this in to the coefficient of restitution expression we get: 
<math>e=\frac{2\sqrt{mgh_f}}{2\sqrt{mgh_0}}</math> 
<math>e=\frac{\sqrt{h_f}}{\sqrt{h_0}}</math> 

==Connectedness==
'''How is this topic connected to something that you are interested in?''' 
This topic has major connections to sports. As a former high school volleyball player, it is very interesting to think about the collisions that occurred in the game and how the coefficient of restitution is related to those collisions. 
'''How is it connected to your major (Mechanical Engineering)?''' 
The coefficient of restitution is a key concept in the study of Dynamics which is the foundation of many aspects of mechanical engineering. It also has to do with applications of mechanical engineering regarding materials. 

'''Is there an interesting industrial application?'''
In sports the coefficient of restitution is an important concept especially in sports like golf or tennis where collisions are a frequent and essential part of the game. In these sports sometimes limitations are put on how high the coefficient of restitution can be for certain clubs or bats. The maximum coefficient of restitution allowed in golf for clubs is .82. There have been studies done on the coefficients for different sports balls. These studies usually follow a procedure where different balls are dropped and left to bounce. Then data regarding their initial and final velocities or how high they bounced is recorded. The images below show some of the data from one of these experiments.

[[File:sportsgraph.png|This graph shows the initial and final velocities of many different balls from different sports.]]
[[File:sportschart.png|This chart shows the relationship between different coefficients of restitution and the initial and final velocities of sports balls.]]

== See also ==

Are there related topics or categories in this wiki resource for the curious reader to explore? How does this topic fit into that context?

===Further reading===

Books, Articles or other print media on this topic

===External links===

==References==

This section contains the the references you used while writing this page

[[Category:Which Category did you place this in?]]

Coefficient of Restitution

2015-12-06T02:02:47Z

Mmoreno30: /* Middling */

Claimed by Maria Moreno

The coefficient of restitution measures the amount of Kinetic Energy remains and how much energy is lost after a collision. It also shows the degree of elasticity of a collision. It is defined as the ratio of the difference in initial velocities divided by the difference in final velocities. The coefficient of restitution of a collision is usually between 0 and 1 except in rare cases when energy is released by a collision.
[[File:coe2.png|right]]

==The Main Idea==

The coefficient of restitution is a ratio that describes the degree of elasticity of a collision. It is used to solve problems dealing with collisions that are not perfectly elastic or inelastic.
The equation that describes the coefficient of restitution involved dividing the difference in the final velocities by the difference in the initial velocity.

Consider objects A and B with initial velocities vAi and vBi and final velocities vAf and vBf.
The coefficient of restitution, e is determined with the following formula:

<math> e = \frac{v_{Bf}-v_{Af}}{v_{Ai}-v_{Bi}} </math>

===A Mathematical Model===
Consider a perfectly elastic collision between objects A and B where vAi and vBi refer to the initial velocities of A and B and vAf and vBf refer to the final velocities of A and B.

Start with the conservation of momentum principle which states that 
<math>\vec{p_i} = \vec{p_f}</math> and 
<math>m_A\vec{v_{Ai}}+m_A\vec{v_{Bi}}=m_A\vec{v_{Af}}+m_A\vec{v_{Bf}}</math> (1) 

Remember that for a perfectly elastic collision kinetic energy is also conserved meaning: 
<math>KE_i = KE_f</math> and 
<math>\frac{1}{2}m_A v^2_{Ai}+\frac{1}{2}m_A v^2_Bi=\frac{1}{2}m_A v^2_{Af}+\frac{1}{2}m_A v^2_{Bf}</math> (2) 

Dividing (2) by (1) yields: 

<math>\vec{v_{Ai}}+\vec{v_{Bi}}=\vec{v_{Af}}+\vec{v_{Bf}}</math> 
<math>1=\frac{v_{Ai}+v_{Bi}}{v_{Af}+v_{Bf}}</math> 

<math>e=\frac{v_{Ai}+v_{Bi}}{v_{Af}+v_{Bf}}</math> 

For perfectly elastic collision e=1.

===Inelastic Collisions===

An inelastic collision is a type of collision where Kinetic Energy is not conserved. In this case the difference in initial velocities will not equal the difference in final velocities. This means that the coefficient of restitution will not equal 1.

Consider the same collision as previously described, except this collision is inelastic and Kinetic Energy is not conserved. Since some of the kinetic energy is dissipated: 
<math>\frac{1}{2}m_A v^2_{Ai}+\frac{1}{2}m_A v^2_Bi<\frac{1}{2}m_A v^2_{Af}+\frac{1}{2}m_A v^2_{Bf}</math> 
<math>\frac{v_{Ai}+v_{Bi}}{v_{Af}+v_{Bf}}<1</math> 

<math>e=\frac{v_{Ai}+v_{Bi}}{v_{Af}+v_{Bf}}<1</math> 

A perfectly inelastic collision (for example when two objects collide and then stick together) has a coefficient of restitution of 0.

===A Computational Model===

==Examples==
===Simple===
A ball is traveling at <math>4 \frac{m}{s}</math> and collides with a ball. After the collision the ball travels at <math>2.4\frac{m}{s}</math> in the opposite direction. Find the coefficient of restitution of the collision. 
Start with the definition of the coefficient of restitution: 
<math>e=\frac{v_{wall_f}-v_{ball_f}}{v_{ball_i}-v_{wall_i}}</math> 
Then plug in values and solve: 
<math>e=\frac{0-v_{ball_f}}{v_{ball_i}-0}</math> 
<math>e=\frac{-(-2.4)}{4}</math> 
<math>e=.6</math>

===Middling===
A hockey puck travels at a constant velocity <math>5\frac{m}{s}</math> and is hit by another puck with a velocity of <math>8\frac{m}{s}</math>. After the collision, it travels in the opposite direction at 4m/s. This collision has a coefficient of restitution of .55. Find the velocity of the second puck after the collision. 
Start with the definition of the coefficient of restitution. We are calling the velocity of the first puck <math>v_1</math> and the velocity of the second puck <math>v_2</math>. 
<math>e=\frac{v_{1f}-v_{2f}}{v_{2i}-v_{1i}}</math> 
Solve for <math>v_{2f}</math> 

<math>v_{1f}-e(v_{2i}-v_{1i})</math>

===Difficult===
[[File:Cor1.jpg|right]]
Bouncing Ball Example:

Consider a ball dropped from initial height <math>h_0</math>. When it bounces it reaches a height <math>h_f</math>. Find the coefficient of restitution of this collision. 
<math>e=\frac{v_{floor_f}-v_{ball_f}}{v_{ball_i}-v_{floor_i}}</math> 
The floor's velocity is always 0. 
<math>e=\frac{v_{ball_f}}{v_{ball_i}}</math> 
Recall the law of conservation of energy: 
<math> \Delta KE = \Delta PE</math> 
For the ball from h0 to h=0
<math> \frac{1}{2}mv_{ball_i}^2=mgh_0</math> 
<math> \frac{1}{2}v_{ball_i}^2=gh_0</math> 
<math> v_{ball_i}=2\sqrt{gh_0} </math> 
The same concept can be applied to the second bounce. 
<math>v_{ball_f}=2\sqrt{gh_f}</math> 
Plugging this in to the coefficient of restitution expression we get: 
<math>e=\frac{2\sqrt{mgh_f}}{2\sqrt{mgh_0}}</math> 
<math>e=\frac{\sqrt{h_f}}{\sqrt{h_0}}</math> 

==Connectedness==
'''How is this topic connected to something that you are interested in?''' 
This topic has major connections to sports. As a former high school volleyball player, it is very interesting to think about the collisions that occurred in the game and how the coefficient of restitution is related to those collisions. 
'''How is it connected to your major (Mechanical Engineering)?''' 
The coefficient of restitution is a key concept in the study of Dynamics which is the foundation of many aspects of mechanical engineering. It also has to do with applications of mechanical engineering regarding materials. 

'''Is there an interesting industrial application?'''
In sports the coefficient of restitution is an important concept especially in sports like golf or tennis where collisions are a frequent and essential part of the game. In these sports sometimes limitations are put on how high the coefficient of restitution can be for certain clubs or bats. The maximum coefficient of restitution allowed in golf for clubs is .82. There have been studies done on the coefficients for different sports balls. These studies usually follow a procedure where different balls are dropped and left to bounce. Then data regarding their initial and final velocities or how high they bounced is recorded. The images below show some of the data from one of these experiments.

[[File:sportsgraph.png|This graph shows the initial and final velocities of many different balls from different sports.]]
[[File:sportschart.png|This chart shows the relationship between different coefficients of restitution and the initial and final velocities of sports balls.]]

== See also ==

Are there related topics or categories in this wiki resource for the curious reader to explore? How does this topic fit into that context?

===Further reading===

Books, Articles or other print media on this topic

===External links===

==References==

This section contains the the references you used while writing this page

[[Category:Which Category did you place this in?]]

Coefficient of Restitution

2015-12-06T01:59:24Z

Mmoreno30: /* Difficult */

Claimed by Maria Moreno

The coefficient of restitution measures the amount of Kinetic Energy remains and how much energy is lost after a collision. It also shows the degree of elasticity of a collision. It is defined as the ratio of the difference in initial velocities divided by the difference in final velocities. The coefficient of restitution of a collision is usually between 0 and 1 except in rare cases when energy is released by a collision.
[[File:coe2.png|right]]

==The Main Idea==

The coefficient of restitution is a ratio that describes the degree of elasticity of a collision. It is used to solve problems dealing with collisions that are not perfectly elastic or inelastic.
The equation that describes the coefficient of restitution involved dividing the difference in the final velocities by the difference in the initial velocity.

Consider objects A and B with initial velocities vAi and vBi and final velocities vAf and vBf.
The coefficient of restitution, e is determined with the following formula:

<math> e = \frac{v_{Bf}-v_{Af}}{v_{Ai}-v_{Bi}} </math>

===A Mathematical Model===
Consider a perfectly elastic collision between objects A and B where vAi and vBi refer to the initial velocities of A and B and vAf and vBf refer to the final velocities of A and B.

Start with the conservation of momentum principle which states that 
<math>\vec{p_i} = \vec{p_f}</math> and 
<math>m_A\vec{v_{Ai}}+m_A\vec{v_{Bi}}=m_A\vec{v_{Af}}+m_A\vec{v_{Bf}}</math> (1) 

Remember that for a perfectly elastic collision kinetic energy is also conserved meaning: 
<math>KE_i = KE_f</math> and 
<math>\frac{1}{2}m_A v^2_{Ai}+\frac{1}{2}m_A v^2_Bi=\frac{1}{2}m_A v^2_{Af}+\frac{1}{2}m_A v^2_{Bf}</math> (2) 

Dividing (2) by (1) yields: 

<math>\vec{v_{Ai}}+\vec{v_{Bi}}=\vec{v_{Af}}+\vec{v_{Bf}}</math> 
<math>1=\frac{v_{Ai}+v_{Bi}}{v_{Af}+v_{Bf}}</math> 

<math>e=\frac{v_{Ai}+v_{Bi}}{v_{Af}+v_{Bf}}</math> 

For perfectly elastic collision e=1.

===Inelastic Collisions===

An inelastic collision is a type of collision where Kinetic Energy is not conserved. In this case the difference in initial velocities will not equal the difference in final velocities. This means that the coefficient of restitution will not equal 1.

Consider the same collision as previously described, except this collision is inelastic and Kinetic Energy is not conserved. Since some of the kinetic energy is dissipated: 
<math>\frac{1}{2}m_A v^2_{Ai}+\frac{1}{2}m_A v^2_Bi<\frac{1}{2}m_A v^2_{Af}+\frac{1}{2}m_A v^2_{Bf}</math> 
<math>\frac{v_{Ai}+v_{Bi}}{v_{Af}+v_{Bf}}<1</math> 

<math>e=\frac{v_{Ai}+v_{Bi}}{v_{Af}+v_{Bf}}<1</math> 

A perfectly inelastic collision (for example when two objects collide and then stick together) has a coefficient of restitution of 0.

===A Computational Model===

==Examples==
===Simple===
A ball is traveling at <math>4 \frac{m}{s}</math> and collides with a ball. After the collision the ball travels at <math>2.4\frac{m}{s}</math> in the opposite direction. Find the coefficient of restitution of the collision. 
Start with the definition of the coefficient of restitution: 
<math>e=\frac{v_{wall_f}-v_{ball_f}}{v_{ball_i}-v_{wall_i}}</math> 
Then plug in values and solve: 
<math>e=\frac{0-v_{ball_f}}{v_{ball_i}-0}</math> 
<math>e=\frac{-(-2.4)}{4}</math> 
<math>e=.6</math>

===Middling===
A hockey puck travels at a constant velocity <math>5\frac{m}{s}</math> and is hit by another puck with a velocity of <math>8\frac{m}{s}</math>. After the collision, it travels in the opposite direction at 4m/s. This collision has a coefficient of restitution of .55. Find the velocity of the second puck. 
Start with the definition of the coefficient of restitution. We are calling the velocity of the first puck <math>v_1</math> and the velocity of the second puck <math>v_2</math>. 
<math>e=\frac{v_{1f}-v_{2f}}{v_{2i}-v_{1i}}</math> 
Solve for <math>v_2</math> 

<math></math>

===Difficult===
[[File:Cor1.jpg|right]]
Bouncing Ball Example:

Consider a ball dropped from initial height <math>h_0</math>. When it bounces it reaches a height <math>h_f</math>. Find the coefficient of restitution of this collision. 
<math>e=\frac{v_{floor_f}-v_{ball_f}}{v_{ball_i}-v_{floor_i}}</math> 
The floor's velocity is always 0. 
<math>e=\frac{v_{ball_f}}{v_{ball_i}}</math> 
Recall the law of conservation of energy: 
<math> \Delta KE = \Delta PE</math> 
For the ball from h0 to h=0
<math> \frac{1}{2}mv_{ball_i}^2=mgh_0</math> 
<math> \frac{1}{2}v_{ball_i}^2=gh_0</math> 
<math> v_{ball_i}=2\sqrt{gh_0} </math> 
The same concept can be applied to the second bounce. 
<math>v_{ball_f}=2\sqrt{gh_f}</math> 
Plugging this in to the coefficient of restitution expression we get: 
<math>e=\frac{2\sqrt{mgh_f}}{2\sqrt{mgh_0}}</math> 
<math>e=\frac{\sqrt{h_f}}{\sqrt{h_0}}</math> 

==Connectedness==
'''How is this topic connected to something that you are interested in?''' 
This topic has major connections to sports. As a former high school volleyball player, it is very interesting to think about the collisions that occurred in the game and how the coefficient of restitution is related to those collisions. 
'''How is it connected to your major (Mechanical Engineering)?''' 
The coefficient of restitution is a key concept in the study of Dynamics which is the foundation of many aspects of mechanical engineering. It also has to do with applications of mechanical engineering regarding materials. 

'''Is there an interesting industrial application?'''
In sports the coefficient of restitution is an important concept especially in sports like golf or tennis where collisions are a frequent and essential part of the game. In these sports sometimes limitations are put on how high the coefficient of restitution can be for certain clubs or bats. The maximum coefficient of restitution allowed in golf for clubs is .82. There have been studies done on the coefficients for different sports balls. These studies usually follow a procedure where different balls are dropped and left to bounce. Then data regarding their initial and final velocities or how high they bounced is recorded. The images below show some of the data from one of these experiments.

[[File:sportsgraph.png|This graph shows the initial and final velocities of many different balls from different sports.]]
[[File:sportschart.png|This chart shows the relationship between different coefficients of restitution and the initial and final velocities of sports balls.]]

== See also ==

Are there related topics or categories in this wiki resource for the curious reader to explore? How does this topic fit into that context?

===Further reading===

Books, Articles or other print media on this topic

===External links===

==References==

This section contains the the references you used while writing this page

[[Category:Which Category did you place this in?]]

Coefficient of Restitution

2015-12-06T01:57:20Z

Mmoreno30: /* Difficult */

Claimed by Maria Moreno

The coefficient of restitution measures the amount of Kinetic Energy remains and how much energy is lost after a collision. It also shows the degree of elasticity of a collision. It is defined as the ratio of the difference in initial velocities divided by the difference in final velocities. The coefficient of restitution of a collision is usually between 0 and 1 except in rare cases when energy is released by a collision.
[[File:coe2.png|right]]

==The Main Idea==

The coefficient of restitution is a ratio that describes the degree of elasticity of a collision. It is used to solve problems dealing with collisions that are not perfectly elastic or inelastic.
The equation that describes the coefficient of restitution involved dividing the difference in the final velocities by the difference in the initial velocity.

Consider objects A and B with initial velocities vAi and vBi and final velocities vAf and vBf.
The coefficient of restitution, e is determined with the following formula:

<math> e = \frac{v_{Bf}-v_{Af}}{v_{Ai}-v_{Bi}} </math>

===A Mathematical Model===
Consider a perfectly elastic collision between objects A and B where vAi and vBi refer to the initial velocities of A and B and vAf and vBf refer to the final velocities of A and B.

Start with the conservation of momentum principle which states that 
<math>\vec{p_i} = \vec{p_f}</math> and 
<math>m_A\vec{v_{Ai}}+m_A\vec{v_{Bi}}=m_A\vec{v_{Af}}+m_A\vec{v_{Bf}}</math> (1) 

Remember that for a perfectly elastic collision kinetic energy is also conserved meaning: 
<math>KE_i = KE_f</math> and 
<math>\frac{1}{2}m_A v^2_{Ai}+\frac{1}{2}m_A v^2_Bi=\frac{1}{2}m_A v^2_{Af}+\frac{1}{2}m_A v^2_{Bf}</math> (2) 

Dividing (2) by (1) yields: 

<math>\vec{v_{Ai}}+\vec{v_{Bi}}=\vec{v_{Af}}+\vec{v_{Bf}}</math> 
<math>1=\frac{v_{Ai}+v_{Bi}}{v_{Af}+v_{Bf}}</math> 

<math>e=\frac{v_{Ai}+v_{Bi}}{v_{Af}+v_{Bf}}</math> 

For perfectly elastic collision e=1.

===Inelastic Collisions===

An inelastic collision is a type of collision where Kinetic Energy is not conserved. In this case the difference in initial velocities will not equal the difference in final velocities. This means that the coefficient of restitution will not equal 1.

Consider the same collision as previously described, except this collision is inelastic and Kinetic Energy is not conserved. Since some of the kinetic energy is dissipated: 
<math>\frac{1}{2}m_A v^2_{Ai}+\frac{1}{2}m_A v^2_Bi<\frac{1}{2}m_A v^2_{Af}+\frac{1}{2}m_A v^2_{Bf}</math> 
<math>\frac{v_{Ai}+v_{Bi}}{v_{Af}+v_{Bf}}<1</math> 

<math>e=\frac{v_{Ai}+v_{Bi}}{v_{Af}+v_{Bf}}<1</math> 

A perfectly inelastic collision (for example when two objects collide and then stick together) has a coefficient of restitution of 0.

===A Computational Model===

==Examples==
===Simple===
A ball is traveling at <math>4 \frac{m}{s}</math> and collides with a ball. After the collision the ball travels at <math>2.4\frac{m}{s}</math> in the opposite direction. Find the coefficient of restitution of the collision. 
Start with the definition of the coefficient of restitution: 
<math>e=\frac{v_{wall_f}-v_{ball_f}}{v_{ball_i}-v_{wall_i}}</math> 
Then plug in values and solve: 
<math>e=\frac{0-v_{ball_f}}{v_{ball_i}-0}</math> 
<math>e=\frac{-(-2.4)}{4}</math> 
<math>e=.6</math>

===Middling===
A hockey puck travels at a constant velocity <math>5\frac{m}{s}</math> and is hit by another puck with a velocity of <math>8\frac{m}{s}</math>. After the collision, it travels in the opposite direction at 4m/s. This collision has a coefficient of restitution of .55. Find the velocity of the second puck. 
Start with the definition of the coefficient of restitution. We are calling the velocity of the first puck <math>v_1</math> and the velocity of the second puck <math>v_2</math>. 
<math>e=\frac{v_{1f}-v_{2f}}{v_{2i}-v_{1i}}</math> 
Solve for <math>v_2</math> 

<math></math>

===Difficult===
[[File:Cor1.jpg|right]]
Bouncing Ball Example:

Consider a ball dropped from initial height <math>h_0</math>. When it bounces it reaches a height <math>h_f</math>. Find the coefficient of restitution of this collision. 
<math>e=\frac{v_{floor_f}-v_{ball_f}}{v_{ball_i}-v_{floor_i}}</math> 
The floor's velocity is always 0. 
<math>e=\frac{v_{ball_f}}{v_{ball_i}}</math> 
Recall the law of conservation of energy: 
<math> \Delta KE = \Delta PE</math> 
For the ball from h0 to h=0
<math> \frac{1}{2}mv_{ball_i}^2=mgh_0</math> 
<math> \frac{1}{2}v_{ball_i}^2=gh_0</math> 
<math> v_{ball_i}=2\sqrt{gh_0} </math> 
The same concept can be applied to the second bounce. 
<math>v_{ball_f}=2\sqrt{gh_f}</math>

==Connectedness==
'''How is this topic connected to something that you are interested in?''' 
This topic has major connections to sports. As a former high school volleyball player, it is very interesting to think about the collisions that occurred in the game and how the coefficient of restitution is related to those collisions. 
'''How is it connected to your major (Mechanical Engineering)?''' 
The coefficient of restitution is a key concept in the study of Dynamics which is the foundation of many aspects of mechanical engineering. It also has to do with applications of mechanical engineering regarding materials. 

'''Is there an interesting industrial application?'''
In sports the coefficient of restitution is an important concept especially in sports like golf or tennis where collisions are a frequent and essential part of the game. In these sports sometimes limitations are put on how high the coefficient of restitution can be for certain clubs or bats. The maximum coefficient of restitution allowed in golf for clubs is .82. There have been studies done on the coefficients for different sports balls. These studies usually follow a procedure where different balls are dropped and left to bounce. Then data regarding their initial and final velocities or how high they bounced is recorded. The images below show some of the data from one of these experiments.

[[File:sportsgraph.png|This graph shows the initial and final velocities of many different balls from different sports.]]
[[File:sportschart.png|This chart shows the relationship between different coefficients of restitution and the initial and final velocities of sports balls.]]

== See also ==

Are there related topics or categories in this wiki resource for the curious reader to explore? How does this topic fit into that context?

===Further reading===

Books, Articles or other print media on this topic

===External links===

==References==

This section contains the the references you used while writing this page

[[Category:Which Category did you place this in?]]

Coefficient of Restitution

2015-12-06T01:50:56Z

Mmoreno30: /* Difficult */

Claimed by Maria Moreno

The coefficient of restitution measures the amount of Kinetic Energy remains and how much energy is lost after a collision. It also shows the degree of elasticity of a collision. It is defined as the ratio of the difference in initial velocities divided by the difference in final velocities. The coefficient of restitution of a collision is usually between 0 and 1 except in rare cases when energy is released by a collision.
[[File:coe2.png|right]]

==The Main Idea==

The coefficient of restitution is a ratio that describes the degree of elasticity of a collision. It is used to solve problems dealing with collisions that are not perfectly elastic or inelastic.
The equation that describes the coefficient of restitution involved dividing the difference in the final velocities by the difference in the initial velocity.

Consider objects A and B with initial velocities vAi and vBi and final velocities vAf and vBf.
The coefficient of restitution, e is determined with the following formula:

<math> e = \frac{v_{Bf}-v_{Af}}{v_{Ai}-v_{Bi}} </math>

===A Mathematical Model===
Consider a perfectly elastic collision between objects A and B where vAi and vBi refer to the initial velocities of A and B and vAf and vBf refer to the final velocities of A and B.

Start with the conservation of momentum principle which states that 
<math>\vec{p_i} = \vec{p_f}</math> and 
<math>m_A\vec{v_{Ai}}+m_A\vec{v_{Bi}}=m_A\vec{v_{Af}}+m_A\vec{v_{Bf}}</math> (1) 

Remember that for a perfectly elastic collision kinetic energy is also conserved meaning: 
<math>KE_i = KE_f</math> and 
<math>\frac{1}{2}m_A v^2_{Ai}+\frac{1}{2}m_A v^2_Bi=\frac{1}{2}m_A v^2_{Af}+\frac{1}{2}m_A v^2_{Bf}</math> (2) 

Dividing (2) by (1) yields: 

<math>\vec{v_{Ai}}+\vec{v_{Bi}}=\vec{v_{Af}}+\vec{v_{Bf}}</math> 
<math>1=\frac{v_{Ai}+v_{Bi}}{v_{Af}+v_{Bf}}</math> 

<math>e=\frac{v_{Ai}+v_{Bi}}{v_{Af}+v_{Bf}}</math> 

For perfectly elastic collision e=1.

===Inelastic Collisions===

An inelastic collision is a type of collision where Kinetic Energy is not conserved. In this case the difference in initial velocities will not equal the difference in final velocities. This means that the coefficient of restitution will not equal 1.

Consider the same collision as previously described, except this collision is inelastic and Kinetic Energy is not conserved. Since some of the kinetic energy is dissipated: 
<math>\frac{1}{2}m_A v^2_{Ai}+\frac{1}{2}m_A v^2_Bi<\frac{1}{2}m_A v^2_{Af}+\frac{1}{2}m_A v^2_{Bf}</math> 
<math>\frac{v_{Ai}+v_{Bi}}{v_{Af}+v_{Bf}}<1</math> 

<math>e=\frac{v_{Ai}+v_{Bi}}{v_{Af}+v_{Bf}}<1</math> 

A perfectly inelastic collision (for example when two objects collide and then stick together) has a coefficient of restitution of 0.

===A Computational Model===

==Examples==
===Simple===
A ball is traveling at <math>4 \frac{m}{s}</math> and collides with a ball. After the collision the ball travels at <math>2.4\frac{m}{s}</math> in the opposite direction. Find the coefficient of restitution of the collision. 
Start with the definition of the coefficient of restitution: 
<math>e=\frac{v_{wall_f}-v_{ball_f}}{v_{ball_i}-v_{wall_i}}</math> 
Then plug in values and solve: 
<math>e=\frac{0-v_{ball_f}}{v_{ball_i}-0}</math> 
<math>e=\frac{-(-2.4)}{4}</math> 
<math>e=.6</math>

===Middling===
A hockey puck travels at a constant velocity <math>5\frac{m}{s}</math> and is hit by another puck with a velocity of <math>8\frac{m}{s}</math>. After the collision, it travels in the opposite direction at 4m/s. This collision has a coefficient of restitution of .55. Find the velocity of the second puck. 
Start with the definition of the coefficient of restitution. We are calling the velocity of the first puck <math>v_1</math> and the velocity of the second puck <math>v_2</math>. 
<math>e=\frac{v_{1f}-v_{2f}}{v_{2i}-v_{1i}}</math> 
Solve for <math>v_2</math> 

<math></math>

===Difficult===
[[File:Cor1.jpg|right]]
Bouncing Ball Example:

Consider a ball dropped from initial height <math>h_0</math>. When it bounces it reaches a height <math>h_1</math>. Find the coefficient of restitution of this collision. 
<math>e=\frac{v_{floor_f}-v_{ball_f}}{v_{ball_i}-v_{floor_i}}</math> 
The floor's velocity is always 0. 
<math>e=\frac{v_{ball_f}}{v_{ball_i}}</math> 
Recall the law of conservation of energy: 
<math> \Delta KE = \Delta PE</math>

==Connectedness==
'''How is this topic connected to something that you are interested in?''' 
This topic has major connections to sports. As a former high school volleyball player, it is very interesting to think about the collisions that occurred in the game and how the coefficient of restitution is related to those collisions. 
'''How is it connected to your major (Mechanical Engineering)?''' 
The coefficient of restitution is a key concept in the study of Dynamics which is the foundation of many aspects of mechanical engineering. It also has to do with applications of mechanical engineering regarding materials. 

'''Is there an interesting industrial application?'''
In sports the coefficient of restitution is an important concept especially in sports like golf or tennis where collisions are a frequent and essential part of the game. In these sports sometimes limitations are put on how high the coefficient of restitution can be for certain clubs or bats. The maximum coefficient of restitution allowed in golf for clubs is .82. There have been studies done on the coefficients for different sports balls. These studies usually follow a procedure where different balls are dropped and left to bounce. Then data regarding their initial and final velocities or how high they bounced is recorded. The images below show some of the data from one of these experiments.

[[File:sportsgraph.png|This graph shows the initial and final velocities of many different balls from different sports.]]
[[File:sportschart.png|This chart shows the relationship between different coefficients of restitution and the initial and final velocities of sports balls.]]

== See also ==

Are there related topics or categories in this wiki resource for the curious reader to explore? How does this topic fit into that context?

===Further reading===

Books, Articles or other print media on this topic

===External links===

==References==

This section contains the the references you used while writing this page

[[Category:Which Category did you place this in?]]

Coefficient of Restitution

2015-12-06T01:50:35Z

Mmoreno30: /* Difficult */

Claimed by Maria Moreno

The coefficient of restitution measures the amount of Kinetic Energy remains and how much energy is lost after a collision. It also shows the degree of elasticity of a collision. It is defined as the ratio of the difference in initial velocities divided by the difference in final velocities. The coefficient of restitution of a collision is usually between 0 and 1 except in rare cases when energy is released by a collision.
[[File:coe2.png|right]]

==The Main Idea==

The coefficient of restitution is a ratio that describes the degree of elasticity of a collision. It is used to solve problems dealing with collisions that are not perfectly elastic or inelastic.
The equation that describes the coefficient of restitution involved dividing the difference in the final velocities by the difference in the initial velocity.

Consider objects A and B with initial velocities vAi and vBi and final velocities vAf and vBf.
The coefficient of restitution, e is determined with the following formula:

<math> e = \frac{v_{Bf}-v_{Af}}{v_{Ai}-v_{Bi}} </math>

===A Mathematical Model===
Consider a perfectly elastic collision between objects A and B where vAi and vBi refer to the initial velocities of A and B and vAf and vBf refer to the final velocities of A and B.

Start with the conservation of momentum principle which states that 
<math>\vec{p_i} = \vec{p_f}</math> and 
<math>m_A\vec{v_{Ai}}+m_A\vec{v_{Bi}}=m_A\vec{v_{Af}}+m_A\vec{v_{Bf}}</math> (1) 

Remember that for a perfectly elastic collision kinetic energy is also conserved meaning: 
<math>KE_i = KE_f</math> and 
<math>\frac{1}{2}m_A v^2_{Ai}+\frac{1}{2}m_A v^2_Bi=\frac{1}{2}m_A v^2_{Af}+\frac{1}{2}m_A v^2_{Bf}</math> (2) 

Dividing (2) by (1) yields: 

<math>\vec{v_{Ai}}+\vec{v_{Bi}}=\vec{v_{Af}}+\vec{v_{Bf}}</math> 
<math>1=\frac{v_{Ai}+v_{Bi}}{v_{Af}+v_{Bf}}</math> 

<math>e=\frac{v_{Ai}+v_{Bi}}{v_{Af}+v_{Bf}}</math> 

For perfectly elastic collision e=1.

===Inelastic Collisions===

An inelastic collision is a type of collision where Kinetic Energy is not conserved. In this case the difference in initial velocities will not equal the difference in final velocities. This means that the coefficient of restitution will not equal 1.

Consider the same collision as previously described, except this collision is inelastic and Kinetic Energy is not conserved. Since some of the kinetic energy is dissipated: 
<math>\frac{1}{2}m_A v^2_{Ai}+\frac{1}{2}m_A v^2_Bi<\frac{1}{2}m_A v^2_{Af}+\frac{1}{2}m_A v^2_{Bf}</math> 
<math>\frac{v_{Ai}+v_{Bi}}{v_{Af}+v_{Bf}}<1</math> 

<math>e=\frac{v_{Ai}+v_{Bi}}{v_{Af}+v_{Bf}}<1</math> 

A perfectly inelastic collision (for example when two objects collide and then stick together) has a coefficient of restitution of 0.

===A Computational Model===

==Examples==
===Simple===
A ball is traveling at <math>4 \frac{m}{s}</math> and collides with a ball. After the collision the ball travels at <math>2.4\frac{m}{s}</math> in the opposite direction. Find the coefficient of restitution of the collision. 
Start with the definition of the coefficient of restitution: 
<math>e=\frac{v_{wall_f}-v_{ball_f}}{v_{ball_i}-v_{wall_i}}</math> 
Then plug in values and solve: 
<math>e=\frac{0-v_{ball_f}}{v_{ball_i}-0}</math> 
<math>e=\frac{-(-2.4)}{4}</math> 
<math>e=.6</math>

===Middling===
A hockey puck travels at a constant velocity <math>5\frac{m}{s}</math> and is hit by another puck with a velocity of <math>8\frac{m}{s}</math>. After the collision, it travels in the opposite direction at 4m/s. This collision has a coefficient of restitution of .55. Find the velocity of the second puck. 
Start with the definition of the coefficient of restitution. We are calling the velocity of the first puck <math>v_1</math> and the velocity of the second puck <math>v_2</math>. 
<math>e=\frac{v_{1f}-v_{2f}}{v_{2i}-v_{1i}}</math> 
Solve for <math>v_2</math> 

<math></math>

===Difficult===
[[File:Cor1.jpg|right]]
Bouncing Ball Example:

Consider a ball dropped from initial height <math>h_0</math>. When it bounces it reaches a height <math>h_1</math>. Find the coefficient of restitution of this collision. 
<math>e=\frac{v_{floor_f}-v_{ball_f}}{v_{ball_i}-v_{floor_i}}</math> 
The floor's velocity is always 0. 
<math>e=\frac{v_{ball_f}}{v_{ball_i}}</math> 
Recall the law of conservation of energy: 
<math> \delta KE = \delta PE</math>

==Connectedness==
'''How is this topic connected to something that you are interested in?''' 
This topic has major connections to sports. As a former high school volleyball player, it is very interesting to think about the collisions that occurred in the game and how the coefficient of restitution is related to those collisions. 
'''How is it connected to your major (Mechanical Engineering)?''' 
The coefficient of restitution is a key concept in the study of Dynamics which is the foundation of many aspects of mechanical engineering. It also has to do with applications of mechanical engineering regarding materials. 

'''Is there an interesting industrial application?'''
In sports the coefficient of restitution is an important concept especially in sports like golf or tennis where collisions are a frequent and essential part of the game. In these sports sometimes limitations are put on how high the coefficient of restitution can be for certain clubs or bats. The maximum coefficient of restitution allowed in golf for clubs is .82. There have been studies done on the coefficients for different sports balls. These studies usually follow a procedure where different balls are dropped and left to bounce. Then data regarding their initial and final velocities or how high they bounced is recorded. The images below show some of the data from one of these experiments.

[[File:sportsgraph.png|This graph shows the initial and final velocities of many different balls from different sports.]]
[[File:sportschart.png|This chart shows the relationship between different coefficients of restitution and the initial and final velocities of sports balls.]]

== See also ==

Are there related topics or categories in this wiki resource for the curious reader to explore? How does this topic fit into that context?

===Further reading===

Books, Articles or other print media on this topic

===External links===

==References==

This section contains the the references you used while writing this page

[[Category:Which Category did you place this in?]]

Coefficient of Restitution

2015-12-06T01:49:45Z

Mmoreno30: /* Examples */

Claimed by Maria Moreno

The coefficient of restitution measures the amount of Kinetic Energy remains and how much energy is lost after a collision. It also shows the degree of elasticity of a collision. It is defined as the ratio of the difference in initial velocities divided by the difference in final velocities. The coefficient of restitution of a collision is usually between 0 and 1 except in rare cases when energy is released by a collision.
[[File:coe2.png|right]]

==The Main Idea==

The coefficient of restitution is a ratio that describes the degree of elasticity of a collision. It is used to solve problems dealing with collisions that are not perfectly elastic or inelastic.
The equation that describes the coefficient of restitution involved dividing the difference in the final velocities by the difference in the initial velocity.

Consider objects A and B with initial velocities vAi and vBi and final velocities vAf and vBf.
The coefficient of restitution, e is determined with the following formula:

<math> e = \frac{v_{Bf}-v_{Af}}{v_{Ai}-v_{Bi}} </math>

===A Mathematical Model===
Consider a perfectly elastic collision between objects A and B where vAi and vBi refer to the initial velocities of A and B and vAf and vBf refer to the final velocities of A and B.

Start with the conservation of momentum principle which states that 
<math>\vec{p_i} = \vec{p_f}</math> and 
<math>m_A\vec{v_{Ai}}+m_A\vec{v_{Bi}}=m_A\vec{v_{Af}}+m_A\vec{v_{Bf}}</math> (1) 

Remember that for a perfectly elastic collision kinetic energy is also conserved meaning: 
<math>KE_i = KE_f</math> and 
<math>\frac{1}{2}m_A v^2_{Ai}+\frac{1}{2}m_A v^2_Bi=\frac{1}{2}m_A v^2_{Af}+\frac{1}{2}m_A v^2_{Bf}</math> (2) 

Dividing (2) by (1) yields: 

<math>\vec{v_{Ai}}+\vec{v_{Bi}}=\vec{v_{Af}}+\vec{v_{Bf}}</math> 
<math>1=\frac{v_{Ai}+v_{Bi}}{v_{Af}+v_{Bf}}</math> 

<math>e=\frac{v_{Ai}+v_{Bi}}{v_{Af}+v_{Bf}}</math> 

For perfectly elastic collision e=1.

===Inelastic Collisions===

An inelastic collision is a type of collision where Kinetic Energy is not conserved. In this case the difference in initial velocities will not equal the difference in final velocities. This means that the coefficient of restitution will not equal 1.

Consider the same collision as previously described, except this collision is inelastic and Kinetic Energy is not conserved. Since some of the kinetic energy is dissipated: 
<math>\frac{1}{2}m_A v^2_{Ai}+\frac{1}{2}m_A v^2_Bi<\frac{1}{2}m_A v^2_{Af}+\frac{1}{2}m_A v^2_{Bf}</math> 
<math>\frac{v_{Ai}+v_{Bi}}{v_{Af}+v_{Bf}}<1</math> 

<math>e=\frac{v_{Ai}+v_{Bi}}{v_{Af}+v_{Bf}}<1</math> 

A perfectly inelastic collision (for example when two objects collide and then stick together) has a coefficient of restitution of 0.

===A Computational Model===

==Examples==
===Simple===
A ball is traveling at <math>4 \frac{m}{s}</math> and collides with a ball. After the collision the ball travels at <math>2.4\frac{m}{s}</math> in the opposite direction. Find the coefficient of restitution of the collision. 
Start with the definition of the coefficient of restitution: 
<math>e=\frac{v_{wall_f}-v_{ball_f}}{v_{ball_i}-v_{wall_i}}</math> 
Then plug in values and solve: 
<math>e=\frac{0-v_{ball_f}}{v_{ball_i}-0}</math> 
<math>e=\frac{-(-2.4)}{4}</math> 
<math>e=.6</math>

===Middling===
A hockey puck travels at a constant velocity <math>5\frac{m}{s}</math> and is hit by another puck with a velocity of <math>8\frac{m}{s}</math>. After the collision, it travels in the opposite direction at 4m/s. This collision has a coefficient of restitution of .55. Find the velocity of the second puck. 
Start with the definition of the coefficient of restitution. We are calling the velocity of the first puck <math>v_1</math> and the velocity of the second puck <math>v_2</math>. 
<math>e=\frac{v_{1f}-v_{2f}}{v_{2i}-v_{1i}}</math> 
Solve for <math>v_2</math> 

<math></math>

===Difficult===
[[File:Cor1.jpg|right]]
Bouncing Ball Example:

Consider a ball dropped from initial height <math>h_0</math>. When it bounces it reaches a height <math>h_1</math>. Find the coefficient of restitution of this collision. 
<math>e=\frac{v_{floor_f}-v_{ball_f}}{v_{ball_i}-v_{floor_i}}</math> 
The floor's velocity is always 0. 
<math>e=\frac{v_{ball_f}}{v_{ball_i}}</math> 
Recall the law of conservation of energy: 
<math> delta KE = delta PE</math>

==Connectedness==
'''How is this topic connected to something that you are interested in?''' 
This topic has major connections to sports. As a former high school volleyball player, it is very interesting to think about the collisions that occurred in the game and how the coefficient of restitution is related to those collisions. 
'''How is it connected to your major (Mechanical Engineering)?''' 
The coefficient of restitution is a key concept in the study of Dynamics which is the foundation of many aspects of mechanical engineering. It also has to do with applications of mechanical engineering regarding materials. 

'''Is there an interesting industrial application?'''
In sports the coefficient of restitution is an important concept especially in sports like golf or tennis where collisions are a frequent and essential part of the game. In these sports sometimes limitations are put on how high the coefficient of restitution can be for certain clubs or bats. The maximum coefficient of restitution allowed in golf for clubs is .82. There have been studies done on the coefficients for different sports balls. These studies usually follow a procedure where different balls are dropped and left to bounce. Then data regarding their initial and final velocities or how high they bounced is recorded. The images below show some of the data from one of these experiments.

[[File:sportsgraph.png|This graph shows the initial and final velocities of many different balls from different sports.]]
[[File:sportschart.png|This chart shows the relationship between different coefficients of restitution and the initial and final velocities of sports balls.]]

== See also ==

Are there related topics or categories in this wiki resource for the curious reader to explore? How does this topic fit into that context?

===Further reading===

Books, Articles or other print media on this topic

===External links===

==References==

This section contains the the references you used while writing this page

[[Category:Which Category did you place this in?]]

Coefficient of Restitution

2015-12-06T01:33:26Z

Mmoreno30: /* Examples */

Claimed by Maria Moreno

The coefficient of restitution measures the amount of Kinetic Energy remains and how much energy is lost after a collision. It also shows the degree of elasticity of a collision. It is defined as the ratio of the difference in initial velocities divided by the difference in final velocities. The coefficient of restitution of a collision is usually between 0 and 1 except in rare cases when energy is released by a collision.
[[File:coe2.png|right]]

==The Main Idea==

The coefficient of restitution is a ratio that describes the degree of elasticity of a collision. It is used to solve problems dealing with collisions that are not perfectly elastic or inelastic.
The equation that describes the coefficient of restitution involved dividing the difference in the final velocities by the difference in the initial velocity.

Consider objects A and B with initial velocities vAi and vBi and final velocities vAf and vBf.
The coefficient of restitution, e is determined with the following formula:

<math> e = \frac{v_{Bf}-v_{Af}}{v_{Ai}-v_{Bi}} </math>

===A Mathematical Model===
Consider a perfectly elastic collision between objects A and B where vAi and vBi refer to the initial velocities of A and B and vAf and vBf refer to the final velocities of A and B.

Start with the conservation of momentum principle which states that 
<math>\vec{p_i} = \vec{p_f}</math> and 
<math>m_A\vec{v_{Ai}}+m_A\vec{v_{Bi}}=m_A\vec{v_{Af}}+m_A\vec{v_{Bf}}</math> (1) 

Remember that for a perfectly elastic collision kinetic energy is also conserved meaning: 
<math>KE_i = KE_f</math> and 
<math>\frac{1}{2}m_A v^2_{Ai}+\frac{1}{2}m_A v^2_Bi=\frac{1}{2}m_A v^2_{Af}+\frac{1}{2}m_A v^2_{Bf}</math> (2) 

Dividing (2) by (1) yields: 

<math>\vec{v_{Ai}}+\vec{v_{Bi}}=\vec{v_{Af}}+\vec{v_{Bf}}</math> 
<math>1=\frac{v_{Ai}+v_{Bi}}{v_{Af}+v_{Bf}}</math> 

<math>e=\frac{v_{Ai}+v_{Bi}}{v_{Af}+v_{Bf}}</math> 

For perfectly elastic collision e=1.

===Inelastic Collisions===

An inelastic collision is a type of collision where Kinetic Energy is not conserved. In this case the difference in initial velocities will not equal the difference in final velocities. This means that the coefficient of restitution will not equal 1.

Consider the same collision as previously described, except this collision is inelastic and Kinetic Energy is not conserved. Since some of the kinetic energy is dissipated: 
<math>\frac{1}{2}m_A v^2_{Ai}+\frac{1}{2}m_A v^2_Bi<\frac{1}{2}m_A v^2_{Af}+\frac{1}{2}m_A v^2_{Bf}</math> 
<math>\frac{v_{Ai}+v_{Bi}}{v_{Af}+v_{Bf}}<1</math> 

<math>e=\frac{v_{Ai}+v_{Bi}}{v_{Af}+v_{Bf}}<1</math> 

A perfectly inelastic collision (for example when two objects collide and then stick together) has a coefficient of restitution of 0.

===A Computational Model===

==Examples==
===Simple===
A ball is traveling at <math>4 \frac{m}{s}</math> and collides with a ball. After the collision the ball travels at <math>2.4\frac{m}{s}</math> in the opposite direction. Find the coefficient of restitution of the collision. 
Start with the definition of the coefficient of restitution: 
<math>e=\frac{v_{wall_f}-v_{ball_f}}{v_{ball_i}-v_{wall_i}}</math> 
Then plug in values and solve: 
<math>e=\frac{0-v_{ball_f}}{v_{ball_i}-0}</math> 
<math>e=\frac{-(-2.4)}{4}</math> 
<math>e=.6</math>

===Middling===
A hockey puck travels at a constant velocity <math>5\frac{m}{s}</math> and is hit by another puck with a velocity of <math>8\frac{m}{s}</math>. After the collision, it travels in the opposite direction at 4m/s. This collision has a coefficient of restitution of .55. Find the velocity of the second puck. 
Start with the definition of the coefficient of restitution. We are calling the velocity of the first puck <math>v_1</math> and the velocity of the second puck <math>v_2</math>. 
<math>e=\frac{v_{1f}-v_{2f}}{v_{2i}-v_{1i}}</math> 
Solve for <math>v_2</math> 
<math></math>

===Difficult===
[[File:Cor1.jpg|right]]
Bouncing Ball Example:

Consider a ball dropped from initial height <math>h_0</math>. When it bounces it reaches a height <math>h_1</math>. Find the coefficient of restitution of this collision.

==Connectedness==
'''How is this topic connected to something that you are interested in?''' 
This topic has major connections to sports. As a former high school volleyball player, it is very interesting to think about the collisions that occurred in the game and how the coefficient of restitution is related to those collisions. 
'''How is it connected to your major (Mechanical Engineering)?''' 
The coefficient of restitution is a key concept in the study of Dynamics which is the foundation of many aspects of mechanical engineering. It also has to do with applications of mechanical engineering regarding materials. 

'''Is there an interesting industrial application?'''
In sports the coefficient of restitution is an important concept especially in sports like golf or tennis where collisions are a frequent and essential part of the game. In these sports sometimes limitations are put on how high the coefficient of restitution can be for certain clubs or bats. The maximum coefficient of restitution allowed in golf for clubs is .82. There have been studies done on the coefficients for different sports balls. These studies usually follow a procedure where different balls are dropped and left to bounce. Then data regarding their initial and final velocities or how high they bounced is recorded. The images below show some of the data from one of these experiments.

[[File:sportsgraph.png|This graph shows the initial and final velocities of many different balls from different sports.]]
[[File:sportschart.png|This chart shows the relationship between different coefficients of restitution and the initial and final velocities of sports balls.]]

== See also ==

Are there related topics or categories in this wiki resource for the curious reader to explore? How does this topic fit into that context?

===Further reading===

Books, Articles or other print media on this topic

===External links===

==References==

This section contains the the references you used while writing this page

[[Category:Which Category did you place this in?]]

Coefficient of Restitution

2015-12-06T01:21:57Z

Mmoreno30: /* Connectedness */

Claimed by Maria Moreno

The coefficient of restitution measures the amount of Kinetic Energy remains and how much energy is lost after a collision. It also shows the degree of elasticity of a collision. It is defined as the ratio of the difference in initial velocities divided by the difference in final velocities. The coefficient of restitution of a collision is usually between 0 and 1 except in rare cases when energy is released by a collision.
[[File:coe2.png|right]]

==The Main Idea==

The coefficient of restitution is a ratio that describes the degree of elasticity of a collision. It is used to solve problems dealing with collisions that are not perfectly elastic or inelastic.
The equation that describes the coefficient of restitution involved dividing the difference in the final velocities by the difference in the initial velocity.

Consider objects A and B with initial velocities vAi and vBi and final velocities vAf and vBf.
The coefficient of restitution, e is determined with the following formula:

<math> e = \frac{v_{Bf}-v_{Af}}{v_{Ai}-v_{Bi}} </math>

===A Mathematical Model===
Consider a perfectly elastic collision between objects A and B where vAi and vBi refer to the initial velocities of A and B and vAf and vBf refer to the final velocities of A and B.

Start with the conservation of momentum principle which states that 
<math>\vec{p_i} = \vec{p_f}</math> and 
<math>m_A\vec{v_{Ai}}+m_A\vec{v_{Bi}}=m_A\vec{v_{Af}}+m_A\vec{v_{Bf}}</math> (1) 

Remember that for a perfectly elastic collision kinetic energy is also conserved meaning: 
<math>KE_i = KE_f</math> and 
<math>\frac{1}{2}m_A v^2_{Ai}+\frac{1}{2}m_A v^2_Bi=\frac{1}{2}m_A v^2_{Af}+\frac{1}{2}m_A v^2_{Bf}</math> (2) 

Dividing (2) by (1) yields: 

<math>\vec{v_{Ai}}+\vec{v_{Bi}}=\vec{v_{Af}}+\vec{v_{Bf}}</math> 
<math>1=\frac{v_{Ai}+v_{Bi}}{v_{Af}+v_{Bf}}</math> 

<math>e=\frac{v_{Ai}+v_{Bi}}{v_{Af}+v_{Bf}}</math> 

For perfectly elastic collision e=1.

===Inelastic Collisions===

An inelastic collision is a type of collision where Kinetic Energy is not conserved. In this case the difference in initial velocities will not equal the difference in final velocities. This means that the coefficient of restitution will not equal 1.

Consider the same collision as previously described, except this collision is inelastic and Kinetic Energy is not conserved. Since some of the kinetic energy is dissipated: 
<math>\frac{1}{2}m_A v^2_{Ai}+\frac{1}{2}m_A v^2_Bi<\frac{1}{2}m_A v^2_{Af}+\frac{1}{2}m_A v^2_{Bf}</math> 
<math>\frac{v_{Ai}+v_{Bi}}{v_{Af}+v_{Bf}}<1</math> 

<math>e=\frac{v_{Ai}+v_{Bi}}{v_{Af}+v_{Bf}}<1</math> 

A perfectly inelastic collision (for example when two objects collide and then stick together) has a coefficient of restitution of 0.

===A Computational Model===

==Examples==

Be sure to show all steps in your solution and include diagrams whenever possible

===Simple===
A ball is traveling at <math>4 \frac{m}{s}</math> and collides with a ball. After the collision the ball travels at <math>2.4\frac{m}{s}</math> in the opposite direction. Find the coefficient of restitution of the collision. 
Start with the definition of the coefficient of restitution: 
<math>e=\frac{v_{wall_f}-v_{ball_f}}{v_{ball_i}-v_{wall_i}}</math> 
Then plug in values and solve: 
<math>e=\frac{0-v_{ball_f}}{v_{ball_i}-0}</math> 
<math>e=\frac{-(-2.4)}{4}</math> 
<math>e=.6</math>

===Middling===
A hockey puck travels at a constant velocity <math>5\frac{m}{s}</math> and is hit by another puck with a velocity of <math>8\frac{m}{s}</math>. After the collision, it travels in the opposite direction at 4m/s. This collision has a coefficient of restitution of .55. Find the velocity of the second puck. 
Start with the definition of the coefficient of restitution. We are calling the velocity of the first puck <math>v_1</math> and the velocity of the second puck <math>v_2</math>. 
<math>e=\frac{v_{1f}-v_{2f}}{v_{2i}-v_{1i}}</math> 
Solve for <math>v_2</math> 
<math></math>

===Difficult===
[[File:Cor1.jpg|right]]
Bouncing Ball Example:

Consider a ball dropped from initial height <math>h_0</math>. When it bounces it reaches a height <math>h_1</math>. Find the coefficient of restitution of this collision.

==Connectedness==
'''How is this topic connected to something that you are interested in?''' 
This topic has major connections to sports. As a former high school volleyball player, it is very interesting to think about the collisions that occurred in the game and how the coefficient of restitution is related to those collisions. 
'''How is it connected to your major (Mechanical Engineering)?''' 
The coefficient of restitution is a key concept in the study of Dynamics which is the foundation of many aspects of mechanical engineering. It also has to do with applications of mechanical engineering regarding materials. 

'''Is there an interesting industrial application?'''
In sports the coefficient of restitution is an important concept especially in sports like golf or tennis where collisions are a frequent and essential part of the game. In these sports sometimes limitations are put on how high the coefficient of restitution can be for certain clubs or bats. The maximum coefficient of restitution allowed in golf for clubs is .82. There have been studies done on the coefficients for different sports balls. These studies usually follow a procedure where different balls are dropped and left to bounce. Then data regarding their initial and final velocities or how high they bounced is recorded. The images below show some of the data from one of these experiments.

[[File:sportsgraph.png|This graph shows the initial and final velocities of many different balls from different sports.]]
[[File:sportschart.png|This chart shows the relationship between different coefficients of restitution and the initial and final velocities of sports balls.]]

== See also ==

Are there related topics or categories in this wiki resource for the curious reader to explore? How does this topic fit into that context?

===Further reading===

Books, Articles or other print media on this topic

===External links===

==References==

This section contains the the references you used while writing this page

[[Category:Which Category did you place this in?]]

Coefficient of Restitution

2015-12-06T01:20:15Z

Mmoreno30: /* Connectedness */

Claimed by Maria Moreno

The coefficient of restitution measures the amount of Kinetic Energy remains and how much energy is lost after a collision. It also shows the degree of elasticity of a collision. It is defined as the ratio of the difference in initial velocities divided by the difference in final velocities. The coefficient of restitution of a collision is usually between 0 and 1 except in rare cases when energy is released by a collision.
[[File:coe2.png|right]]

==The Main Idea==

The coefficient of restitution is a ratio that describes the degree of elasticity of a collision. It is used to solve problems dealing with collisions that are not perfectly elastic or inelastic.
The equation that describes the coefficient of restitution involved dividing the difference in the final velocities by the difference in the initial velocity.

Consider objects A and B with initial velocities vAi and vBi and final velocities vAf and vBf.
The coefficient of restitution, e is determined with the following formula:

<math> e = \frac{v_{Bf}-v_{Af}}{v_{Ai}-v_{Bi}} </math>

===A Mathematical Model===
Consider a perfectly elastic collision between objects A and B where vAi and vBi refer to the initial velocities of A and B and vAf and vBf refer to the final velocities of A and B.

Start with the conservation of momentum principle which states that 
<math>\vec{p_i} = \vec{p_f}</math> and 
<math>m_A\vec{v_{Ai}}+m_A\vec{v_{Bi}}=m_A\vec{v_{Af}}+m_A\vec{v_{Bf}}</math> (1) 

Remember that for a perfectly elastic collision kinetic energy is also conserved meaning: 
<math>KE_i = KE_f</math> and 
<math>\frac{1}{2}m_A v^2_{Ai}+\frac{1}{2}m_A v^2_Bi=\frac{1}{2}m_A v^2_{Af}+\frac{1}{2}m_A v^2_{Bf}</math> (2) 

Dividing (2) by (1) yields: 

<math>\vec{v_{Ai}}+\vec{v_{Bi}}=\vec{v_{Af}}+\vec{v_{Bf}}</math> 
<math>1=\frac{v_{Ai}+v_{Bi}}{v_{Af}+v_{Bf}}</math> 

<math>e=\frac{v_{Ai}+v_{Bi}}{v_{Af}+v_{Bf}}</math> 

For perfectly elastic collision e=1.

===Inelastic Collisions===

An inelastic collision is a type of collision where Kinetic Energy is not conserved. In this case the difference in initial velocities will not equal the difference in final velocities. This means that the coefficient of restitution will not equal 1.

Consider the same collision as previously described, except this collision is inelastic and Kinetic Energy is not conserved. Since some of the kinetic energy is dissipated: 
<math>\frac{1}{2}m_A v^2_{Ai}+\frac{1}{2}m_A v^2_Bi<\frac{1}{2}m_A v^2_{Af}+\frac{1}{2}m_A v^2_{Bf}</math> 
<math>\frac{v_{Ai}+v_{Bi}}{v_{Af}+v_{Bf}}<1</math> 

<math>e=\frac{v_{Ai}+v_{Bi}}{v_{Af}+v_{Bf}}<1</math> 

A perfectly inelastic collision (for example when two objects collide and then stick together) has a coefficient of restitution of 0.

===A Computational Model===

==Examples==

Be sure to show all steps in your solution and include diagrams whenever possible

===Simple===
A ball is traveling at <math>4 \frac{m}{s}</math> and collides with a ball. After the collision the ball travels at <math>2.4\frac{m}{s}</math> in the opposite direction. Find the coefficient of restitution of the collision. 
Start with the definition of the coefficient of restitution: 
<math>e=\frac{v_{wall_f}-v_{ball_f}}{v_{ball_i}-v_{wall_i}}</math> 
Then plug in values and solve: 
<math>e=\frac{0-v_{ball_f}}{v_{ball_i}-0}</math> 
<math>e=\frac{-(-2.4)}{4}</math> 
<math>e=.6</math>

===Middling===
A hockey puck travels at a constant velocity <math>5\frac{m}{s}</math> and is hit by another puck with a velocity of <math>8\frac{m}{s}</math>. After the collision, it travels in the opposite direction at 4m/s. This collision has a coefficient of restitution of .55. Find the velocity of the second puck. 
Start with the definition of the coefficient of restitution. We are calling the velocity of the first puck <math>v_1</math> and the velocity of the second puck <math>v_2</math>. 
<math>e=\frac{v_{1f}-v_{2f}}{v_{2i}-v_{1i}}</math> 
Solve for <math>v_2</math> 
<math></math>

===Difficult===
[[File:Cor1.jpg|right]]
Bouncing Ball Example:

Consider a ball dropped from initial height <math>h_0</math>. When it bounces it reaches a height <math>h_1</math>. Find the coefficient of restitution of this collision.

==Connectedness==
'''How is this topic connected to something that you are interested in?''' 
This topic has major connections to sports. As a former high school volleyball player, it is very interesting to think about the collisions that occurred in the game and how the coefficient of restitution is related to those collisions. 
'''How is it connected to your major (Mechanical Engineering)?''' 
The coefficient of restitution is a key concept in the study of Dynamics which is the foundation of many aspects of mechanical engineering. 

'''Is there an interesting industrial application?'''
In sports the coefficient of restitution is an important concept especially in sports like golf or tennis where collisions are a frequent and essential part of the game. In these sports sometimes limitations are put on how high the coefficient of restitution can be for certain clubs or bats. The maximum coefficient of restitution allowed in golf for clubs is .82. There have been studies done on the coefficients for different sports balls. These studies usually follow a procedure where different balls are dropped and left to bounce. Then data regarding their initial and final velocities or how high they bounced is recorded. The images below show some of the data from one of these experiments.

[[File:sportsgraph.png|This graph shows the initial and final velocities of many different balls from different sports.]]
[[File:sportschart.png|This chart shows the relationship between different coefficients of restitution and the initial and final velocities of sports balls.]]

== See also ==

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===Further reading===

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==References==

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[[Category:Which Category did you place this in?]]

Coefficient of Restitution

2015-12-06T01:18:40Z

Mmoreno30: /* Middling */

Coefficient of Restitution

2015-12-06T01:16:14Z

Mmoreno30: /* Inelastic Collisions */

Coefficient of Restitution

2015-12-06T01:16:01Z

Mmoreno30: /* Inelastic Collisions */

Coefficient of Restitution

2015-12-06T01:15:36Z

Mmoreno30: /* Inelastic Collisions */

Coefficient of Restitution

2015-12-06T01:15:02Z

Mmoreno30: /* Connectedness */

Coefficient of Restitution

2015-12-06T01:13:32Z

Mmoreno30: /* Connectedness */

Coefficient of Restitution

2015-12-06T01:11:39Z

Mmoreno30: /* Connectedness */

Coefficient of Restitution

2015-12-06T01:07:20Z

Mmoreno30: /* Simple */

Coefficient of Restitution

2015-12-06T01:06:09Z

Mmoreno30: /* Examples */

Coefficient of Restitution

2015-12-05T23:56:46Z

Mmoreno30: /* Difficult */

Coefficient of Restitution

2015-12-05T23:56:31Z

Mmoreno30: /* Difficult */

Coefficient of Restitution

2015-12-05T23:55:13Z

Mmoreno30: /* Simple */