Head-on Collision of Equal Masses - Revision history

Sydney: /* Connectedness */

2019-07-08T20:43:46Z

Connectedness

← Older revision		Revision as of 16:43, 8 July 2019
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	==Connectedness==		==Connectedness==
	[[File:Billiards Rack.JPG\|Billiards Rack\|left\|200 px]] A real world application of head-on collisions of equal mass are billiard balls (playing pool). All the balls have equal mass, and one ball is shot towards another, which is stationary. When the two collide, the ball that is hit will move at a 90 angle from the direction the striking ball comes from. Keeping this mind and with a little practice one's level of playing billiards should increase.		[[File:Billiards Rack.JPG\|Billiards Rack\|left\|200 px]] A real world application of head-on collisions of equal mass are billiard balls (playing pool). All the balls have equal mass, and one ball is shot towards another, which is stationary. When the two collide, the ball that is hit will move at a 180 degree angle from the direction the striking ball comes from. Keeping this mind and with a little practice one's level of playing billiards should increase.

	Another fun application of collisions of equal mass is bumper cars. While the tracks are not typically low enough friction to discount it, the cars approximately all have the same mass and collide frequently. Changes in momentum with collisions are easy to exemplify with bumper cars. For example, it is easy to observe how two bumper cars going the same speed collide and bounce backwards; while one fast car hitting a stationary one causes a larger transfer of energy.		Another fun application of collisions of equal mass is bumper cars. While the tracks are not typically low enough friction to discount it, the cars approximately all have the same mass and collide frequently. Changes in momentum with collisions are easy to exemplify with bumper cars. For example, it is easy to observe how two bumper cars going the same speed collide and bounce backwards; while one fast car hitting a stationary one causes a larger transfer of energy.

Sydney: /* Difficult */

2019-07-08T20:41:35Z

Difficult

← Older revision		Revision as of 16:41, 8 July 2019
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	'''Answer'''		'''Answer'''

	This question requires ~~a lot of steps with~~ a lot of information to keep straight. Let's work backwards from the final question to find the information we need.		This question requires a lot of information to keep straight. Let's work backwards from the final question to find the information we need.

	We can use the kinematic equation <math>d = \frac{v_0+v_f}{2}t</math> to find the distance to the pocket. We have <math>t = 2.5</math> and <math>v_f = v_0</math> since friction is negligible. Now we have to find <math>v_0</math> by starting with the first collision.		We can use the kinematic equation <math>d = \frac{v_0+v_f}{2}t</math> to find the distance to the pocket. We have <math>t = 2.5</math> and <math>v_f = v_0</math> since friction is negligible. Now we have to find <math>v_0</math> by starting with the first collision.

Sydney: /* Main Idea */

2019-07-08T20:38:05Z

Main Idea

← Older revision		Revision as of 16:38, 8 July 2019
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	A collision is a brief interaction between large forces. A head-on collision could be between two carts rolling or sliding on a track with low friction, billiard balls, hockey pucks, or vehicles hitting each other head-on.		A collision is a brief interaction between large forces. A head-on collision could be between two carts rolling or sliding on a track with low friction, billiard balls, hockey pucks, or vehicles hitting each other head-on.

	In terms of the two carts of equal masses example, the two carts are the system. The Momentum Principle tells us that after the collision the total final momentum ''p<~~sub~~>1xf~~</sub>''~~ + ~~''p<sub>~~2xf</~~sub~~>'' must equal the initial total momentum ''p<~~sub~~>1xi</~~sub~~>''. Before the collision, nonzero energy terms included the kinetic energy of cart 1, ''K<~~sub~~>1i</~~sub~~>'', and the internal energies of both carts. After the collision there is an increase in the internal energy of both carts and kinetic energy of both carts, ''K<~~sub~~>1f~~</sub>''~~ + ~~''K<sub>~~2f</~~sub~~>''.		In terms of the two carts of equal masses example, the two carts are the system. The Momentum Principle tells us that after the collision the total final momentum ''<math>p_{1xf} + p_{2xf}</math>'' must equal the initial total momentum ''<math>p_{1xi}</math>''. Before the collision, nonzero energy terms included the kinetic energy of cart 1, ''<math>K_{1i}</math>'', and the internal energies of both carts. After the collision there is an increase in the internal energy of both carts and kinetic energy of both carts, ''<math>K_{1f} + K_{2f}</math>''.

	===A Mathematical Model===		===A Mathematical Model===

Sydney: /* Difficult */

2019-07-07T05:03:44Z

Difficult

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 '''Question'''
 '''Answer'''
 ==Connectedness==

Sydney: /* A Computational Model */

2019-07-07T01:52:41Z

A Computational Model

← Older revision		Revision as of 21:52, 6 July 2019
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	===A Computational Model===		===A Computational Model===

	This ~~simulation shows the head on collision of two balls of equal mass:~~ https://trinket.io/embed/glowscript/37540ee8e0.		This [https://trinket.io/embed/glowscript/37540ee8e0 simulation] shows the head on collision of two balls of equal mass.

	==Examples==		==Examples==

Sydney: /* Middling */

2019-07-07T01:49:37Z

Middling

← Older revision		Revision as of 21:49, 6 July 2019
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	::<math> {53212.11}={3174}{v_{f}}</math>		::<math> {53212.11}={3174}{v_{f}}</math>
	::<math> {v_{f}}={16.77m/s}</math>		::<math> {v_{f}}={16.77m/s}</math>



	The cars move at a speed of 16.77 meters per second immediately after the crash. Note that this does not take into account the energy lost in the form of heat or deformation of the car.		The cars move at a speed of 16.77 meters per second immediately after the crash. Note that this does not take into account the energy lost in the form of heat or deformation of the car.


	===Difficult===		===Difficult===

Sydney at 00:19, 6 July 2019

2019-07-06T00:19:22Z

← Older revision		Revision as of 20:19, 5 July 2019
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	Another fun application of collisions of equal mass is bumper cars. While the tracks are not typically low enough friction to discount it, the cars approximately all have the same mass and collide frequently. Changes in momentum with collisions are easy to exemplify with bumper cars. For example, it is easy to observe how two bumper cars going the same speed collide and bounce backwards; while one fast car hitting a stationary one causes a larger transfer of energy.		Another fun application of collisions of equal mass is bumper cars. While the tracks are not typically low enough friction to discount it, the cars approximately all have the same mass and collide frequently. Changes in momentum with collisions are easy to exemplify with bumper cars. For example, it is easy to observe how two bumper cars going the same speed collide and bounce backwards; while one fast car hitting a stationary one causes a larger transfer of energy.

			==History==
			The idea of inelastic collisions is rooted in the conservation of linear momentum, which is implied by Newton's Laws. Newton published his book, the Philosophiae Naturalis Principia Mathematica, in 1687. This book, commonly known as Principia, described Newton’s findings on collisions and laid a foundation for the different types of collisions we study today.

			The history of collisions in modern physics starts with Rutherford’s Gold Foil experiment. He found that while they did not touch, the alpha particles and the gold nucleus collided elastically. It is essential that the collision was elastic, in that it gives us the famous scatting pattern and led to the discovery of the positively charged nucleus.

	== See also ==		== See also ==

Sydney: /* A Mathematical Model */

2019-07-01T15:50:18Z

A Mathematical Model

← Older revision		Revision as of 11:50, 1 July 2019
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	'''1. Elastic Head-on Collisions of Equal Masses'''		'''1. Elastic Head-on Collisions of Equal Masses'''

	~~[[File:Head on equal mass inelastic .png\|right\|600px]]~~
	Momentum Principle:		Momentum Principle:
	::<math>{p_{xf}}={p_{xi}}+{F_{net,x}}{Δt}</math>		::<math>{p_{xf}}={p_{xi}}+{F_{net,x}}{Δt}</math>
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	:: <math>({K_{1f}}+{E_{int1f}})+({K_{1f}}+{E_{int2f}})=({K_{1i}}+{E_{int1i}})+({K_{2i}}+{E_{int2i}})+{0}+{0} </math>		:: <math>({K_{1f}}+{E_{int1f}})+({K_{1f}}+{E_{int2f}})=({K_{1i}}+{E_{int1i}})+({K_{2i}}+{E_{int2i}})+{0}+{0} </math>
	:: <math>{K_{1f}}+{K_{2f}}={K_{1i}}+{0} </math>		:: <math>{K_{1f}}+{K_{2f}}={K_{1i}}+{0} </math>
			[[File:Head on equal mass inelastic .png\|right\|500px]]
	Since ''K=p<sup>2</sup>/2m'', we can combine the momentum and energy equations:		Since ''K=p<sup>2</sup>/2m'', we can combine the momentum and energy equations:
	::<math>{\frac{p^{2}_{1xf}}{2m}}+{\frac{p^{2}_{2xf}}{2m}}= {\frac{(p_{1xf}+{p}_{2xf})^2}{2m}}</math>		::<math>{\frac{p^{2}_{1xf}}{2m}}+{\frac{p^{2}_{2xf}}{2m}}= {\frac{(p_{1xf}+{p}_{2xf})^2}{2m}}</math>

Sydney: /* A Mathematical Model */

2019-07-01T15:48:28Z

A Mathematical Model

← Older revision		Revision as of 11:48, 1 July 2019
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	'''1. Elastic Head-on Collisions of Equal Masses'''		'''1. Elastic Head-on Collisions of Equal Masses'''

			[[File:Head on equal mass inelastic .png\|right\|600px]]
	Momentum Principle:		Momentum Principle:
	::<math>{p_{xf}}={p_{xi}}+{F_{net,x}}{Δt}</math>		::<math>{p_{xf}}={p_{xi}}+{F_{net,x}}{Δt}</math>

Sydney: /* A Mathematical Model */

2019-07-01T15:32:00Z

A Mathematical Model

← Older revision		Revision as of 11:32, 1 July 2019
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	'''2. Maximally Inelastic Collision of Equal Masses'''		'''2. Maximally Inelastic Collision of Equal Masses'''

	[[File:Head on equal mass.png\|right\|~~400px~~]]		[[File:Head on equal mass.png\|right\|600px]]
	Momentum Principle:		Momentum Principle:
	::<math>{p_{1xf}}+{p_{2xf}}={p_{1xi}}</math>		::<math>{p_{1xf}}+{p_{2xf}}={p_{1xi}}</math>