Coefficient of Restitution: Difference between revisions
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<math>\vec{v_{Ai}}+\vec{v_{Bi}}=\vec{v_{Af}}+\vec{v_{Bf}}</math><br> | <math>\vec{v_{Ai}}+\vec{v_{Bi}}=\vec{v_{Af}}+\vec{v_{Bf}}</math><br> | ||
<math>1=\frac{\vec{v_{Ai}}+\vec{v_{Bi}}}{\vec{v_{Af}}+\vec{v_{Bf}}}</math> | <math>1=\frac{\vec{v_{Ai}}+\vec{v_{Bi}}}{\vec{v_{Af}}+\vec{v_{Bf}}}</math><br> | ||
<math>e=\frac{\vec{v_{Ai}}+\vec{v_{Bi}}}{\vec{v_{Af}}+\vec{v_{Bf}}}</math><br> | |||
For perfectly elastic collision <i>e</e>=1. | |||
===A Computational Model=== | ===A Computational Model=== | ||
Revision as of 14:04, 5 December 2015
Claimed by Maria Moreno
The coefficient of restitution measures the elasticity of a collision.
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]\displaystyle{ e = \frac{v_{Bf}-v_{Af}}{v_{Ai}-v_{Bi}} }[/math]
Perfectly Inelastic Collision
Perfectly Elastic Collision
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]\displaystyle{ \vec{p_i} = \vec{p_f} }[/math]
and
[math]\displaystyle{ 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]\displaystyle{ \vec{KE_i} = \vec{KE_f} }[/math]
and
[math]\displaystyle{ \frac{1}{2}m_A \vec{v^2_{Ai}}+\frac{1}{2}m_A \vec{v^2_{Bi}}=\frac{1}{2}m_A \vec{v^2_{Af}}+\frac{1}{2}m_A \vec{v^2_{Bf}} }[/math] (2)
Dividing (2) by (1) yields:
[math]\displaystyle{ \vec{v_{Ai}}+\vec{v_{Bi}}=\vec{v_{Af}}+\vec{v_{Bf}} }[/math]
[math]\displaystyle{ 1=\frac{\vec{v_{Ai}}+\vec{v_{Bi}}}{\vec{v_{Af}}+\vec{v_{Bf}}} }[/math]
[math]\displaystyle{ e=\frac{\vec{v_{Ai}}+\vec{v_{Bi}}}{\vec{v_{Af}}+\vec{v_{Bf}}} }[/math]
For perfectly elastic collision e</e>=1.
A Computational Model
How do we visualize or predict using this topic. Consider embedding some vpython code here Teach hands-on with GlowScript
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Connectedness
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The coefficient of restitution is a key concept in the study of Dynamics which is the foundation of many aspects of mechanical engineering.
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