Coefficient of Restitution

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 v_Ai and v_Bi and final velocities v_Af and v_Bf. The coefficient of restitution, e is determined with the following formula:

[math]\displaystyle{ 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 v_Ai and v_Bi refer to the initial velocities of A and B and v_Af and v_Bf 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=1.

A Computational Model

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Examples

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Difficult

Bouncing Ball Example:

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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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