In class, I showed that for a perfectly elastic collision, u1 + v1= u2 + v2, or alternatively, u2−u1=v1−v2. If object 2 is initially at rest, then u2=0. Then v1−v2=−u1. However, for a partially elastic collision the relative velocity after the collision will have a smaller magnitude than the relative velocity before the collision. We can express this mathematically as v1−v2=−ru1, where r (a number less than one) is called the coefficient of restitution. For some kinds of bodies, the coefficient r is a constant, independent of v1 and v2. Show that in this case the final kinetic energy of the motion relative to the center of mass is less than the initial kinetic energy of this motion by a factor of r^2. Furthermore, derive expressions for v1 and v2 in terms of u1 and r.

In class, I showed that for a perfectly elastic collision, u1 + v1= u2 + v2, or alternatively, u2−u1=v1−v2. If object 2 is initially at rest, then u2=0. Then v1−v2=−u1. However, for a partially elastic collision the relative velocity after the collision will have a smaller magnitude than the relative velocity before the collision. We can express this mathematically as v1−v2=−ru1, where r (a number less than one) is called the coefficient of restitution. For some kinds of bodies, the coefficient r is a constant, independent of v1 and v2. Show that in this case the final kinetic energy of the motion relative to the center of mass is less than the initial kinetic energy of this motion by a factor of r^2. Furthermore, derive expressions for v1 and v2 in terms of u1 and r.

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