Confusion regarding the coefficient of restitution

[parsesnip]

I learned that there are two different definitions for the coefficient of restitution: e = final relative velocity / initial relative velocity and e = √(final KE/initial KE). However, I don't understand how these two definitions will always give the same value.

If one particle with mass m moving with velocity v has a perfectly inelastic collision with another particle of mass m at rest, then both will move together with the velocity v/2. According to the first definition, e = 0 as the final relative velocity is 0. However, according to the second definition, e = √((1/2(2m)(v/2)²)/(1/2mv^2)) which is 1/√2.

Also, Wikipedia says that "A perfectly inelastic collision has a coefficient of 0, but a 0 value does not have to be perfectly inelastic.". Can someone give me an example of a collision with a coefficient of 0 that is not perfectly inelastic? I thought that was the definition of perfect inelasticity.

Also shouldn't the first definition use relative speed instead of relative velocity? For example, if a particle of mass m hits a wall with velocity v, it will rebound with velocity -v, so according to this definition e should be -1.

 

[jbriggs444]

f one particle with mass m moving with velocity v has a perfectly inelastic collision with another particle of mass m at rest, then both will move together with the velocity v/2. According to the first definition, e = 0 as the final relative velocity is 0. However, according to the second definition, e = √((1/2(2m)(v/2)2)/(1/2mv^2)) which is 1/√2.

If you are measuring velocities from the frame of reference where the particles end up together at rest, it is only fair to measure kinetic energy in the same frame. In that frame, final kinetic energy is zero.

The coefficient of restitution is properly measured using energy in the center of mass frame.

 

[Merlin3189]

I learned that there are two different definitions for the coefficient of restitution: e = final relative velocity / initial relative velocity ...

Unless you have a definition wher e is negative, I think it is always |e| = - (final relative velocity)/(initial relative velocity)
<Edit: provided relative velocity is defined exactly the same both before and after. Most texts seem to define them in the opposite sense, which explains why they get both having the same value <edit2: sign rather than value>.>

As far as I can see, it doesn't matter whether one object is a brick wall or another small object, whether one is stationary or both moving, the relative velocity always changes sign.

<Edit: speed would not be useful for the definition, except in one dimension.>

The rest I'm still jugging with!