[AustinSchuh]

Assuming a COR != 1, we get that the kicker should weigh M_ball / COR. What ever initial conditions exist, it is trivial to find a mass of the kicker that will result in the kicker transferring all it's energy to the ball. And for most sane initial conditions, you will want the masses to be within a factor of 2 of each other.

I've been thinking about this for a while, and this post finally got me to sit down and derive what happens when the kicker is spinning and hits a ball. Something bugged me about using equations derived for linear impacts when modeling something that is swinging.

Lets start with the assumption that angular momentum is conserved.

I_a w_a + I_b w_b = I_a w'_a + I_b w'_b

Then define an "angular coefficient of restitution" that's similar to the coefficient of restitution for linear collisions.

-w'_a + w'_b = COR (w_a - w_b)

Solving the linear system of equations gets us that

w'_a = (I_a w_a + I_b w_b + I_b COR (w_b - w_a)) / (I_a + I_b)

w'_b = (I_b w_b + I_a w_a + I_a COR (w_a - w_b)) / (I_a + I_b)

So, when a rotating kicker is hitting something, we want the moments of inertia to be similar. Or, if the COR isn't one, then the moment of inertia of the kicker should be I_ball / COR, where I_ball is the moment of inertia of the ball around the axis that the kicker spins around. Very similar to the original billiard ball case.

Our small piece of metal was the end of one of the old IFI frames that looked like a U. So, it was a ~1/2 lb quite stiff piece of aluminum that had about a 4" radius. It's moment of inertia around the kicker axis would have been quite a bit lower than the ball's around the kicker axle, resulting in a shortened kick distance because the kicker would bounce off the ball instead of kicking it. Which is consistent with what we observed.