I was looking at this mechanics problem:
Looking at the first paragraph, about showing a parallel path after two bounces, I solved it as follows. Consider this diagram:
Resolving velocity perpendicular to the first and second cushions, and using the coefficient of restitution, we have
$$ ev_1\sin{\theta} = v_2\sin{\phi} \\ ev_2\cos{\phi} = v_3\cos{\psi} $$
Also, resolving for conservation of momentum parallel to the cushions:
$$ v_1\cos{\theta} = v_2\cos{\phi} \\ v_2\sin{\phi} = v_3\sin{\psi} $$
Combining these, we can conclude:
$$ ev_1\sin{\theta} = v_3\sin{\psi} \\ ev_1\cos{\theta} = v_3\cos{\psi} $$
Therefore $\tan{\theta} = \tan{\psi}$ and hence $\theta = \psi$.
I'm wondering if anyone has any ideas for a solution to the second paragraph (the paragraph with the puck resting at a point dividing L in the ratio z:1)? I've experimented with a couple of ways, and am pretty sure it reduces to proving that a point always exists for the first bounce that will ensure the puck hits the adjacent cushion on the second bounce.