Note that the case analysis is tricky because the function can be discontinuous. The interesting cases are when a ball’s X velocity becomes zero due to a collision, or (more subtly) two balls with only Y velocity gain X velocity due to an off-center collision. But I think the statement about monotonicity still holds.
Yeah I thought about those two cases as well, but I agree that they are correct. Perhaps we could make the proof a bit simpler by picking the X direction to be one that the balls never travel perpendicular too (although in fact I can’t even think of a proof that such a direction exists).
Yep, that looks pretty airtight to me. Well done!
Note that the case analysis is tricky because the function can be discontinuous. The interesting cases are when a ball’s X velocity becomes zero due to a collision, or (more subtly) two balls with only Y velocity gain X velocity due to an off-center collision. But I think the statement about monotonicity still holds.
Yeah I thought about those two cases as well, but I agree that they are correct. Perhaps we could make the proof a bit simpler by picking the X direction to be one that the balls never travel perpendicular too (although in fact I can’t even think of a proof that such a direction exists).
That wouldn’t help with the first case, because a ball can stop completely. Let’s keep the proof as it is :-)