When revising for a general physics paper in my master course we had to go back and do all these standard collision problems. One frequent question is of the following form: A body with some mass distribution (usually a thin rod of length $L$ and mass $M$) has velocity $v_0$. It then hits a stationary spike that is held in place (e.g. fixed to the earth) at a distance $d$ from its centre of mass. The collision is fully elastic. The exercise is then to calculate the CoM velocity and rotation speed after the collision.
A standard solution to this is to set up two equations:
- angular momentum conservation about the axis perpendicular to $v_0$ and through the collision point
- energy conservation of the rod, since the collision is elastic and the spike is held in place, i.e. the rod's energy does not change
A friend of mine then (either by ingenious intuition or by remembering it from our dynamics course) claimed that angular momentum conservation (which can be slightly annoying to write down) can be replaced by the following condition: the velocity in the spike frame of the point on the rod that touches the spike is reversed.
To my surprise it worked for the specific question we were doing, we got the same result using both calculations. Now my question:
- Is this actually true for inelastic collisions in general or some kind of coincidence? I haven't tried to prove it, but I think if it is true that should be straightforward. This is not my main question though, since I believe it is probably true in general.
- Now assuming that it is true: Why on earth is it true?? As in: is there an easy explanation or does it just come out of the math? The reason I find this so astonishing is that angular momentum conservation and energy conservation apply globally and it is important that the rod is a rigid body. It seems amazing to me that one can replace that by a simple and completely local condition, that tells you how one point of the rigid body will move.