Physics Puzzle Solution: Stuns Shots and the 90-Degrees Rule in Billiard
This is the solution to the ‘Stun Shots and the 90-Degrees Rule in Billiard‘ puzzle.
Conservation of momentum yields:
which can be divided by m to yield a relation between the velocities; taking the square yields:
Conservation of energy gives (experimentally speaking, billiard ball collisions are elastic to a very good approximation):
This can be multiplied by 2 and divided by m, and then subtracted from the previous equation, which leaves us with:
For the right hand side to equal zero the final velocities must either be orthogonal (i.e., at 90 degrees, as we have set out to show), or one of them should be zero. The only case in which one of them will be zero is in a head-on collision, when the cue ball hits the target ball and stops, transferring all of its momentum to the target ball.
Although this solution is exact, the model isn’t. No stun shot is perfect - and don’t forget friction! - and the 90-degrees rule should be taken as a general guideline. An elaborate discussion of the results in the presence of friction can be found in the 1987 paper “Analysis of Billiard Ball Collisions in Two Dimensions” by Wallace and Schroeder, published in volume 56 of the American Journal of Physics (you’ll need a subscription to read the paper).
