Savchenko Problem 1.1.20
Physics · mechanics
Problem
A ball is launched along a pool table with sides a and b from the middle of side b. At what angle to the side of the table must the ball begin to move to return to the same point from which it began its movement?Intuitive explanation
The key insight is that a ball bouncing elastically off walls follows a "mirror image" path—you can "unfold" the reflections by imagining copies of the table reflected across each wall. The ball travels in a straight line through this unfolded space, and it returns to its starting point when that straight-line path closes on itself. This happens when the ball's trajectory has traveled integer multiples of the table dimensions, making the geometry of the unfolded path form a closed loop that brings it back to the original point.
Intuitive explanation
The key insight is that a ball bouncing elastically off walls follows a "mirror image" path—you can "unfold" the reflections by imagining copies of the table reflected across each wall. The ball travels in a straight line through this unfolded space, and it returns to its starting point when that straight-line path closes on itself. This happens when the ball's trajectory has traveled integer multiples of the table dimensions, making the geometry of the unfolded path form a closed loop that brings it back to the original point.