A ball with radius $r\ll w$ (can be regarded as a point) is propelled from the bottom-left corner at an angle $5°\leq\alpha\leq 85°$ with respect the horizontal axis (as depicted in the figure). The movement is constrained to a plane, so there is no depth involved. The velocity of the ball is irrelevant.
There are no dissipative effects, so the ball will keep bouncing forever with no loss of energy. In other words, the collisions (ball-walls) are perfectly elastic (the angles are conserved before and after a collision).
- Find the angle $\alpha$ that maximizes the number of bounces until the ball returns to the first collision point. How many bounces $n(h,w,\alpha)$ will it take for this to happen? If you think there is no angle $\alpha$ for which the ball returns to the first collision point, demonstrate it.
NOTE: The first collision point is the point $(x_0,y_0)$ at which the ball first touches a wall.