In this problem we consider a small ball bouncing back and forth between two points. In all parts below, the acceleration of gravity is g, collisions are perfectly elastic, air resistance is negligible, and the impact points are at the same height. The diagrams are not drawn to scale.
a. Consider a ball bouncing between two inclined planes, which each make an angle θ<90° to the horizontal. The ball has speed v0 at the impact points, which are separated by a distance D.
i. The ball can bounce back and forth along the same path, as shown.
[Figure: Two inclined planes making angle θ with horizontal, separated by distance D]
For what values of θ is this motion possible? For these values, what is v0?
ii. The ball can also take one trajectory while traveling to the right, and a separate trajectory when traveling back. Let α=0 be the angle between the paths at the impact points.
[Figure: Two inclined planes making angle θ with horizontal, separated by distance D, with angle α between paths]
For what values of θ and α is this motion possible? For these values, what is v0?
b. Now suppose the ball bounces within a hemispherical well of radius of curvature R. As in part a.ii, it alternates between two distinct paths, with flight times t1 and t2=t1.
[Figure: Hemispherical well of radius R]
Find all of the possible values of R, in terms of t1 and t2.
c. Finally, suppose the well has a sinusoidal shape, described by y(x)=Lsin(2πx/L). The ball takes two distinct paths with flight times t1 and t2=t1, and the horizontal distance between the impact points is less than πL. Find all of the possible values of L, in terms of t1 and t2.