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.

Part A. Two plains (3.5 points)

Consider a ball bouncing between two inclined planes, which each make an angle $\theta < 90^\circ$ to the horizontal. The ball has speed $v_0$ at the impact points, which are separated by a distance $D$.

The ball can bounce back and forth along the same path, as shown.

A1  ^1.50 For what values of $\theta$ is this motion possible? For these values, what is $v_0$?

The ball can also take one trajectory while traveling to the right, and a separate trajectory when traveling back. Let $\phi \neq 0$ be the angle between the paths at the impact points.

A2  ^2.00 For what values of $\theta$ and $\phi$ is this motion possible? For these values, what is $v_0$?

Part B. Hemispherical well (3.0 points)

Now suppose the ball bounces within a hemispherical well of radius of curvature $R$. As in question A2, it alternates between two distinct paths, with flight times $t_1$ and $t_2 \neq t_1$.

B1  ^3.00 Find all of the possible values of $R$, in terms of $t_1$ and $t_2$.

Part C. Sinusoidal well (3.5 points)

Finally, suppose the well has a sinusoidal shape, described by $y(x)= L \sin \dfrac{2x}{L}$. The ball takes two distinct paths with flight times $t_1$ and $t_2 \neq t_1$, and the horizontal distance between the impact points is less than $\pi L$.

C1  ^3.50 Find all of the possible values of $L$, in terms of $t_1$ and $t_2$.