5.6 Worked example: Reflection of balls
We consider the reflection of a ball from the ground, the lower side of a table, and back. The ball is considered to be a sphere with radius $R$, mass $m$, and moments of inertia $m \alpha R^2$ (by symmetry they all agree). Its velocity at time $t_0$ will be denoted as $\dot{\mathbf z}_0$. It has no spin initially. $\mathbf\omega_0 = \mathbf 0$. The velocity and the spin after the $n^{\textrm{th}}$ collision will be denoted as $\dot{\mathbf z}_n$ and $\mathbf\omega_n$. We will disregard gravity such that the ball travels on a straight path in between collisions.
a) Sketch the setup, and the parameters adopted for the first collision: The positive $x$ axis will be parallel to the floor and the origin will be put into the location of the collision. Its direction will be chosen such that the ball moves in the $x$-$z$ plane. Take note of all quantities needed to discuss the angular momentum with respect to the origin.
b) Upon collision there is a force normal to the floor, $\mathbf F_{\perp}$, and a force tangential to the floor, $\mathbf F_{\parallel}$. The spin of the ball will only change due to the tangential force. The normal force $\mathbf F_{\perp}$ acts in the same way as for point particles. The velocity in vertical direction reverses direction and preserved its modulus. Denote the velocity component in horizontal direction as $v_n = \hat x \cdot \mathbf{\dot z}$, and demonstrate that conservation of energy and angular momentum imply that \begin{align*} v_{n}^2 + \alpha R^2 \omega_n^2 &= v_{n+1}^2 + \alpha R^2 \omega_{n+1}^2 \\ v_{n} - \alpha R \: \omega_n &= v_{n+1} - \alpha R \: \omega_{n+1} \, . \end{align*} Show that the tangential velocity component will therefore also reverse its direction and preserves the modulus, \begin{align*} v_{n} + R \: \omega_n &= - ( v_{n+1} + R \: \omega_{n+1} ) \, . \end{align*}
c) 
Determine $v_1( v_0, \omega_0)$ and $\omega_1( v_0, \omega_0)$ for the initial conditions specified above.
Now, we determine $v_2( v_1, \omega_1)$ and $\omega_2( v_1, \omega_1)$ by shifting the origin of the coordinate systems
to the point where the next collision will arise, and we rotate by $\pi$ to account for the fact that we collide at the lower side of the table.
What does this imply for $v_1$ and $\omega_1$?
Continue the iteration, and plot $v_1$, $v_2$ and $v_3$ as function of $\alpha$.
Discuss the result for a sphere with uniform mass distribution (what does this imply for $\omega$?), and a sphere with $\omega=1/3$.
Hint:
d) What changes in this discussion when the ball has a spin initially?