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--- book:chap5:5.6_worked_example [2022/01/03 14:58] – created jv
+++ book:chap5:5.6_worked_example [2022/01/07 21:37] (current) – abril
@@ Line 1: / Line 1: @@
-FIXME draft with missing figures and references :!:
 ===== 5.6  Worked example: Reflection of balls  =====
-<WRAP id=section_workedExample-ballReflections />
 We consider the reflection of a ball from the ground, the lower side of a table, and back.
@@ Line 8: / Line 5: @@
 (by symmetry they all agree).
 Its velocity at time $t_0$ will be denoted as $\dot{\mathbf z}_0$.
-It has no spin initially.\ $\mathbf\omega_0 = \mathbf 0$.
+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.
-  -  Sketch the setup, and the parameters adopted for the first collision:
+**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.
-  -  Upon collision there is a force normal to the floor, $\mathbf F_{\perp}$, and a force tangential to the floor, $\mathbf F_{\parallel}$.
+**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.
@@ Line 30: / Line 28: @@
     v_{n}   + R \: \omega_n   &= - ( v_{n+1}  + R \: \omega_{n+1} ) \, .
 \end{align*}
-  -  **$\star$** Determine  $v_1( v_0, \omega_0)$ and $\omega_1( v_0, \omega_0)$ for the initial conditions specified above.
+**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 $\omega=1/3$.
+Discuss the result for a sphere with uniform mass distribution (what does this imply for $\omega$?), and a sphere with $\omega=1/3$.\\
+++Hint:|  For the plot one conveniently implements the recursion, rather than explicitly calculating $v_3$.++
-**Hint: **  For the plot one conveniently implements the recursion, rather than explicitly calculating $v_3$.
+**d)** What changes in this discussion when the ball has a spin initially?
-  -  What changes in this discussion when the ball has a spin initially?