THE UNIVERSITY OF BRITISH COLUMBIA
 
Science 1 Physics
 
Assignment # 5:
 
Advanced Mechanics
 
Wed. 11 Oct. 2000 - finish by Wed. 18 Oct.
 

  
Figure: :   Superball
\begin{figure} \begin{center} \epsfig{file=PS/SuperBall.ps,height=1.4in} \end{center}\end{figure}

A ``superball'' is a hard rubber ball with a nearly perfect coefficient of restitution and a huge coefficient of static friction with hard, smooth surfaces. As a result, when it bounces off such a surface we may treat the collision as perfectly elastic (no loss of energy) and assume that the point (of the ball) in contact with the surface during the collision does not slip, no matter how acute the angle $\theta$ of its trajectory. To simplify the problem, we will also assume that the following experiment is done in a weightless environment, although the qualitative results are unchanged by gravity.


  
Figure: :   First Bounce:
\begin{figure} \begin{center} \epsfig{file=../PS/SuperBall-bounce.ps,height=1.5in} \end{center}\end{figure}

 
The ball is thrown at the floor with an initial velocity $\vec{v}$1 making an angle $\theta_1$ with the positive $\hat{x}$ direction or with the floor.1 The ball is not rotating initially ( $\omega_1 = 0$). After the collision we expect $\omega_2$ to be negative.2

Assuming that the ball is a uniform solid sphere, calculate the angle $\theta_2$ and the speed v2 in terms of $\theta_1$ and $v_1 \equiv \vert$$\vec{v}$1|.

Under the Table:
If the ball was thrown under a table, as shown, with the smooth, dry underside of the tabletop parallel to the floor, show that it will always return to the thrower's hand along the same trajectory it followed in the first two bounces (shown in the figure).

Superball Design:
Does it matter if the ball is really a uniform solid sphere? If it were a hollow sphere or a rubber sphere with a lead centre, would you still obtain the same qualitative result? What criteria are important? Explain.

. . . and Tipler Ch. 8:

(4$^{\rm th}$ Edition) problems 9, 86, 97 and 124

. . . and Tipler Ch. 9:

(4$^{\rm th}$ Edition) problems 5, 31, 53, 86 and 92