7.
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{361d263e-0ee1-47e9-8fc2-0f127f1c2d7e-24_553_951_258_557}
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\caption{Figure 2}
\end{figure}
Figure 2 represents the plan view of part of a horizontal floor, where \(A B\) and \(C D\) represent fixed vertical walls, with \(A B\) parallel to \(C D\).
A small ball is projected along the floor towards wall \(A B\). Immediately before hitting wall \(A B\), the ball is moving with speed \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an angle \(\alpha\) to \(A B\), where \(0 < \alpha < \frac { \pi } { 2 }\)
The ball hits wall \(A B\) and then hits wall \(C D\).
After the impact with wall \(C D\), the ball is moving at angle \(\frac { 1 } { 2 } \alpha\) to \(C D\).
The coefficient of restitution between the ball and wall \(A B\) is \(\frac { 2 } { 3 }\)
The coefficient of restitution between the ball and wall \(C D\) is also \(\frac { 2 } { 3 }\)
The floor and the walls are modelled as being smooth. The ball is modelled as a particle.
- Show that \(\tan \left( \frac { 1 } { 2 } \alpha \right) = \frac { 1 } { 3 }\)
- Find the percentage of the initial kinetic energy of the ball that is lost as a result of the two impacts.