4.
\begin{figure}[h]
\includegraphics[alt={},max width=\textwidth]{0cec16c3-23a0-4620-a80f-b5d4e014e2fc-12_81_1383_255_342}
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\caption{Figure 1}
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Three particles, \(P , Q\) and \(R\), lie at rest on a smooth horizontal plane. The particles are in a straight line with \(Q\) between \(P\) and \(R\), as shown in Figure 1 .
Particle \(P\) is projected towards \(Q\) with speed \(u\). At the same time, \(R\) is projected with speed \(\frac { 1 } { 2 } u\) away from \(Q\), in the direction \(Q R\).
Particle \(P\) has mass \(m\) and particle \(Q\) has mass \(2 m\).
The coefficient of restitution between \(P\) and \(Q\) is \(e\).
- Show that the speed of \(Q\) immediately after the collision between \(P\) and \(Q\) is
$$\frac { u ( 1 + e ) } { 3 }$$
It is given that \(e > \frac { 1 } { 2 }\)
- Determine whether there is a collision between \(Q\) and \(R\).
- Determine the direction of motion of \(P\) immediately after the collision between \(P\) and \(Q\).
- Find, in terms of \(m , u\) and \(e\), the total kinetic energy lost in the collision between \(P\) and \(Q\), simplifying your answer.
- Explain how using \(e = 1\) could be used to check your answer to part (d).