5. A particle \(P\) moves in a straight line. At time \(t\) seconds its displacement from a fixed point \(O\) on the line is \(x\) metres. The motion of \(P\) is modelled by the differential equation $$\frac { \mathrm { d } ^ { 2 } x } { \mathrm {~d} t ^ { 2 } } + 2 \frac { \mathrm {~d} x } { \mathrm {~d} t } + 2 x = 12 \cos 2 t - 6 \sin 2 t$$ When \(t = 0 , P\) is at rest at \(O\).

1. Find, in terms of \(t\), the displacement of \(P\) from \(O\).
2. Show that \(P\) comes to instantaneous rest when \(t = \frac { \pi } { 4 }\).
3. Find, in metres to 3 significant figures, the displacement of \(P\) from \(O\) when \(t = \frac { \pi } { 4 }\).
4. Find the approximate period of the motion for large values of \(t\). \section*{6.} \begin{figure}[h]
   \captionsetup{labelformat=empty} \caption{Figure 2} \includegraphics[alt={},max width=\textwidth]{618fdb9c-cc0b-4a80-a148-4311c908c94e-5_534_923_388_541}
   \end{figure} A small ball \(Q\) of mass \(2 m\) is at rest at the point \(B\) on a smooth horizontal plane. A second small ball \(P\) of mass \(m\) is moving on the plane with speed \(\frac { 13 } { 12 } u\) and collides with \(Q\). Both the balls are smooth, uniform and of the same radius. The point \(C\) is on a smooth vertical wall \(W\) which is at a distance \(d _ { 1 }\) from \(B\), and \(B C\) is perpendicular to \(W\). A second smooth vertical wall is perpendicular to \(W\) and at a distance \(d _ { 2 }\) from \(B\). Immediately before the collision occurs, the direction of motion of \(P\) makes an angle \(\alpha\) with \(B C\), as shown in Fig. 2, where tan \(\alpha = \frac { 5 } { 12 }\). The line of centres of \(P\) and \(Q\) is parallel to \(B C\). After the collision \(Q\) moves towards \(C\) with speed \(\frac { 3 } { 5 } u\).
5. Show that, after the collision, the velocity components of \(P\) parallel and perpendicular to \(C B\) are \(\frac { 1 } { 5 } u\) and \(\frac { 5 } { 12 } u\) respectively.
6. Find the coefficient of restitution between \(P\) and \(Q\).
7. Show that when \(Q\) reaches \(C , P\) is at a distance \(\frac { 4 } { 3 } d _ { 1 }\) from \(W\). For each collision between a ball and a wall the coefficient of restitution is \(\frac { 1 } { 2 }\).
   Given that the balls collide with each other again,
8. show that the time between the two collisions of the balls is \(\frac { 15 d _ { 1 } } { u }\),
9. find the ratio \(d _ { 1 } : d _ { 2 }\). \section*{END}