5 A particle \(P\) of mass \(2 m\) is moving on a smooth horizontal surface with speed \(u\) when it collides directly with a particle \(Q\) of mass \(k m\) whose speed is \(3 u\) in the opposite direction. As a result of the collision, the directions of motion of both particles are reversed and the speed of \(P\) is halved.

1. Find, in terms of \(u\) and \(k\), the speed of \(Q\) after the collision. Hence write down the range of possible values of \(k\).
2. Calculate the magnitude of the impulse which \(Q\) exerts on \(P\).
3. Given that \(k = \frac { 1 } { 2 }\), calculate the coefficient of restitution between \(P\) and \(Q\). \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{982647bd-8514-40cf-b4ee-674f51df32c5-3_472_1143_221_242} \captionsetup{labelformat=empty} \caption{Fig. 1}
   \end{figure} One end of a light inextensible string is attached to a point \(P\). The other end is attached to a point \(Q , 1.96 \mathrm {~m}\) vertically below \(P\). A small smooth bead \(B\), of mass 0.3 kg , is threaded on the string and moves in a horizontal circle with centre \(Q\) and radius \(1.96 \mathrm {~m} . B\) rotates about \(Q\) with constant angular speed \(\omega\) rad s \(^ { - 1 }\) (see Fig. 1).