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.\\
(i) Find, in terms of $u$ and $k$, the speed of $Q$ after the collision. Hence write down the range of possible values of $k$.\\
(ii) Calculate the magnitude of the impulse which $Q$ exerts on $P$.\\
(iii) Given that $k = \frac { 1 } { 2 }$, calculate the coefficient of restitution between $P$ and $Q$.

\begin{figure}[h]
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\caption{Fig. 1}
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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).\\

\hfill \mbox{\textit{OCR M2 2008 Q5 [9]}}