\includegraphics{figure_7}

A particle $P$ is attached to a fixed point $O$ by a light inextensible string of length $0.7$ m. A particle $Q$ is in equilibrium suspended from $O$ by an identical string. With the string $OP$ taut and horizontal, $P$ is projected vertically downwards with speed $6$ m s$^{-1}$ so that it strikes $Q$ directly (see diagram). $P$ is brought to rest by the collision and $Q$ starts to move with speed $4.9$ m s$^{-1}$.

\begin{enumerate}[label=(\roman*)]
\item Find the speed of $P$ immediately before the collision. Hence find the coefficient of restitution between $P$ and $Q$. [3]
\item Given that the speed of $Q$ is $v$ m s$^{-1}$ when $OQ$ makes an angle $\theta$ with the downward vertical, find an expression for $v^2$ in terms of $\theta$, and show that the tension in the string $OQ$ is $14.7m(1 + 2\cos\theta)$ N, where $m$ kg is the mass of $Q$. [6]
\item Find the radial and transverse components of the acceleration of $Q$ at the instant that the string $OQ$ becomes slack. [4]
\item Show that $V^2 = 0.8575$, where $V$ m s$^{-1}$ is the speed of $Q$ when it reaches its greatest height (after the string $OQ$ becomes slack). Hence find the greatest height reached by $Q$ above its initial position. [4]
\end{enumerate}

\hfill \mbox{\textit{OCR M3 2010 Q7 [17]}}