7.

\begin{figure}[h]
\begin{center}
  \includegraphics[alt={},max width=\textwidth]{f6ab162c-8473-4464-ad62-87a359d85ab3-12_499_833_262_664}
\captionsetup{labelformat=empty}
\caption{Figure 6}
\end{center}
\end{figure}

A particle $P$ of mass $5 m$ is attached to one end of a light inextensible string of length $a$. The other end of the string is attached to a fixed point $O$. The particle is held at the point $A$, where $O A = a$ and $O A$ is horizontal, as shown in Figure 6. The particle is projected vertically downwards with speed $\sqrt { } \left( \frac { 9 a g } { 5 } \right)$. When the string makes an angle $\theta$ with the downward vertical through $O$ and the string is still taut, the tension in the string is $T$.
\begin{enumerate}[label=(\alph*)]
\item Show that $T = 3 m g ( 5 \cos \theta + 3 )$.

At the instant when the particle reaches the point $B$ the string becomes slack.
\item Find the speed of $P$ at $B$.

At time $t = 0 , P$ is at $B$.

At time $t$, before the string becomes taut once more, the coordinates of $P$ are $( x , y )$ referred to horizontal and vertical axes with origin $O$. The $x$-axis is directed along $O A$ produced and the $y$-axis is vertically upward.
\item Find
\begin{enumerate}[label=(\roman*)]
\item $x$ in terms of $t , a$ and $g$,
\item $y$ in terms of $t , a$ and $g$.
\end{enumerate}\end{enumerate}

\hfill \mbox{\textit{Edexcel M3 2013 Q7 [16]}}