\textbf{In this question you should take the acceleration due to gravity to be $10 \text{ms}^{-2}$.}

\includegraphics{figure_12}

A small ball $P$ is projected from a point $A$ with speed $39 \text{ms}^{-1}$ at an angle of elevation $\theta$, where $\sin \theta = \frac{5}{13}$ and $\cos \theta = \frac{12}{13}$. Point $A$ is $20 \text{m}$ vertically above a point $B$ on horizontal ground. The ball first lands at a point $C$ on the horizontal ground (see diagram).

The ball $P$ is modelled as a particle moving freely under gravity.

\begin{enumerate}[label=(\alph*)]
\item Find the maximum height of $P$ above the ground during its motion. [3]
\end{enumerate}

The time taken for $P$ to travel from $A$ to $C$ is $7$ seconds.

\begin{enumerate}[label=(\alph*)]
\setcounter{enumi}{1}
\item Determine the value of $T$. [3]

\item State \textbf{one} limitation of the model, other than air resistance or the wind, that could affect the answer to part (b). [1]
\end{enumerate}

At the instant that $P$ is projected, a second small ball $Q$ is released from rest at $B$ and moves towards $C$ along the horizontal ground.

At time $t$ seconds, where $t \geq 0$, the velocity $v \text{ms}^{-1}$ of $Q$ is given by
$$v = kt^3 + 6t^2 + \frac{3}{2}t,$$
where $k$ is a positive constant.

\begin{enumerate}[label=(\alph*)]
\setcounter{enumi}{3}
\item Given that $P$ and $Q$ collide at $C$, determine the acceleration of $Q$ immediately before this collision. [6]
\end{enumerate}

\hfill \mbox{\textit{OCR H240/03 2023 Q12 [13]}}