A small ball $B$ moves in the plane of a fixed horizontal axis $Ox$, which lies on horizontal ground, and a fixed vertically upwards axis $Oy$. $B$ is projected from $O$ with a velocity whose components along $Ox$ and $Oy$ are $U \mathrm{m s}^{-1}$ and $V \mathrm{m s}^{-1}$, respectively. The units of $x$ and $y$ are metres.

$B$ is modelled as a particle moving freely under gravity.

\begin{enumerate}[label=(\alph*)]
\item Show that the path of $B$ has equation $2U^2 y = 2UVx - gx^2$. [3]
\end{enumerate}

During its motion, $B$ just clears a vertical wall of height $\frac{1}{3}a$ m at a horizontal distance $a$ m from $O$. $B$ strikes the ground at a horizontal distance $3a$ m beyond the wall.

\begin{enumerate}[label=(\alph*)]
\setcounter{enumi}{1}
\item Determine the angle of projection of $B$. Give your answer in degrees correct to 3 significant figures. [5]
\item Given that the speed of projection of $B$ is $54.6 \mathrm{m s}^{-1}$, determine the value of $a$. [2]
\item Hence find the maximum height of $B$ above the ground during its motion. [3]
\item State one refinement of the model, other than including air resistance, that would make it more realistic. [1]
\end{enumerate}

\hfill \mbox{\textit{OCR H240/03 2022 Q13 [14]}}