\includegraphics{figure_11}

A particle $P$ moves freely under gravity in the plane of a fixed horizontal axis $Ox$, which lies on horizontal ground, and a fixed vertical axis $Oy$. $P$ is projected from $O$ with a velocity whose components along $Ox$ and $Oy$ are $U$ and $V$, respectively. $P$ returns to the ground at a point $C$.

\begin{enumerate}[label=(\alph*)]
\item Determine, in terms of $U$, $V$ and $g$, the distance $OC$. [4]

\includegraphics{figure_11b}

$P$ passes through two points $A$ and $B$, each at a height $h$ above the ground and a distance $d$ apart, as shown in the diagram.

\item Write down the horizontal and vertical components of the velocity of $P$ at $A$. [2]
\item Hence determine an expression for $d$ in terms of $U$, $V$, $g$ and $h$. [3]
\item Given that the direction of motion of $P$ as it passes through $A$ is inclined to the horizontal at an angle $\theta$, where $\tan \theta = \frac{1}{2}$, determine an expression for $V$ in terms of $g$, $d$ and $h$. [4]
\end{enumerate}

\hfill \mbox{\textit{OCR H240/03 2020 Q11 [13]}}