4. Ryan is playing a game of snooker. The horizontal table is modelled as the horizontal $x - y$ plane with the point $O$ as the origin and unit vectors parallel to the $x$-axis and the $y$-axis denoted by $\mathbf { i }$ and $\mathbf { j }$ respectively. All balls on the table have a common mass $m \mathrm {~kg}$. The table and the four sides, called cushions, are modelled as smooth surfaces.

The dimensions of the table, in metres, are as shown in the diagram.\\
\includegraphics[max width=\textwidth, alt={}, center]{3578a810-46da-4d9e-a98f-248be72a517a-5_663_1138_667_482}

Initially, all balls are stationary. Ryan strikes ball $A$ so that it collides with ball $B$. Before the collision, $A$ has velocity $( - \mathbf { i } + 8 \mathbf { j } ) \mathrm { ms } ^ { - 1 }$ and, after the collision, it has velocity $( 2 \mathbf { i } + \mathbf { j } ) \mathrm { ms } ^ { - 1 }$.
\begin{enumerate}[label=(\alph*)]
\item Show that the velocity of ball $B$ after the collision is $( - 3 \mathbf { i } + 7 \mathbf { j } ) \mathrm { ms } ^ { - 1 }$.

After the collision with ball $A$, ball $B$ hits the cushion at point $C$ before rebounding and moving towards the pocket at $P$. The cushion is parallel to the vector $\mathbf { i }$ and the coefficient of restitution between the cushion and ball $B$ is $\frac { 5 } { 7 }$.
\item Calculate the velocity of ball $B$ after impact with the cushion.
\item Find, in terms of $m$, the magnitude of the impulse exerted on ball $B$ by the cushion at $C$, stating your units clearly.
\item Given that $C$ has position vector $( x \mathbf { i } + 1 \cdot 75 \mathbf { j } ) \mathrm { m }$,
\begin{enumerate}[label=(\roman*)]
\item determine the time taken between the ball hitting the cushion at $C$ and entering the pocket at $P$,
\item find the value of $x$.
\end{enumerate}\item Describe one way in which the model used could be refined. Explain how your refinement would affect your answer to (d)(i).
\end{enumerate}

\hfill \mbox{\textit{WJEC Further Unit 6 2019 Q4 [15]}}