5. In this question, $\mathbf { i }$ and $\mathbf { j }$ represent unit vectors due east and due north respectively.

Two smooth spheres $P$ and $Q$, of equal radii, are moving on a smooth horizontal surface. The mass of $P$ is 2 kg and the mass of $Q$ is 6 kg . The velocity of $P$ is $( 8 \mathbf { i } - 6 \mathbf { j } ) \mathrm { ms } ^ { - 1 }$ and the velocity of $Q$ is $( 4 \mathbf { i } + 10 \mathbf { j } ) \mathrm { ms } ^ { - 1 }$. At a particular instant, $Q$ is positioned 12 m east and 48 m south of $P$.
\begin{enumerate}[label=(\alph*)]
\item Prove that $P$ and $Q$ will collide.

At the instant the spheres collide, the line joining their centres is parallel to the vector $\mathbf { j }$. Immediately after the collision, sphere $Q$ has speed $5 \mathrm {~ms} ^ { - 1 }$.
\item Determine the coefficient of restitution between the spheres and hence calculate the velocity of $P$ immediately after the collision.
\item Find the magnitude of the impulse required to stop sphere $P$ after the collision.
\end{enumerate}

\hfill \mbox{\textit{WJEC Further Unit 6 2023 Q5 [16]}}