A smooth sphere $P$ of mass $2m$ is moving in a straight line with speed $u$ on a smooth horizontal table. Another smooth sphere $Q$ of mass $m$ is at rest on the table. The sphere $P$ collides directly with $Q$. The coefficient of restitution between $P$ and $Q$ is $\frac{1}{3}$. The spheres are modelled as particles.

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\item Show that, immediately after the collision, the speeds of $P$ and $Q$ are $\frac{2}{9}u$ and $\frac{8}{9}u$ respectively. [7]
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After the collision, $Q$ strikes a fixed vertical wall which is perpendicular to the direction of motion of $P$ and $Q$. The coefficient of restitution between $Q$ and the wall is $e$. When $P$ and $Q$ collide again, $P$ is brought to rest.

\begin{enumerate}[label=(\alph*)]
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\item Find the value of $e$. [7]
\item Explain why there must be a third collision between $P$ and $Q$. [1]
\end{enumerate}

\hfill \mbox{\textit{Edexcel M2 2003 Q6 [15]}}