\includegraphics{figure_2}

A smooth sphere $P$ of mass $2m$ is moving in a straight line with speed $v$ 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.

\begin{enumerate}[label=(\alph*)]
\item Show that, immediately after the collision, the speeds of $P$ and $Q$ are $\frac{3}{4}u$ and $\frac{5}{4}u$ respectively.
[7]
\end{enumerate}

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*)]
\setcounter{enumi}{1}
\item Find the value of $e$.
[7]
\end{enumerate}

\begin{enumerate}[label=(\alph*)]
\setcounter{enumi}{2}
\item Explain why there must be a third collision between $P$ and $Q$.
[1]
\end{enumerate}

Show that $GX = \frac{44}{63}a$.
[6]

The mass of the lamina is $M$. A particle of mass $λM$ is attached to the lamina at $C$. The lamina is suspended from $B$ and hangs freely under gravity with $AB$ horizontal.

\begin{enumerate}[label=(\alph*)]
\setcounter{enumi}{1}
\item Find the value of $λ$.
[3]
\end{enumerate}

TURN OVER FOR QUESTION 7

\hfill \mbox{\textit{Edexcel M2  Q4 [24]}}