A smooth uniform sphere $A$, of mass $m$, is moving with velocity $8u$ in a straight line on a smooth horizontal table. A smooth uniform sphere $B$, of mass $4m$, has the same radius as $A$ and is moving on the table with velocity $u$.

\includegraphics{figure_4}

The sphere $A$ collides directly with the sphere $B$.

The coefficient of restitution between $A$ and $B$ is $e$.

\begin{enumerate}[label=(\alph*)]
\item \begin{enumerate}[label=(\roman*)]
\item Find, in terms of $u$ and $e$, the velocities of $A$ and $B$ immediately after the collision. [6 marks]

\item The direction of motion of $A$ is reversed by the collision. Show that $e > a$, where $a$ is a constant to be determined. [2 marks]
\end{enumerate}

\item Subsequently, $B$ collides with a fixed smooth vertical wall which is at right angles to the direction of motion of $A$ and $B$. The coefficient of restitution between $B$ and the wall is $\frac{2}{5}$.

The sphere $B$ collides with $A$ again after rebounding from the wall.

Show that $e < b$, where $b$ is a constant to be determined. [3 marks]

\item Given that $e = \frac{4}{7}$, find, in terms of $m$ and $u$, the magnitude of the impulse exerted on $B$ by the wall. [3 marks]
\end{enumerate}

\hfill \mbox{\textit{AQA M3 2016 Q4 [14]}}