Two particles $A$ and $B$ move on a smooth horizontal table. The mass of $A$ is $m$, and the mass of $B$ is $4m$. Initially $A$ is moving with speed $u$ when it collides directly with $B$, which is at rest on the table. As a result of the collision, the direction of motion of $A$ is reversed. The coefficient of restitution between the particles is $e$.

\begin{enumerate}[label=(\alph*)]
\item Find expressions for the speed of $A$ and the speed of $B$ immediately after the collision.
[7]
\end{enumerate}

In the subsequent motion, $B$ strikes a smooth vertical wall and rebounds. The wall is perpendicular to the direction of motion of $B$. The coefficient of restitution between $B$ and the wall is $\frac{4}{5}$. Given that there is a second collision between $A$ and $B$,

\begin{enumerate}[label=(\alph*)]
\setcounter{enumi}{1}
\item show that $\frac{1}{4} < e < \frac{9}{16}$.
[5]
\end{enumerate}

Given that $e = \frac{1}{2}$,

\begin{enumerate}[label=(\alph*)]
\setcounter{enumi}{2}
\item find the total kinetic energy lost in the first collision between $A$ and $B$.
[3]
\end{enumerate}

\hfill \mbox{\textit{Edexcel M2 2006 Q8 [15]}}