Three particles $A$, $B$ and $C$ are free to move in the same straight line on a large smooth horizontal surface. Their masses are 3.3 kg, 2.2 kg and 1 kg respectively. The coefficient of restitution in collisions between any two of them is $e$.

Initially, $B$ and $C$ are at rest and $A$ is moving towards $B$ with speed $u \text{ ms}^{-1}$ (see diagram). $A$ collides directly with $B$ and $B$ then goes on to collide directly with $C$.

\includegraphics{figure_4}

\begin{enumerate}[label=(\alph*)]
\item The velocities of $A$ and $B$ immediately after the first collision are denoted by $v_A \text{ ms}^{-1}$ and $v_B \text{ ms}^{-1}$ respectively.

$\bullet$ Show that $v_A = \frac{u(3-2e)}{5}$.

$\bullet$ Find an expression for $v_B$ in terms of $u$ and $e$. [4]

\item Find an expression in terms of $u$ and $e$ for the velocity of $B$ immediately after its collision with $C$. [4]
\end{enumerate}

After the collision between $B$ and $C$ there is a further collision between $A$ and $B$.

\begin{enumerate}[label=(\alph*)]
\setcounter{enumi}{2}
\item Determine the range of possible values of $e$. [4]
\end{enumerate}

\hfill \mbox{\textit{OCR FM1 AS 2021 Q4 [12]}}