Problem source

Three collinear, non-touching particles $A$, $B$ and $C$ have masses $a$, $b$ and $c$,
respectively, and  are 
at rest on a smooth horizontal surface. 
The particle $A$ is given an initial velocity $u$ towards~$B$. 
These particles collide, giving $B$ a velocity $v$ towards $C$. 
These two particles then collide, giving $C$ a velocity $w$. 
The coefficient of 
restitution is $e$ in both collisions. 
Determine an expression for $v$, and show that 
\[
\displaystyle w = \frac {abu \l 1+e \r^2}{\l a + b \r \l b+c \r}\;.
\]
Determine the final velocities of each of the three particles in the cases:
\begin{questionparts}
\item $\displaystyle \frac ab  = \frac bc = e\,$;
\item $\displaystyle \frac ba  = \frac cb  = e\,$.
\end{questionparts}