\begin{enumerate}
  \item Particle $P$ has mass $4 m$ and particle $Q$ has mass $2 m$.
\end{enumerate}

The particles are moving in opposite directions along the same straight line on a smooth horizontal surface.

Particle $P$ collides directly with particle $Q$.\\
Immediately before the collision, the speed of $P$ is $2 u$ and the speed of $Q$ is $3 u$.\\
Immediately after the collision, the speed of $P$ is $x$ and the speed of $Q$ is $y$.\\
The direction of motion of each particle is reversed as a result of the collision.\\
The total kinetic energy of $P$ and $Q$ after the collision is half of the total kinetic energy of $P$ and $Q$ before the collision.\\
(a) Show that $y = \frac { 8 } { 3 } u$

The coefficient of restitution between $P$ and $Q$ is $e$.\\
(b) Find the value of $e$.

After the collision, $Q$ hits a smooth fixed vertical wall that is perpendicular to the direction of motion of $Q$.

Particle $Q$ rebounds.\\
The coefficient of restitution between $Q$ and the wall is $f$.\\
Given that there is no second collision between $P$ and $Q$,\\
(c) find the range of possible values of $f$.

Given that $f = \frac { 1 } { 4 }$\\
(d) find, in terms of $m$ and $u$, the magnitude of the impulse received by $Q$ as a result of its impact with the wall.

\hfill \mbox{\textit{Edexcel M2 2023 Q7 [14]}}