\includegraphics{figure_2}

Figure 1 represents the plan of part of a smooth horizontal floor, where $W_1$ and $W_2$ are two fixed parallel vertical walls. The walls are $3$ metres apart.

A particle lies at rest at a point $O$ on the floor between the two walls, where the point $O$ is $d$ metres, $0 < d \leq 3$, from $W_1$.

At time $t = 0$, the particle is projected from $O$ towards $W_1$ with speed $u\text{ms}^{-1}$ in a direction perpendicular to the walls.

The coefficient of restitution between the particle and each wall is $\frac{2}{3}$.

The particle returns to $O$ at time $t = T$ seconds, having bounced off each wall once.

\begin{enumerate}[label=(\alph*)]
\item Show that $T = \frac{45 - 5d}{4u}$.
[6]

\item The value of $u$ is fixed, the particle still hits each wall once but the value of $d$ can now vary.

Find the least possible value of $T$, giving your answer in terms of $u$. You must give a reason for your answer.
[2]
\end{enumerate}

\hfill \mbox{\textit{SPS SPS FM Mechanics 2021 Q3 [8]}}