1.

\begin{figure}[h]
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  \includegraphics[alt={},max width=\textwidth]{a871044a-17c5-440d-8d8f-886939603dd4-02_307_889_244_589}
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\caption{Figure 1}
\end{center}
\end{figure}

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, $0 < d \leqslant 3$, from $W _ { 1 }$

At time $t = 0$, the particle is projected from $O$ towards $W _ { 1 }$ with speed $u \mathrm {~m} \mathrm {~s} ^ { - 1 }$ 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 $T = \frac { 45 - 5 d } { 4 u }$

The value of $u$ is fixed, the particle still hits each wall once but the value of $d$ can now vary.
\item Find the least possible value of $T$, giving your answer in terms of $u$. You must give a reason for your answer.
\end{enumerate}

\hfill \mbox{\textit{Edexcel FM1 2019 Q1 [8]}}