5 A particle $P$ of mass 4.5 kg is free to move along the $x$-axis. In a model of the motion it is assumed that $P$ is acted on by two forces:

\begin{itemize}
  \item a constant force of magnitude $f \mathrm {~N}$ in the positive $x$ direction;
  \item a resistance to motion, $R \mathrm {~N}$, whose magnitude is proportional to the speed of $P$.
\end{itemize}

At time $t$ seconds the velocity of $P$ is $v \mathrm {~ms} ^ { - 1 }$. When $t = 0 , P$ is at the origin $O$ and is moving in the positive direction with speed $u \mathrm {~ms} ^ { - 1 }$, and when $v = 5 , R = 2$.
\begin{enumerate}[label=(\alph*)]
\item Show that, according to the model, $\frac { d v } { d t } = \frac { 10 f - 4 v } { 45 }$.
\item \begin{enumerate}[label=(\roman*)]
\item By solving the differential equation in part (a), show that $\mathrm { v } = \frac { 1 } { 2 } \left( 5 \mathrm { f } - ( 5 \mathrm { f } - 2 \mathrm { u } ) \mathrm { e } ^ { - \frac { 4 } { 45 } \mathrm { t } } \right)$.
\item Describe briefly how, according to the model, the speed of $P$ varies over time in each of the following cases.

\begin{itemize}
\end{enumerate}\item $\mathrm { u } < 2.5 \mathrm { f }$
  \item $\mathrm { u } = 2.5 \mathrm { f }$
  \item $u > 2.5 f$
\item In the case where $\mathrm { u } = 2 \mathrm { f }$, find in terms of $f$ the exact displacement of $P$ from $O$ when $t = 9$.
\end{itemize}
\end{enumerate}

\hfill \mbox{\textit{OCR Further Mechanics 2019 Q5 [14]}}