2 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, $\frac { \mathrm { d } v } { \mathrm {~d} t } = \frac { 10 f - 4 v } { 45 }$.
\item \begin{enumerate}[label=(\roman*)]
\item By solving the differential equation in part (a), show that $v = \frac { 1 } { 2 } \left( 5 f - ( 5 f - 2 u ) \mathrm { e } ^ { - \frac { 4 } { 45 } t } \right)$.
\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 $u < 2.5 f$
  \item $u = 2.5 f$
  \item $u > 2.5 f$
\item In the case where $u = 2 f$, find in terms of $f$ the exact displacement of $P$ from $O$ when $t = 9$.\\
\includegraphics[max width=\textwidth, alt={}, center]{439ec1f0-60b8-4272-98cc-0c30d116c4cf-03_314_652_137_660}
\end{itemize}

The flat surface of a smooth solid hemisphere of radius $r$ is fixed to a horizontal plane on a planet where the acceleration due to gravity is denoted by $\gamma$. $O$ is the centre of the flat surface of the hemisphere.

A particle $P$ is held at a point on the surface of the hemisphere such that the angle between $O P$ and the upward vertical through $O$ is $\alpha$, where $\cos \alpha = \frac { 3 } { 4 }$.\\
$P$ is then released from rest. $F$ is the point on the plane where $P$ first hits the plane (see diagram).\\
(a) Find an exact expression for the distance $O F$.

The acceleration due to gravity on and near the surface of the planet Earth is roughly $6 \gamma$.\\
(b) Explain whether $O F$ would increase, decrease or remain unchanged if the action were repeated on the planet Earth.
\end{enumerate}

\hfill \mbox{\textit{OCR Further Mechanics 2021 Q2 [19]}}