2 A particle $P$ of mass 2.4 kg is moving in a straight line $O A$ on a horizontal plane. $P$ is acted on by a force of magnitude 30 N in the direction of motion. The distance $O A$ is 10 m .
\begin{enumerate}[label=(\alph*)]
\item Find the work done by this force as $P$ moves from $O$ to $A$.

The motion of $P$ is resisted by a constant force of magnitude $R \mathrm {~N}$. The velocity of $P$ increases from $12 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ at $O$ to $18 \mathrm {~ms} ^ { - 1 }$ at $A$.
\item Find the value of $R$.
\item Find the average power used in overcoming the resistance force on $P$ as it moves from $O$ to $A$.

When $P$ reaches $A$ it collides directly with a particle $Q$ of mass 1.6 kg which was at rest at $A$ before the collision. The impulse exerted on $Q$ by $P$ as a result of the collision is 17.28 Ns .
\item \begin{enumerate}[label=(\roman*)]
\item Find the speed of $Q$ after the collision.
\item Hence show that the collision is inelastic.

It is required to model the motion of a car of mass $m \mathrm {~kg}$ travelling at a constant speed $v \mathrm {~m} \mathrm {~s} ^ { - 1 }$ around a circular portion of banked track. The track is banked at $30 ^ { \circ }$ (see diagram).\\
\includegraphics[max width=\textwidth, alt={}, center]{b9741472-f230-4e2d-9c8b-47f7e168e938-03_355_565_269_274}

In a model, the following modelling assumptions are made.

\begin{itemize}
\end{enumerate}\item The track is smooth.
  \item The car is a particle.
  \item The car follows a horizontal circular path with radius $r \mathrm {~m}$.\\
(a) Show that, according to the model, $\sqrt { 3 } v ^ { 2 } = g r$.
\end{itemize}

For a particular portion of banked track, $r = 24$.\\
(b) Find the value of $v$ as predicted by the model.

A car is being driven on this portion of the track at the constant speed calculated in part (b). The driver finds that in fact he can drive a little slower or a little faster than this while still moving in the same horizontal circle.\\
(c) Explain

\begin{itemize}
  \item how this contrasts with what the model predicts,
  \item how to improve the model to account for this.
\end{itemize}
\end{enumerate}

\hfill \mbox{\textit{OCR FM1 AS 2021 Q2 [14]}}