2 A car of mass 1500 kg has an engine with maximum power 60 kW . When the car is travelling at $10 \mathrm {~m} \mathrm {~s} ^ { - 1 }$ along a straight horizontal road using maximum power, its acceleration is $3.3 \mathrm {~ms} ^ { - 2 }$.

In an initial model of the motion of the car it is assumed that the resistance to motion is constant.
\begin{enumerate}[label=(\alph*)]
\item Using this initial model, find the greatest possible steady speed of the car along the road.

In a refined model the resistance to motion is assumed to be proportional to the speed of the car.
\item Using this refined model, find the greatest possible steady speed of the car along the road.

The greatest possible steady speed of the car on the road is measured and found to be $21.6 \mathrm {~ms} ^ { - 1 }$.
\item Explain what this value means about the models used in parts (a) and (b).\\
\includegraphics[max width=\textwidth, alt={}, center]{aa25b8a6-9a5a-4de2-9534-18db8a175c34-03_583_378_169_255}

As shown in the diagram, $A B$ is a long thin rod which is fixed vertically with $A$ above $B$. One end of a light inextensible string of length 1 m is attached to $A$ and the other end is attached to a particle $P$ of mass $m _ { 1 } \mathrm {~kg}$. One end of another light inextensible string of length 1 m is also attached to $P$. Its other end is attached to a small smooth ring $R$, of mass $m _ { 2 } \mathrm {~kg}$, which is free to move on $A B$.

Initially, $P$ moves in a horizontal circle of radius 0.6 m with constant angular velocity $\omega \mathrm { rads } ^ { - 1 }$. The magnitude of the tension in string $A P$ is denoted by $T _ { 1 } \mathrm {~N}$ while that in string $P R$ is denoted by $T _ { 2 } \mathrm {~N}$.\\
(a) By considering forces on $R$, express $T _ { 2 }$ in terms of $m _ { 2 }$.\\
(b) Show that
\begin{enumerate}[label=(\roman*)]
\item $T _ { 1 } = \frac { 49 } { 4 } \left( m _ { 1 } + m _ { 2 } \right)$,
\item $\omega ^ { 2 } = \frac { 49 \left( m _ { 1 } + 2 m _ { 2 } \right) } { 4 m _ { 1 } }$.\\
(c) Deduce that, in the case where $m _ { 1 }$ is much bigger than $m _ { 2 } , \omega \approx 3.5$.

In a different case, where $m _ { 1 } = 2.5$ and $m _ { 2 } = 2.8 , P$ slows down. Eventually the system comes to rest with $P$ and $R$ hanging in equilibrium.
\end{enumerate}\item Find the total energy lost by $P$ and $R$ as the angular velocity of $P$ changes from the initial value of $\omega \mathrm { rads } ^ { - 1 }$ to zero.
\end{enumerate}

\hfill \mbox{\textit{OCR FM1 AS 2021 Q2 [11]}}