6 The battery on Carol and Martin's car is flat so the car will not start. They hope to be able to "bump start" the car by letting it run down a hill and engaging the engine when the car is going fast enough. Fig. 6.1 shows the road leading away from their house, which is at A . The road is straight, and at all times the car is steered directly along it.

\begin{itemize}
  \item From A to B the road is horizontal.
  \item Between B and C, it goes up a hill with a uniform slope of $1.5 ^ { \circ }$ to the horizontal.
  \item Between C and D the road goes down a hill with a uniform slope of $3 ^ { \circ }$ to the horizontal. CD is 100 m . (This is the part of the road where they hope to get the car started.)
  \item From D to E the road is again horizontal.
\end{itemize}

\begin{figure}[h]
\begin{center}
  \includegraphics[alt={},max width=\textwidth]{f87e062a-fdf2-45cf-8bc0-d05683b28e1a-4_241_1134_808_450}
\captionsetup{labelformat=empty}
\caption{Fig. 6.1}
\end{center}
\end{figure}

The mass of the car is 750 kg , Carol's mass is 50 kg and Martin's mass is 80 kg .\\
Throughout the rest of this question, whenever Martin pushes the car, he exerts a force of 300 N along the line of the car.\\
(i) Between A and B , Martin pushes the car and Carol sits inside to steer it. The car has an acceleration of $0.25 \mathrm {~m} \mathrm {~s} ^ { - 2 }$.

Show that the resistance to the car's motion is 100 N .

Throughout the rest of this question you should assume that the resistance to motion is constant at 100 N .\\
(ii) They stop at B and then Martin tries to push the car up the hill BC.

Show that Martin cannot push the car up the hill with Carol inside it but can if she gets out.\\
Find the acceleration of the car when Martin is pushing it and Carol is standing outside.\\
(iii) While between B and C , Carol opens the window of the car and pushes it from outside while steering with one hand. Carol is able to exert a force of 150 N parallel to the surface of the road but at an angle of $30 ^ { \circ }$ to the line of the car. This is illustrated in Fig. 6.2.

\begin{figure}[h]
\begin{center}
  \includegraphics[alt={},max width=\textwidth]{f87e062a-fdf2-45cf-8bc0-d05683b28e1a-4_218_426_2133_831}
\captionsetup{labelformat=empty}
\caption{Fig. 6.2}
\end{center}
\end{figure}

Find the acceleration of the car.\\
(iv) At C, both Martin and Carol get in the car and, starting from rest, let it run down the hill under gravity. If the car reaches a speed of $8 \mathrm {~ms} ^ { - 1 }$ they can get the engine to start.

Does the car reach this speed before it reaches D ?

\hfill \mbox{\textit{OCR MEI M1 2015 Q6 [18]}}