8 A car of mass 800 kg travels up a line of greatest slope of a straight road inclined at \(5 ^ { \circ }\) to the horizontal. The power developed by the car is constant and equal to 25 kW . The resistance to the motion of the car is constant and equal to 750 N . The car passes through a point A on the road with speed \(7 \mathrm {~ms} ^ { - 1 }\).

1. Find
   • the acceleration of the car at A ,
   • the greatest steady speed at which the car can travel up the hill.
   The car later passes through a point B on the road where \(\mathrm { AB } = 131 \mathrm {~m}\). The time taken to travel from A to B is 10.4 s .
2. Calculate the speed of the car at B. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{a69c3e7a-dfc4-438b-84ba-e88c15d421ea-06_442_346_292_251} \captionsetup{labelformat=empty} \caption{Fig. 9}
   \end{figure} A particle P of mass \(m\) is joined to a fixed point O by a light inextensible string of length \(l . \mathrm { P }\) is released from rest with the string taut and making an acute angle \(\alpha\) with the downward vertical, as shown in Fig. 9. At a time \(t\) after P is released the string makes an angle \(\theta\) with the downward vertical and the tension in the string is \(T\). Angles \(\alpha\) and \(\theta\) are measured in radians.
3. Show that $$\left( \frac { \mathrm { d } \theta } { \mathrm {~d} t } \right) ^ { 2 } = \frac { 2 g } { l } \cos \theta + k _ { 1 } ,$$ where \(k _ { 1 }\) is a constant to be determined in terms of \(g , l\) and \(\alpha\).
4. Show that $$T = 3 m g \cos \theta + k _ { 2 } ,$$ where \(k _ { 2 }\) is a constant to be determined in terms of \(m , g\) and \(\alpha\). It is given that \(\alpha\) is small enough for \(\alpha ^ { 2 }\) to be negligible.
5. Find, in terms of \(m\) and \(g\), the approximate tension in the string.
6. Show that the motion of P is approximately simple harmonic.