12 In this question you should take the acceleration due to gravity to be \(10 \mathrm {~ms ^ { - 2 }\).} A small ball \(P\) is projected from a point \(A\) with speed \(39 \mathrm {~ms} ^ { - 1 }\) at an angle of elevation \(\theta\), where \(\sin \theta = \frac { 5 } { 13 }\) and \(\cos \theta = \frac { 12 } { 13 }\). Point \(A\) is 20 m vertically above a point \(B\) on horizontal ground. The ball first lands at a point \(C\) on the horizontal ground (see diagram). The ball \(P\) is modelled as a particle moving freely under gravity.

1. Find the maximum height of \(P\) above the ground during its motion. The time taken for \(P\) to travel from \(A\) to \(C\) is \(T\) seconds.
2. Determine the value of \(T\).
3. State one limitation of the model, other than air resistance or the wind, that could affect the answer to part (b). At the instant that \(P\) is projected, a second small ball \(Q\) is released from rest at \(B\) and moves towards \(C\) along the horizontal ground. At time \(t\) seconds, where \(t \geqslant 0\), the velocity \(v \mathrm {~ms} ^ { - 1 }\) of \(Q\) is given by
   \(v = k t ^ { 3 } + 6 t ^ { 2 } + \frac { 3 } { 2 } t\),
   where \(k\) is a positive constant.
4. Given that \(P\) and \(Q\) collide at \(C\), determine the acceleration of \(Q\) immediately before this collision.
   \includegraphics[max width=\textwidth, alt={}, center]{977ffad6-2440-46bf-9f17-0f30817d2ddf-10_607_803_303_246} The diagram shows a small block \(B\), of mass 2 kg , and a particle \(P\), of mass 4 kg , which are attached to the ends of a light inextensible string. The string is taut and passes over a small smooth pulley fixed at the intersection of a horizontal surface and an inclined plane. The particle can move on the inclined plane, which is rough, and which makes an angle of \(60 ^ { \circ }\) with the horizontal. The block can move on the horizontal surface, which is also rough. The system is released from rest, and in the subsequent motion \(P\) moves down the plane and \(B\) does not reach the pulley. It is given that the coefficient of friction between \(P\) and the inclined plane is twice the coefficient of friction between \(B\) and the horizontal surface.
5. Determine, in terms of \(g\), the tension in the string. When \(P\) is moving at \(2 \mathrm {~ms} ^ { - 1 }\) the string breaks. In the 0.5 seconds after the string breaks \(P\) moves 1.9 m down the plane.
6. Determine the deceleration of \(B\) after the string breaks. Give your answer correct to 3 significant figures.