5. A train of mass \(m\) is moving along a straight horizontal railway line. A time \(t\), the train is moving with speed \(v\) and the resistance to motion has magnitude \(k v\), where \(k\) is a constant. The engine of the train is working at a constant rate \(P\).

1. Show that, when \(v > 0 , \quad m v \frac { \mathrm {~d} v } { \mathrm {~d} t } + k v ^ { 2 } = P\). When \(t = 0\), the speed of the train is \(\frac { 1 } { 3 } \sqrt { \left( \frac { P } { k } \right) }\).
2. Find, in terms of \(m\) and \(k\), the time taken for the train to double its initial speed.
   (8) \section*{6.} \begin{figure}[h]
   \captionsetup{labelformat=empty} \caption{Figure 2} \includegraphics[alt={},max width=\textwidth]{fc091589-cb39-47a4-a8b3-06f5fd5ce06a-4_638_285_315_897}
   \end{figure} Two small smooth spheres \(A\) and \(B\), of equal size and of mass \(m\) and \(2 m\) respectively, are moving initially with the same speed \(U\) on a smooth horizontal floor. The spheres collide when their centres are on a line \(L\). Before the collision the spheres are moving towards each other, with their directions of motion perpendicular to each other and each inclined at an angle of \(45 ^ { \circ }\) to the line \(L\), as shown in Figure 2. The coefficient of restitution between the spheres is \(\frac { 1 } { 2 }\).
3. Find the magnitude of the impulse which acts on \(A\) in the collision. \begin{figure}[h]
   \captionsetup{labelformat=empty} \caption{Figure 3} \includegraphics[alt={},max width=\textwidth]{fc091589-cb39-47a4-a8b3-06f5fd5ce06a-4_481_737_1610_792}
   \end{figure} The line \(L\) is parallel to and a distance \(d\) from a smooth vertical wall, as shown in Figure 3.
4. Find, in terms of \(d\), the distance between the points at which the spheres first strike the wall.
   (5)