A particle \(P\) of mass 0.5 kg is released from rest at time \(t = 0\) and falls vertically through a liquid. The motion of \(P\) is resisted by a force of magnitude \(2 v \mathrm {~N}\), where \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) is the speed of \(v\) at time \(t\) seconds.
Show that \(5 \frac { \mathrm {~d} v } { \mathrm {~d} t } = 49 - 20 v\).
Find the speed of \(P\) when \(t = 1\).
(5)
A small smooth sphere \(S\) of mass \(m\) is attached to one end of a light inextensible string of length \(2 a\). The other end of the string is attached to a fixed point \(A\) which is at a distance \(a \sqrt { } 3\) from a smooth vertical wall. The sphere \(S\) hangs at rest in equilibrium. It is then projected horizontally towards the wall with a speed \(\sqrt { \left( \frac { 37 g a } { 5 } \right) }\).
Show that \(S\) strikes the wall with speed \(\sqrt { \left( \frac { 27 g a } { 5 } \right) }\).
(4)
Given that the loss in kinetic energy due to the impact with the wall is \(\frac { 3 m g a } { 5 }\),
find the coefficient of restitution between \(S\) and the wall.
(7)