4 A motor-cycle, whose mass including the rider is 120 kg , is decelerating on a horizontal straight road. The motor-cycle passes a point \(A\) with speed \(40 \mathrm {~ms} ^ { - 1 }\) and when it has travelled a distance of \(x \mathrm {~m}\) beyond \(A\) its speed is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The engine develops a constant power of 8 kW and resistances are modelled by a force of \(0.25 v ^ { 2 } \mathrm {~N}\) opposing the motion.

1. Show that \(\frac { 480 v ^ { 2 } } { v ^ { 3 } - 32000 } \frac { \mathrm {~d} v } { \mathrm {~d} x } = - 1\).
2. Find the speed of the motor-cycle when it has travelled 500 m beyond \(A\).