1 A car of mass \(m\) moves horizontally in a straight line. At time \(t\) the car is a distance \(x\) from a point O and is moving away from O with speed \(v\). There is a force of magnitude \(k v ^ { 2 }\), where \(k\) is a constant, resisting the motion of the car. The car's engine has a constant power \(P\). The terminal speed of the car is \(U\).

1. Show that $$m v ^ { 2 } \frac { \mathrm {~d} v } { \mathrm {~d} x } = P \left( 1 - \frac { v ^ { 3 } } { U ^ { 3 } } \right)$$
2. Show that the distance moved while the car accelerates from a speed of \(\frac { 1 } { 4 } U\) to a speed of \(\frac { 1 } { 2 } U\) is $$\frac { m U ^ { 3 } } { 3 P } \ln A$$ stating the exact value of the constant \(A\). Once the car attains a speed of \(\frac { 1 } { 2 } U\), no further power is supplied by the car's engine.
3. Find, in terms of \(m , P\) and \(U\), the time taken for the speed of the car to reduce from \(\frac { 1 } { 2 } U\) to \(\frac { 1 } { 4 } U\).