Resistance as kv - finding constant k

7 A motorcycle has a maximum power of 72 kilowatts. The motorcycle and its rider are travelling along a straight horizontal road. When they are moving at a speed of \(\mathrm { V } \mathrm { m } \mathrm { s } ^ { - 1 }\), they experience a total resistance force of magnitude \(k V\) newtons, where \(k\) is a constant.

1. The maximum speed of the motorcycle and its rider is \(60 \mathrm {~ms} ^ { - 1 }\). Show that \(k = 20\).
2. When the motorcycle is travelling at \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), the rider allows the motorcycle to freewheel so that the only horizontal force acting is the resistance force. When the motorcycle has been freewheeling for \(t\) seconds, its speed is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and the magnitude of the resistance force is \(20 v\) newtons. The mass of the motorcycle and its rider is 500 kg .
   1. Show that \(\frac { \mathrm { d } v } { \mathrm {~d} t } = - \frac { v } { 25 }\).
   2. Hence find the time that it takes for the speed of the motorcycle to reduce from \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) to \(10 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
      (6 marks)

A car of mass \(1400\text{ kg}\) travelling at a speed of \(v\text{ m s}^{-1}\) experiences a resistive force of magnitude \(40v\text{ N}\). The greatest possible constant speed of the car along a straight level road is \(56\text{ m s}^{-1}\).

1. Find, in kW, the greatest possible power of the car's engine. [2]
2. Find the greatest possible acceleration of the car at an instant when its speed on a straight level road is \(32\text{ m s}^{-1}\). [3]
3. The car travels down a hill inclined at an angle of \(\theta°\) to the horizontal at a constant speed of \(50\text{ m s}^{-1}\). The power of the car's engine is \(60\text{ kW}\). Find the value of \(\theta\). [4]