6 A car, of mass \(m\), is moving along a straight smooth horizontal road. At time \(t\), the car has speed \(v\). As the car moves, it experiences a resistance force of magnitude \(0.05 m v\). No other horizontal force acts on the car.
- Show that
$$\frac { \mathrm { d } v } { \mathrm {~d} t } = - 0.05 v$$
- When \(t = 0\), the speed of the car is \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
Show that \(v = 20 \mathrm { e } ^ { - 0.05 t }\).
- Find the time taken for the speed of the car to reduce to \(10 \mathrm {~ms} ^ { - 1 }\).