6 A cyclist and his bicycle have a total mass of 81 kg . The cyclist starts from rest and rides in a straight line. The cyclist exerts a constant force of 135 N and the motion is opposed by a resistance of magnitude \(9 v \mathrm {~N}\), where \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) is the cyclist's speed at time \(t \mathrm {~s}\) after starting.

1. Show that \(\frac { 9 } { 15 - v } \frac { \mathrm {~d} v } { \mathrm {~d} t } = 1\).
2. Solve this differential equation to show that \(v = 15 \left( 1 - \mathrm { e } ^ { - \frac { 1 } { 9 } t } \right)\).
3. Find the distance travelled by the cyclist in the first 9 s of the motion.