4. A lorry of mass \(M\) is moving along a straight horizontal road. The engine produces a constant driving force of magnitude \(F\). The total resistance to motion is modelled as having magnitude \(k v ^ { 2 }\), where \(k\) is a constant, and \(v\) is the speed of the lorry. Given the lorry moves with constant speed \(V\),

1. show that \(V = \sqrt { \frac { F } { k } }\). Given instead that the lorry starts from rest,
2. show that the distance travelled by the lorry in attaining a speed of \(\frac { 1 } { 2 } V\) is $$\frac { M } { 2 k } \ln \left( \frac { 4 } { 3 } \right)$$