1 A raindrop of mass \(m\) falls vertically from rest under gravity. Initially the mass of the raindrop is \(m _ { 0 }\). As it falls it loses mass by evaporation at a rate \(\lambda m\), where \(\lambda\) is a constant. Its motion is modelled by assuming that the evaporation produces no resultant force on the raindrop. The velocity of the raindrop is \(v\) at time \(t\). The forces on the raindrop are its weight and a resistance force of magnitude \(k m v\), where \(k\) is a constant.

1. Find \(m\) in terms of \(m _ { 0 } , \lambda\) and \(t\).
2. Write down the equation of motion of the raindrop. Solve this differential equation and hence show that \(v = \frac { g } { \lambda - k } \left( \mathrm { e } ^ { ( \lambda - k ) t } - 1 \right)\).
3. Find the velocity of the raindrop when it has lost half of its initial mass.