7. A raindrop absorbs water as it falls vertically under gravity through a cloud. In a model of the motion the cloud is assumed to consist of stationary water particles. At time \(t\), the mass of the raindrop is \(m\) and the speed of the raindrop is \(v\). At time \(t = 0\), the raindrop is at rest. The rate of increase of the mass of the raindrop with respect to time is modelled as being \(m k v\), where \(k\) is a positive constant.

1. Ignoring air resistance, show from first principles, that $$\frac { \mathrm { d } v } { \mathrm {~d} t } = g - k v ^ { 2 }$$
2. Find the time taken for the raindrop to reach a speed of \(\frac { 1 } { 2 } \sqrt { } \left( \frac { g } { k } \right)\)