5 The pilot of a hot air balloon keeps it at a fixed altitude by dropping sand from the balloon. Each grain of sand has mass \(m \mathrm {~kg}\) and is released from rest. When a grain has fallen a distance \(x \mathrm {~m}\), it has speed \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Each grain falls vertically and the only forces acting on it are its weight and air resistance of magnitude \(m k v ^ { 2 } \mathrm {~N}\), where \(k\) is a positive constant.

1. Show that \(\left( \frac { v } { g - k v ^ { 2 } } \right) \frac { \mathrm { d } v } { \mathrm {~d} x } = 1\).
2. Find \(v ^ { 2 }\) in terms of \(k , g\) and \(x\). Hence show that, as \(x\) becomes large, the limiting value of \(v\) is \(\sqrt { \frac { g } { k } }\).
3. Given that the altitude of the balloon is 300 m and that each grain strikes the ground at \(90 \%\) of its limiting velocity, find \(k\).