A parachutist of mass \(m\) kg opens his parachute when he is moving vertically downwards with speed \(50\,\text{m s}^{-1}\). At time \(t\) seconds after opening his parachute, he has fallen a distance \(x\) m from the point where he opened his parachute, and his speed is \(v\,\text{m s}^{-1}\).

The forces acting on him are his weight and a resistive force of magnitude \(mv\) N.

(a) Find an expression for \(v\) in terms of \(t\).

(b) Find an expression for \(x\) in terms of \(t\).

(c) Find the distance that the parachutist has fallen, since opening his parachute, when his speed is \(15\,\text{m s}^{-1}\).