A particle \(P\) of mass \(m\,\text{kg}\) is held at rest at a point \(O\) and released so that it moves vertically under gravity against a resistive force of magnitude \(0.1mv^2\) N, where \(v\,\text{m s}^{-1}\) is the velocity of \(P\) at time \(t\) s.

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

The displacement of \(P\) from \(O\) at time \(t\) is \(x\) m.

(b) Find an expression for \(v^2\) in terms of \(x\).