A particle of mass \(m \mathrm{~kg}\) falls vertically under gravity, from rest. At time \(t \mathrm{~s}, P\) has fallen \(x \mathrm{~m}\) and has velocity \(v \mathrm{~m} \mathrm{~s}^{-1}\). The only forces acting on \(P\) are its weight and a resistance of magnitude \(k m g v \mathrm{~N}\), where \(k\) is a constant.
(a) Find an expression for \(v\) in terms of \(t, g\) and \(k\).

(b) Given that \(k=0.05\), find, in metres, how far \(P\) has fallen when its speed is \(12 \mathrm{~m} \mathrm{~s}^{-1}\).