A particle \(P\) of mass 2 kg moves along a horizontal straight line. The point \(O\) is a fixed point on this line. At time \(t \mathrm{~s}\) the velocity of \(P\) is \(v \mathrm{~ms}^{-1}\) and the displacement of \(P\) from \(O\) is \(x \mathrm{~m}\).
A force of magnitude \(\left(8 x-\frac{128}{x^{3}}\right) \mathrm{N}\) acts on \(P\) in the direction \(O P\). When \(t=0, x=8\) and \(v=-15\).
(a) Show that \(v=-\frac{2}{x}\left(x^{2}-4\right)\).
(b) Find an expression for \(x\) in terms of \(t\).