A particle \(P\) moving in a straight line has displacement \(x \mathrm{~m}\) from a fixed point \(O\) on the line at time \(t \mathrm{~s}\). The acceleration of \(P\), in \(\mathrm{ms}^{-2}\), is given by \(\frac{200}{x^{2}}-\frac{100}{x^{3}}\) for \(x\gt 0\). When \(t=0, x=1\) and \(P\) has velocity \(10 \mathrm{~ms}^{-1}\) directed towards \(O\).
(a) Show that the velocity \(v \mathrm{~m} \mathrm{~s}^{-1}\) of \(P\) is given by \(v=\frac{10(1-2 x)}{x}\).
(b) Show that \(x\) and \(t\) are related by the equation \(\mathrm{e}^{-40 t}=(2 x-1) \mathrm{e}^{2 x-2}\) and deduce what happens to \(x\) as \(t\) becomes large.