A particle \(P\) of mass \(m \mathrm{~kg}\) moves in a horizontal straight line against a resistive force of magnitude \(m k v^{2} \mathrm{~N}\), where \(v \mathrm{~ms}^{-1}\) is the speed of \(P\) after it has moved a distance \(x \mathrm{~m}\) and \(k\) is a positive constant. The initial speed of \(P\) is \(u \mathrm{~ms}^{-1}\).
(a) Show that \(x=\frac{1}{k} \ln 2\) when \(v=\frac{1}{2} u\).

Beginning at the instant when the speed of \(P\) is \(\frac{1}{2} u\), an additional force acts on \(P\). This force has magnitude \(\frac{5 m}{v} \mathrm{~N}\) and acts in the direction of increasing \(x\).
(b) Show that when the speed of \(P\) has increased again to \(u \mathrm{~ms}^{-1}\), the total distance travelled by \(P\) is given by an expression of the form
\(\frac{1}{3 k} \ln \left(\frac{A-k u^{3}}{B-k u^{3}}\right),\)
stating the values of the constants \(A\) and \(B\).