4. A particle \(P\) of mass \(m \mathrm {~kg}\) moves along a straight line under the action of a force of magnitude \(\frac { k m } { x ^ { 2 } } \mathrm {~N}\), where \(k\) is a constant, directed towards a fixed point \(O\) on the line, where \(O P = x \mathrm {~m} P\) starts from rest at \(A\), at a distance \(a \mathrm {~m}\) from \(O\). When \(O P = x \mathrm {~m}\), the speed of \(P\) is \(v \mathrm {~ms} ^ { - 1 }\).

1. Show that \(v = \sqrt { \frac { 2 k ( a - x ) } { a x } }\).
   \(B\) is the point half-way between \(O\) and \(A\). When \(k = \frac { 1 } { 2 }\) and \(a = 1\), the time taken by \(P\) to travel from \(A\) to \(B\) is \(T\) seconds
   Assuming the result that, for \(0 \leq x \leq 1 , \int \sqrt { \frac { x } { 1 - x } } \mathrm {~d} x = \arcsin ( \sqrt { } x ) - \sqrt { } \left( x - x ^ { 2 } \right) +\) constant,
2. find the value of \(T\). \section*{MECHANICS 3 (A) TEST PAPER 9 Page 2}