4. The acceleration \(a \mathrm {~ms} ^ { - 2 }\) of a particle \(P\) moving in a straight line away from a fixed point \(O\) is given by \(a = \frac { k } { 1 + t }\), where \(t \mathrm {~s}\) is the time that has elapsed since \(P\) left \(O\), and \(k\) is a constant.

1. By solving a suitable differential equation, find an expression for the velocity \(v \mathrm {~ms} ^ { - 1 }\) of \(P\) in terms of \(t , k\) and another constant \(c\). Given that \(v = 0\) when \(t = 0\) and that \(v = 4\) when \(t = 2\),
2. show that \(v \ln 3 = 4 \ln ( 1 + t )\).
3. Calculate the time when \(P\) has a speed of \(8 \mathrm {~ms} ^ { - 1 }\). \section*{MECHANICS 3 (A)TEST PAPER 4 Page 2}