5 A particle \(P\) of mass 3 kg moves on the \(x\)-axis under the action of a single force acting in the positive \(x\)-direction. At time \(t \mathrm {~s}\), where \(t \geqslant 0\), the displacement of \(P\) is \(x \mathrm {~m}\) and its velocity is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The magnitude of the force acting is inversely proportional to \(( t + 1 ) ^ { 2 }\). Initially \(P\) is at rest at the point where \(x = 1\). When \(t = 1 , v = 2\).

1. Show that \(\frac { \mathrm { dv } } { \mathrm { dt } } = \frac { \mathrm { k } } { 3 ( \mathrm { t } + 1 ) ^ { 2 } }\) where \(k\) is a constant.
2. Find an expression for \(v\) in terms of \(t\).
3. Find an expression for \(x\) in terms of \(t\). As \(t\) increases, \(v\) approaches a limiting value, \(\mathrm { V } _ { \mathrm { T } }\).
4. Determine how far \(P\) is from its initial position at the instant when \(v\) is \(95 \%\) of \(\mathrm { V } _ { \mathrm { T } }\).