A particle \(P\) of mass 0.5 kg moves along the positive \(x\)-axis. It moves away from the origin \(O\) under the action of a single force directed away from \(O\). When \(O P = x\) metres, the magnitude of the force is \(\frac { 3 } { ( x + 1 ) ^ { 3 } } \mathrm {~N}\) and the speed of \(P\) is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
Initially \(P\) is at rest at \(O\).
Show that \(v ^ { 2 } = 6 \left( 1 - \frac { 1 } { ( x + 1 ) ^ { 2 } } \right)\).
Show that the speed of \(P\) never reaches \(\sqrt { } 6 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
Find \(x\) when \(P\) has been moving for 2 seconds.