1. A particle \(P\) is moving along the \(x\)-axis. At time \(t\) seconds, where \(t \geqslant 0 , P\) is \(x\) metres from the origin \(O\) and is moving with speed \(v \mathrm {~ms} ^ { - 1 }\)

The acceleration of \(P\) has magnitude \(\frac { 2 } { ( 2 x + 1 ) ^ { 3 } } \mathrm {~ms} ^ { - 2 }\) and is directed towards \(O\)
When \(t = 0 , P\) passes through \(O\) in the positive \(x\) direction with speed \(1 \mathrm {~ms} ^ { - 1 }\)

1. Find \(v\) in terms of \(x\)
2. Show that \(x = \frac { 1 } { 2 } ( \sqrt { ( 4 t + 1 ) } - 1 )\)