4 A particle \(P\) of mass 8 kg moves in a straight line on a smooth horizontal plane. At time \(t \mathrm {~s}\) the displacement of \(P\) from a fixed point \(O\) on the line is \(x \mathrm {~m}\) and the velocity of \(P\) is \(v \mathrm {~ms} ^ { - 1 }\). Initially, \(P\) is at rest at \(O\).
\(P\) is acted on by a horizontal force, directed along the line away from \(O\), with magnitude proportional to \(\sqrt { 9 + v ^ { 2 } }\). When \(v = 1.25\), the magnitude of this force is 13 N .

1. Show that \(\frac { 1 } { \sqrt { 9 + v ^ { 2 } } } \frac { \mathrm {~d} v } { \mathrm {~d} t } = \frac { 1 } { 2 }\).
2. Find an expression for \(v\) in terms of \(t\) for \(t \geqslant 0\).
3. Find an expression for \(x\) in terms of \(t\) for \(t \geqslant 0\).