4 A particle \(P\) of mass 0.4 kg is projected horizontally with speed \(2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) from a fixed point \(O\) on a smooth horizontal surface. At time \(t \mathrm {~s}\) after projection \(P\) is \(x \mathrm {~m}\) from \(O\) and is moving away from \(O\) with speed \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\). There is a force of magnitude \(1.6 v ^ { 2 } \mathrm {~N}\) resisting the motion of \(P\).

1. Find an expression for \(\frac { \mathrm { d } v } { \mathrm {~d} x }\) in terms of \(v\), and hence show that \(v = 2 \mathrm { e } ^ { - 4 x }\).
2. Find the distance travelled by \(P\) in the 0.5 seconds after it leaves \(O\).