3 A particle \(P\) of mass 0.4 kg is projected horizontally along a smooth horizontal plane from a point \(O\). At time \(t \mathrm {~s}\) after projection the velocity of \(P\) is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\). A force of magnitude \(0.8 t \mathrm {~N}\) directed away from \(O\) acts on \(P\) and a force of magnitude \(2 \mathrm { e } ^ { - t } \mathrm {~N}\) opposes the motion of \(P\).

1. Show that \(\frac { \mathrm { d } v } { \mathrm {~d} t } = 2 t - 5 \mathrm { e } ^ { - t }\).
2. Given that \(v = 8\) when \(t = 1\), express \(v\) in terms of \(t\).
3. Find the speed of projection of \(P\).