13 In this question the unit vectors \(\mathbf { i }\) and \(\mathbf { j }\) are in the directions east and north respectively.
At time \(t\) seconds, where \(t \geqslant 0\), a particle \(P\) of mass 2 kg is moving on a smooth horizontal surface under the action of a constant horizontal force \(( - 8 \mathbf { i } - 54 \mathbf { j } ) \mathrm { N }\) and a variable horizontal force \(\left( 4 t \mathbf { i } + 6 ( 2 t - 1 ) ^ { 2 } \mathbf { j } \right) \mathrm { N }\).

1. Determine the value of \(t\) when the forces acting on \(P\) are in equilibrium. It is given that \(P\) is at rest when \(t = 0\).
2. Determine the speed of \(P\) at the instant when \(P\) is moving due north.
3. Determine the distance between the positions of \(P\) when \(t = 0\) and \(t = 3\).