2 A particle moves in a horizontal plane under the action of a single force, \(\mathbf { F }\) newtons.
The unit vectors \(\mathbf { i }\) and \(\mathbf { j }\) are directed east and north respectively.
At time \(t\) seconds, the velocity of the particle, \(\mathbf { v } \mathrm { m } \mathrm { s } ^ { - 1 }\), is given by $$\mathbf { v } = \left( 8 t - t ^ { 4 } \right) \mathbf { i } + 6 \mathrm { e } ^ { - 3 t } \mathbf { j }$$

1. Find an expression for the acceleration of the particle at time \(t\).
2. The mass of the particle is 2 kg .
   1. Find an expression for the force \(\mathbf { F }\) acting on the particle at time \(t\).
   2. Find the magnitude of \(\mathbf { F }\) when \(t = 1\).
3. Find the value of \(t\) when \(\mathbf { F }\) acts due south.
4. When \(t = 0\), the particle is at the point with position vector \(( 3 \mathbf { i } - 5 \mathbf { j } )\) metres. Find an expression for the position vector, \(\mathbf { r }\) metres, of the particle at time \(t\).
   [0pt] [4 marks]