15 At time \(t = 0\), a parachutist jumps out of an airplane that is travelling horizontally.
The velocity, \(\mathbf { v } \mathrm { m } \mathrm { s } ^ { - 1 }\), of the parachutist at time \(t\) seconds is given by: $$\mathbf { v } = \left( 40 \mathrm { e } ^ { - 0.2 t } \right) \mathbf { i } + 50 \left( \mathrm { e } ^ { - 0.2 t } - 1 \right) \mathbf { j }$$ The unit vectors \(\mathbf { i }\) and \(\mathbf { j }\) are horizontal and vertical respectively.
Assume that the parachutist is at the origin when \(t = 0\)
Model the parachutist as a particle. 15

1. Find an expression for the position vector of the parachutist at time \(t\).
   [0pt] [4 marks] 15
2. The parachutist opens her parachute when she has travelled 100 metres horizontally.
   Find the vertical displacement of the parachutist from the origin when she opens her parachute.
   [0pt] [4 marks]
   15
3. Carefully, explaining the steps that you take, deduce the value of \(g\) used in the formulation of this model.