3 In this question \(\mathbf { i }\) and \(\mathbf { j }\) are perpendicular unit vectors and \(c\) is a positive real number.
The resultant of two forces \(c \mathbf { i N }\) and \(- \mathbf { i } + 2 \sqrt { c } \mathbf { j N }\) is denoted by \(R \mathrm {~N}\).

1. Show that the magnitude of \(R\) is \(c + 1\). A car of mass 900 kg travels along a straight horizontal road with constant resistance to motion of magnitude \(( c + 1 ) \mathrm { N }\). The car passes through point A on the road with speed \(6 \mathrm {~ms} ^ { - 1 }\), and 8 seconds later passes through a point B on the same road. The power developed by the car while travelling from A to B is zero. Furthermore, while travelling between A and B, the car's direction of motion is unchanged.
2. Determine the range of possible values of \(c\). The car later passes through a point C on the road. While travelling between B and C the power developed by the car is modelled as constant and equal to 18 kW . The car passes through C with speed \(5 \mathrm {~ms} ^ { - 1 }\) and acceleration \(3.5 \mathrm {~ms} ^ { - 2 }\).
3. Determine the value of \(c\).
4. Suggest how one of the modelling assumptions made in this question could be improved.