6 In this question you must show detailed reasoning. In this question, positions are given relative to a fixed origin, O . The unit vectors \(\mathbf { i }\) and \(\mathbf { j }\) are in the \(x\) - and \(y\)-directions respectively in a horizontal plane. Distances are measured in centimetres and the time, \(t\), is measured in seconds, where \(0 \leqslant t \leqslant 5\). A small radio-controlled toy car C moves on a horizontal surface which contains O.
The acceleration of C is given by \(2 \mathbf { i } + t \mathbf { j } \mathrm {~cm} \mathrm {~s} ^ { - 2 }\).
When \(t = 4\), the displacement of C from O is \(16 \mathbf { i } + \frac { 32 } { 3 } \mathbf { j } \mathrm {~cm}\), and the velocity of C is \(8 \mathbf { i } \mathrm {~cm} \mathrm {~s} ^ { - 1 }\).
Determine a cartesian equation for the path of C for \(0 \leqslant t \leqslant 5\). You are not required to simplify your answer.