8 The displacement, \(x \mathrm {~m}\), from the origin O of a particle on the \(x\)-axis is given by
$$x = 10 + 36 t + 3 t ^ { 2 } - 2 t ^ { 3 }$$
where \(t\) is the time in seconds and \(- 4 \leqslant t \leqslant 6\).
- Write down the displacement of the particle when \(t = 0\).
- Find an expression in terms of \(t\) for the velocity, \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\), of the particle.
- Find an expression in terms of \(t\) for the acceleration of the particle.
- Find the maximum value of \(v\) in the interval \(- 4 \leqslant t \leqslant 6\).
- Show that \(v = 0\) only when \(t = - 2\) and when \(t = 3\). Find the values of \(x\) at these times.
- Calculate the distance travelled by the particle from \(t = 0\) to \(t = 4\).
- Determine how many times the particle passes through O in the interval \(- 4 \leqslant t \leqslant 6\).