4 A particle moves so that at time \(t\) seconds its velocity \(\mathbf { v } \mathrm { m } \mathrm { s } ^ { - 1 }\) is given by
$$\mathbf { v } = \left( 4 t ^ { 3 } - 12 t + 3 \right) \mathbf { i } + 5 \mathbf { j } + 8 t \mathbf { k }$$
- When \(t = 0\), the position vector of the particle is \(( - 5 \mathbf { i } + 6 \mathbf { k } )\) metres.
Find the position vector of the particle at time \(t\).
- Find the acceleration of the particle at time \(t\).
- Find the magnitude of the acceleration of the particle at time \(t\). Do not simplify your answer.
- Hence find the time at which the magnitude of the acceleration is a minimum.
- The particle is moving under the action of a single variable force \(\mathbf { F }\) newtons. The mass of the particle is 7 kg .
Find the minimum magnitude of \(\mathbf { F }\).