**AS Mathematics, Berry J., Fentem R., Francis B., Graham T., Harper Collins 2000, Ex 24. P1 Q6 [Modified]**

Application of Vectors: Motion

A rocket was launched at a fireworks display. At time $t$ seconds, the position of the rockets, in metres relative to the launch location (let this be the origin $O$), was $\mathbf{r}=20t\mathbf{i}+\left[ {{(t-4)}^{3}}+64 \right]\mathbf{j}$ where $\mathbf{i}$ and $\mathbf{j}$ are horizontal and vertical unit vectors respectively. The rocket exploded when $t=4$.

(i)

Write down the height of the rocket when it exploded.

(ii)

Find the displacement of the rocket from $t=3$ to $t=4$.

(iii)