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 \(t=4\).

1. Write down the height of the rocket when it exploded.
2. Find the displacement of the rocket from \(t=3\) to \(t=4\).
3. Show that the Cartesian equation representing the path of the rocket was \(y={{\left( \frac{x}{20}-4 \right)}^{3}}+64\) and sketch the path of the rocket. Hence state the direction of movement of the rocket when it exploded.