$($b$)$ A small rocket, with a booster attached, blasts off and heads straight upward. At a

height of $5km$ and velocity of $200m{s}^{-1},$ it releases the booster. You may neglect

air resistance and assume the booster acts under the influence of acceleration due

to gravity. Take $g=9\mathrm{.}81m{s}^{-2}.$

$($i$)$ What is the maximum height the booster attains?

$($ii$)$ What is the velocity of the booster at a height of $6km?$ Explain the significance

of the sign of your answer.

$($iii$)$ Suppose that as the booster travels back down through the height of $6km$ its

rocket reignites for $1s$ imparting an additional upward acceleration $a.$ Determine

the value of a above which the booster reverses its motion.