$($a$)$ A person drives from town $A$ to town $B$ and there are two roads available. The first

road is of length $l$ and the driver can cruise at constant speed $v.$ The second road is

of length $\frac{3}{4}l$ and a certain proportion $()$ of the road is under snow. Without snow

on the road they can travel with speed $v$ but with snow they can only travel with

speed $\frac{1}{2}v.$

$($i$)$ Write down and simplify expressions for the travel time on each road.

$($ii$)$ What is the value of $$ below which it is faster to travel on the second road?

$($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.