A catapult launches a test rocket vertically upward

from a well, giving the rocket an initial speed of $80\mathrm{.}0$

m$/$s at ground level. The engines then fire, and the

rocket accelerates upward at $4\mathrm{.}00$ m$/$s$2$ until it

reaches an altitude of $1000$ m$.$ At that point, its

engines fail and the rocket goes into free fall, with

an acceleration of $-9\mathrm{.}80$ m$/$s$2.($a$)$ For what time

interval is the rocket in motion above the ground?

$($b$)$ What is its maximum altitude? $($c$)$ What is its

velocity just before it hits the ground? $($You will

need to consider the motion while the engine is

operating and the free$-$fall motion separately.$)$