A missile silo is used to launch interplanetary rockets vertically upward out of the silo, giving the rocket an initial speed of $80\mathrm{.}6$ m$/$s at ground level. As the rocket clears the silo, the engines fire, and the rocket accelerates upward at $3\mathrm{.}80$ m$/$s$2$ until it reaches an altitude of $1,040$ m$.$ At that point its engines fail, and the rocket goes into free fall, with an acceleration of $9\mathrm{.}80$ m$/$s$2.($You will need to consider the motion while the engine is operating and the free$-$fall motion separately. Due to the nature of this problem, do not use rounded intermediate values in your calculations$).$

A$)$ Determine the velocity of the rocket $($in m$/$s$)$ at the end of the engine burn time and also the burn time $($in s$).$ 

B$)$ Determine the maximum altitude of the rocket $($in m$)$ and the total time $($in s$)$ for the rocket to reach this altitude from ground level.

C$)$ Determine the rocket's velocity $($in m$/$s$)$ just before ground impact and its total time of flight $($in s$).$