$(15$ points$)$ To blast open an asteroid in the outer space, a robot launches a $5\mathrm{.}00$ kg steel ball towards it$.$ The force function of the ball as a function of time is given by $F(t)=60\mathrm{.}0-15\mathrm{.}0t^2.$ The canon applies the force to accelerate the steel ball from rest during the time interval from $0$ to $2\mathrm{.}00$ s $.$

a$.$ Write the velocity as a function of time $t.$

b$.$ Write the power as a function of time $t.$ You don't need to expand the factors.

c$.$ Find the total work done from $0$ to $2\mathrm{.}00$ s in J$.$

d$.$ Find the total impulse exerted on the steel ball in Ns$.$

e$.$ If the robot's mass is $400.kg,$ what is its recoil speed in $\frac{m}{s}?$