First, find the equation that models both the velocity and acceleration as function of time.
v(t) = dx/dt = 5t^{2}/2  t^{3}/3
a(t) = dv/dt = 5t  t^{2}
a) The maximum height will occur when velocity is zero (it has reached it zenith).
v = 0 = 5t^{2}/2  t^{3}/3
t = 7.5 s
Substitute time back into the position equation gives,
y_{t=7.5} = 5 (7.5)^{3}/6  (7.5)^{4}/12
= 87.89 ft
b) Use the acceleration equation to find the time when the acceleration will be zero.
a = 0 = 5t  t^{2}
t = 5 s (zero is not a realistic solution)
Substitute time back into the position equation gives,
y_{t=5.0} = 5 (5)^{3}/6  (5)^{4}/12
= 52.08 ft
So, the rocket has a negative acceleration after the rocket goes up only 52.08 ft, but it keeps climbing until it reaches its zenith of 87.89 feet.
