Arocket, initially at rest on the ground, accelerates straight upward from rest with constant acceleration 58.8m/s2 . the acceleration period lasts for time 7.00s until the fuel is exhausted. after that, the rocket is in free fall.
find the maximum height ymax reached by the rocket. ignore air resistance and assume a constant acceleration due to gravity equal to 9.80 m/s2 .

Split the operation in two parts. Part A) constant acceleration 58.8m/s^2, Part B) free fall.

Part A)
Height reached, y = a*[t^2] / 2 = 58.8 m/s^2 * [7.00 s]^2 / 2 = 1440.6 m

Now you need the final speed to use it as initial speed of the next part.

Vf = Vo + at = 0 + 58.8m/s^2 * 7.00 s = 411.6 m/s

Part B) Free fall

Maximum height, y max ==> Vf = 0

Vf = Vo - gt ==> t = [Vo - Vf]/g = 411.6 m/s / 9.8 m/s^2 = 42 s

ymax = yo + Vo*t - g[t^2] / 2

ymax = 1440.6 m + 411.6m/s * 42 s - 9.8m/s^2 * [42s]^2 /2
ymax = 1440.6 m + 17287.2m - 8643.6m = 10084.2 m

ymax = 10084.2m