A rocket is launched vertically upward from Earth s surface at a speed of 5.1 km/s .
4 Answers

A rocket, unlike unpowered projectiles, is propelled for a period of time after launch. In fact, when a rocket is launched it is presumed to have a certain period of upward acceleration. This is different from other projectiles that have only a downward acceleration = g.
So even though a initial vertically upward speed is given…the fact that the word ROCKET is used makes this question numerically impossible to solve w/o more info.
IF, instead of rocket, this object was a cannon fired shell, the time to reach max height = 5100/9.81 = 519.9 s and the max altitude = 1/2gt² = (0.5)(9.81)(519.9)² = 1325688 ≈ 1,330,000 m*
*this answer is based on a constant g = 9.81 m/s², which is not the case for such a high altitude. The avg g for this altitude is more like: g{avg} = 8.50 m/s², which means the computed altitude should be: 1,150,000 m.

As the rocket rises to its maximum altitude, its velocity decreases from its initial velocity to 0 m/s at the rate of 9.8 m/s each second. Use the following equation to determine the maximum height.
vf^2 = vi^2 + 2 * a * d, vf = 0, vi = 5,100 m/s, a = 9.8
0 = 5,100^2 + 2 * 9.8 * d
19.6 * d = 5,100^2
d = 5,100^2 ÷ 19.6
This is approximately 1.327 * 10^6 meters.

h = Voy^2/2g = 25*10^6/20 = 1.25*10^6 m

without knowing burn time its impossible to answer