physics
posted by nate .
A rocket is fired straight up through the atmosphere from the South Pole, burning out at an altitude of 208 km when traveling at 5.8 km/s.
(a) What maximum distance from Earth's surface does it travel before falling back to Earth?

The Earth's radius is Re = 6378 km.
Burnout occurs when R = 6586 km.
At the highest altitude, the burnout kinetic energy equals the gain in potential energy.
(1/2) M V^2 = GM[1/R  1/(R+h)]
Solve for R+h. R+hRe is the distance from the surface of the earth.
M is the Earth's mass and G is the universal constant of gravity.
In case you don't want to look up M and G,you can use the fact that
GM/Re^2 = g = 9.81 m/s^2