posted by Joe .
At the Earth's surface, a projectile is launched straight up at a speed of 9.9 km/s. To what height will it rise? Ignore air resistance and the rotation of the Earth.
(1/2) m v^2 = potential energy
potential energy is mgh for low altitude but that will not work for high altitude where g is less than 9.81 m/s^2
increase in potential = G M m (1/Re - 1/(Re+h)
call z = Re +h
(1/2) v^2 = G M (1/Re - 1/z)
solve for z
then h = z - Re