Question

On the pole of the Earth a body is imparted velocity v0 directed vertically up. Knowing the radius of the Earth and the free-fall acceleration on its surface, find the height to which the body will ascend. The air drag is to be neglected.

Answer 1

let m be the mass of the body and h be the height it reaches.

Applying the law of conservation of energy we have,

$-\frac{GMm}{R}+\frac{1}{2}mv_{o}^{2}=-\frac{GMm}{R+h}+0$

$\frac{GM}{R+h}=\frac{2GM-v_{O}^{2}R}{2R}$

cross multiplying and rearranging,

$h=\frac{Rv_{o}^{2}}{2gR-v_{o}^{2}}$