Question

A mass $m$ is taken to a height $R$ from the surface of the earth and then is given a vertical velocity $v$. Find the minimum value of $v$, so that mass never returns to the surface of the earth.
(Radius of earth is $R$ and mass of the earth $M$).

Answer 1

From energy conservation, since velocity (and hence kinetic energy) at infinity will be zero in this case,

$\frac{1}{2}m{v}^2-\frac{GMm}{R+R}=0$

$v=\sqrt{\frac{GM}{R}}=\sqrt{gR}=\sqrt{9.8\left(6400000\right)}=7.92km/s$

Answer is $7.92km/s$.