Question

What is the kinetic energy needed to project a body of mass $m$ from the surface of the earth to infinity. Radius of earth is $R$ and acceleration due to gravity on earth's surface is $g$.

Answer 1

From energy conservation,

$KE+PE=0$ since at infinity, velocity will be zero for minimum launch velocity. Hence,

$KE-\frac{GMm}{R}=0$

$KE=\frac{GMm}{R}$

Also, since $g=\frac{GM}{R^2}$

$KE=gmR$

Answer is $gmR$