Question

A space-ship is launched into a circular orbit close to the Earth's surface. What additional speed should now be imparted to the spaceship so that orbit to overcome the gravitational pull of the Earth.

Answer 1

Let $△K$ be the additional kinetic energy imparted to the spaceship to overcome the gravitation pull then
$K=\frac{GMm}{2R}$

Total kinetic energy $=\frac{GMm}{2R}+△K=\frac{GMm}{2R}+\frac{GMm}{2R}=\frac{GMm}{R}$
then $\frac{1}{2}m{v}_2^2=\frac{GMm}{R}\to v_2=\sqrt{\frac{2GM}{R}}$

But $v_1=\sqrt{\frac{GM}{R}}$.
So Additional velocity $=v_2-v_1=\sqrt{\frac{2GM}{R}}-\sqrt{\frac{GM}{R}}=(\sqrt{2}-1)\sqrt{\frac{GM}{R}}$.