Question

A particle is kept at rest at a distance $R$ (Earths radius) above the earths surface. The minimum speed with which it should be projected so that it does not return is

• A. $\sqrt{\frac{GM}{R}}$
• B. $\sqrt{\frac{GM}{2R}}$
• C. $\sqrt{\frac{GM}{3R}}$
• D. $\sqrt{\frac{GM}{4R}}$

Answer 1

The correct option is A $\sqrt{\frac{GM}{R}}$

Total energy of the particle at height $R$ from Earths surface when projected with a speed $v$ is
$E_i=\frac{-GMm}{R+R}+\frac{1}{2}m{v}^2$
For this $v$ to be escape speed, the particle should reach a point where gravitational force due to earth is zero.
To find the minimum projection speed to achieve this, final kinetic energy should also be zero.
$E_f=0$
Equating these two equations:
$\Rightarrow \frac{-GMm}{2R}+\frac{1}{2}m{v}_e^2=0\Rightarrow v_e=\sqrt{\frac{GM}{R}}$