Question

A satellite is revolving around earth in a circular orbit of radius $2R$, where $R$ is the earth's radius. A particle is projected from this satellite radially outward relative to satellite with speed $u$. Find the minimum value of $u$ so that particle escapes the gravitational force of earth. $M$ is the mass of earth. (Mass of earth >>> Mass of satellite >>> Mass of particle)

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

Answer 1

The correct option is B $\sqrt{\frac{GM}{2R}}$

The particle already has one component of velocity i.e. $\sqrt{\frac{GM}{2R}}$. If it projected with velocity of $u$ radially outwards, let $u_1$ be its magnitude of velocity.
So,
$u^2+\frac{GM}{2R}=u_1^2$
$\Rightarrow u_1=\sqrt{u^2+\frac{GM}{2R}}.....\left(1\right)$

Using energy conservation, we can say:
$\frac{1}{2}m{u}_1^2-\frac{GMm}{2R}=0\Rightarrow u_1=\sqrt{\frac{GM}{R}}\text{From}\left(1\right)\Rightarrow \frac{GM}{2R}+u^2=\frac{GM}{R}$
$u=\sqrt{\frac{GM}{2R}}$