Question

A particle of mass $m$ is thrown upwards from the surface of the earth, with a velocity $u$. The mass and the radius of the earth are, respectively, $M$ and $R$. $G$ is gravitational constant and $g$ is acceleration due to gravity on the surface of the earth. The minimum value of $u$ so that the particle does not return back to the earth is

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

Answer 1

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

Particle will not return if it is thrown upwards with escape velocity.

Conserving mechanical energy between the instants the particle is at the surface of the earth and at a very large distance

$U_i+K_i=U_f+K_f$

At a very large distance, $U_f=0$ and for the particle to just escape gravity. its final velocity will be $0$, so $K_f=0$.

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

$\Rightarrow v_e=\sqrt{\frac{2GM}{R}}$

Hence, option $\left(c\right)$ is correct.

$\begin{array}{c}\text{Why this question ?}\\ \text{Tip: Escape velocity can be calculated}\\ \text{for any orbit not just the surface of the}\\ \text{planet. For a particle to escape, its total}\\ \text{mechanical energy should be equal to zero.}\end{array}$