Question

A particle is kept at rest at a distance R (earth's radius) above the earth's surface. The minimum speed with which it should be projected so that it does not return is
(a) $\sqrt{\frac{GM}{4R}}$

(b) $\sqrt{\frac{GM}{2R}}$

(c) $\sqrt{\frac{GM}{R}}$

(d) $\sqrt{\frac{2GM}{R}}$

Answer 1

(c) $\sqrt{\frac{GM}{R}}$

Potential energy of the particle at a distance R from the surface of the Earth is ${\left(P.E.\right)}_i=\frac{GMm}{(R+R)}=\frac{1}{2}\frac{GMm}{R}$.
Here, M is the mass of the earth; R is the radius of the earth and m is the mass of the body.

Let the particle be projected with speed v so that it just escapes the gravitational pull of the earth.
So, kinetic energy of the body = $-$[change in the potential energy of the body]

Now, kinetic energy of the body = $-$[final potential energy $-$ initial potential energy]
$$\begin{gathered} \Rightarrow \frac{1}{2}m{v}^2=-\left[\frac{GMm}{\infty }-\frac{GMm}{2R}\right] \\ \Rightarrow v=\sqrt{\frac{GMm}{R}} \end{gathered}$$