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 

Options

• \[\sqrt{\frac{GM}{4R}}\]

• \[\sqrt{\frac{GM}{2R}}\]

• \[\sqrt{\frac{GM}{R}}\]

• \[\sqrt{\frac{2GM}{R}}\]

Answer 1

\[\sqrt{\frac{GM}{R}}\]

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]

\[\Rightarrow \frac{1}{2}m v^2 = - \left[ \frac{GMm}{\infty} - \frac{GMm}{2R} \right]\]

\[ \Rightarrow v = \sqrt{\frac{GMm}{R}}\]