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) √GM4R
(b) √GM2R
(c) √GMR
(d) √2GMR

Answer 1

The answer is option (c).

Explanation:

At a distance R from the earth’s surface, the particle has the potential energy is  

P.E.i = 12GMmR = GMm (R+R),

where M = mass of the earth;  

R = radius of the earth  

m = body mass

Let the projection of the particle be with the speed v. It just escapes the earth’s gravitational pull.

So, the body’s kinetic energy = -[change in the body’s potential energy]

Now, the body’s kinetic energy = -[- initial potential energy + final potential energy]

⇒12mv2 = -GMm∞ - GMm2R

⇒v = GMmR