Question

A satellite is revolving around the earth close to the surface of the earth with kinetic energy E. How much kinetic energy should be given to it so that it escape from the surface of earth?

Answer 1

Let mass of the satellite be m and velocity be v.

Given E = ½mv²

Since the satellite is moving close to the earth so its distance from the centre can be approximately taken as R

Centripetal force balances the gravitational pull thus.

mv²/R = GmM/R²

=> mv² = GmM/R

=> ½mv² = ½GmM/R

=> E = ½GmM/R ---1.

To overcome earth’s gravitational field the satellite needs to have escape velocity v_c

v_c = √(2GM/R)

=> v_c² = 2GM/R

multiplying both sides by ½m

½mv_c² = GmM/R

So the kinetic energy necessary to escape E_c is

E_c = GmM/R ---2.

From 1 and 2.

=> E_c = 2E

E_c – E = E

Thus KE that has to be given more should be E.

That is final KE for escape must be 2E