A rocket is fired with a speed, `v=2sqrt(gR)`, near the earth's surface and directed upwards. (a) Show that it will escape from the earth. (b) Show that in interstellar space its speed is `v=sqrt(2gR)`.

Explanation

(a) As `PE` of the rocket at the surface of the earth is `(-GMm//R)` and at infinity is zero, energy required for escaping from the earth will be
`=0-((GMm)/R)=mgR [ :'g=(GM)/(R^(2))]`
And as initial `KE` of the rocket `1/2 mv^(2)=2mgR` is greater than the energy required for escaping `(=mgR)`, the rocket will escape.
(b) If is the velocity of the rocket in interstellar spacfe (free from gravitational effects) then by conservation of energy,
`1/2m(2sqrt(gR))^(2)-1/2m(sqrt(2gR))^(2)=1/2 mv^(2)`
`v^(2)=4gR-2gR` or `v=sqrt(2gR)`