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