A rocket is accelerated to speed `upsilon = 2sqrt(gR)` near the earth's surface (`R =` radius of earth). Show that very far from earth its speed will be `upsilon = sqrt(2gR)`.

Explanation

Let `upsilon_(1)` be the speed of rocket in interstellar space. Using law of conservation of energy, we have
`(1)/(2) m(2 sqrt(gR))^(2) = (1)/(2) m upsilon_(e)^(2) + (1)/(2) m upsilon_(1)^(2)`
`= (1)/(2) m (2 sqrt(gR))^(2) + (1)/(2) m upsilon_(1)^(2)`
or `upsilon_(1)^(2) = 4 gR - 2 gR = 2gR`
or `upsilon_(1) = sqrt(2gR)`