A satellite is revolving in a circular orbit at a height h from the earth's surface (radius of earth `R,h lt lt R`). The minimum increase in its orbital velocity required so that the satellite could escape from the earth's gravitational field , is close to :(Neglect the effect of atmosphere)

Options

A. `sqrt(2gR)`
B. `sqrt(gR)`
C. `sqrt((gR)/2)`
D. `sqrt(gR)(sqrt(2)-1)`

Answer

D

Explanation

Orbital velocity of the satellite ,
`v=sqrt((GM)/(R+h))=sqrt((GM)/R) [because Rgt gt h]`
Here, M =mass of the earth]
if the escape velocity of the satellite is `v^.`,
`1/2 mv^(.2)=(GMm)/(R+h)` [m=mass of the satellite]
`therefore v^.-v=sqrt((2GM)/R)-sqrt((GM)/R)`
`=sqrt(2gR)-sqrt(gR)=sqrt(gR)(sqrt(2)-1)`