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A satellite is revolving in a circular o...

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.)

A

`sqrt(2gR)`

B

`sqrt(gR)`

C

`sqrt(gR//2)`

D

`sqrt(gR)(sqrt(2)-1)`

Text Solution

Verified by Experts

The correct Answer is:
D
Orbital velocity of the satellite at height h
`v_(0) = sqrt((GM)/((R+h))) = sqrt((GM)/(R )) = sqrt(gR) (because h lt lt R)`
Let v be the velocity, given to the satellite
Using energy conservation principle
`(1)/(2) mv^(2) - (GMm)/((R+h)) = (1)/(2) m(0)^(2) + ((-GMm)/(oo))`
`(1)/(2)mv^(2) = (GMm)/(R+h)`
Given, `h lt lt R`
`v = sqrt((2GM)/(R ))`
`v = sqrt(2gR)`
Increase in velocity `= v - v_(0)`
`= sqrt(2gR) - sqrt(gR) = sqrt(gR)(sqrt(2) - 1)`
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