An artifical satellite of mass 'm' is moving in a circular orbit aroundthe earth. The height of the satellite above the surface of the earth is R. Suppose that it stops suddenly in its orbit and falls freely under gravity. With what speed it will strike the surface of the earth?

Options

A. `sqrt(gR)`
B. `2sqrt(gR)`
C. `3sqrt(gR)`
D. `4sqrt(gR)`

Answer

A

Explanation

Let v be its velocity when it reaches the earth.By the principle of conservation of energy,
Total energy of the satellite on the surface of the earth
= Its total energy in the orbit
`therefore (1)/(2)mv^(2)-(GMm)/(R )=0-(GMm)/(2R)`
`therefore (1)/(2)mv^(2)=(GMm)/(R )-(GMm)/(2R)`
`therefore v^(2)=(2GM)/(R )[1-(1)/(2)]=(GM)/(R )`
But `GM=gR^(2) " " therefore v^(2)=(gR^(2))/(R ) " " therefore v = sqrt(gR)`