A satellite of mass ' m ' is revolving in a circular orbit around the Earth at a height ' R ' from the surface of the Earth. The minimum energy required to transfer the satellite into a higher circular orbit at a height ' 3R ' from the surface of the Earth is : (where ' R ' is the radius of the Earth and ' g ' is the acceleration due to gravity on the surface of the Earth)

Step-by-step solution

For a satellite orbiting the Earth at a distance r from its centre, the total energy is given by E = - GMm 2r . Initial distance from the centre of the Earth, r₁ = R + R = 2R . Initial total energy, E₁ = - GMm 2(2R) = - GMm 4R . Final distance from the centre of the Earth, r₂ = R + 3R = 4R . Final total energy, E₂ = - GMm 2(4R) = - GMm 8R . Minimum energy required to shift the satellite, E = E₂ - E₁ E = - GMm 8R - (- GMm 4R ) = GMm 8R . Using the relation g = GM R^2 , we get GM = gR^2 . Substituting this value, the