A space transportation vehicle releases a 470-kg communications satellite while in a circular orbit 350 km above the surface of the Earth. A rocket engine on the satellite boosts it into an orbit 2350 km above the surface of the Earth. How much energy does the engine have to provide for this boost?

E = 3.194 x 10⁹ J = 3.194 GJ

Explanation:

The formula for the absolute potential energy is:

U = - GMm/2r

where,

G = Gravitational Constant = 6.67 x 10⁺¹¹ N m²/kg²

M = mass of Earth = 5.972 x 10²⁴ kg

m = mass of satellite = 470 kg

r = distance between the center of Earth and satellite

Thus, the energy required from engine will be difference between the potential energies.

E = U₂ - U₁

E = - GMm/2r₂ - ( - GMm/2r₁)

E = (GMm/2) (1/r₁ - 1/r₂)

where,

r₁ = Radius of Earth + 350 km = 6371 km + 350 km = 6721 km = 6.721 x 10⁶ m

r₂=Radius of Earth + 2350 km=6371 km + 2350 km = 8721 km = 8.721 x 10⁶ m

therefore,

E = [ (6.67 x 10⁺¹¹ N m²/kg²) (5.972 x 10²⁴ kg) (470 kg) / 2] (1/6.721 x 10⁶ m - 1/8.721 x 10⁶ m)

E = 3.194 x 10⁹ J = 3.194 GJ