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6 June, 06:16

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?

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  1. 6 June, 06:43
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    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
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