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1 August, 10:39

A 50-kg satellite circles the Earth in an orbit with a period of 120 min. What minimum energy is required to change the orbit to another circular orbit with a period of 180 min? (Earth: radius = 6.4 106 m, mass = 6.0 1024 kg)

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  1. 1 August, 12:30
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    Answer: 2.94*10^8 J

    Explanation:

    Using the relation

    T^2 = (4π^2/GMe) r^3

    Where v = velocity

    r = radius

    T = period

    Me = mass of earth = 6*10^24

    G = gravitational constant = 6.67*10^-11

    4π^2/GMe = 4π^2 / [ (6.67x10^-11 x6.0x10^24) ]

    = 0.9865 x 10^-13

    Therefore,

    T^2 = (0.9865 * 10^-13) * r^3

    r^3 = 1 / (0.9865 * 10^-13) * T^2

    r^3 = (1.014 x 10^13) * T^2

    To find r1 and r2

    T1 = 120min = 120*60 = 7200s

    T2 = 180min = 180*60 = 10800s

    Therefore,

    r1 = [ (1.014 x 10^13) 7200^2]^ (1/3) = 8.07 x 10^6 m

    r2 = [ (1.014 x 10^13) 10800^2]^ (1/3) = 10.57 x 10^6 m

    Required Mechanical energy

    = - GMem/2 [1/r2 - 1/r1]

    = (6.67 x 10^-11 x 6.0 x 10^24 * 50) / 2 * [ (1/8.07 * 10^-6) - (1/10.57 * 10^-6) ]

    = (2001 x 10^7) / 2 * (0.1239 - 0.0945)

    = (1000.5 * 10^7) * 0.0294

    = 29.4147 * 10^7 J

    = 2.94 x 10^8 J.
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