Help please! Consider the geosynchronous satellite of mass 50 kg. It orbits at a
ID: 1312820 • Letter: H
Question
Help please!
Consider the geosynchronous satellite of mass 50 kg. It orbits at a radius of 42,300 km with a speed of 3076 m/s. a) How much energy is needed to lift the satellite from the earth's surface (Re=6,370 km) up to that radius? b) How much kinetic energy does the satellite have as it orbits?
The answers are 2.67 x 109 Joules and 2.37 x 108 Joules. I've tried several different methods but I can't seem to get those numbers. Full ratings will be awarded to whoever can outline how to do this problem!
Explanation / Answer
=> Potential Energy of the Satellite when it is on earth's surface (Ue) = -GMm/Re = -GMm/6370000 (We will consider everything in SI units, therefore i have entered 6370000m)
=> Potential Energy of the Satellite when it is in orbit (Uo) = -GMm/R = -GMm/42300000
Here, the energy required to just lift the satellite is asked, it need not be moving. Let it be X.
So, Ue+X = U0 (After providing the energy required to the satellite it will be put into orbit)
=> -GMm/6370000 + X = -GMm/42300000
=> X = GMm/6370000 - GMm/ 42300000
=> X = GMm((1/6370000)-(1/42300000))
=>X = 6.67 * 10^-11 * 5.97219*10^24 * 50 * ((1/6370000)-(1/42300000))
=> X = 2655870357.3 Joules
=> X = 2.6558703573 * 10^9 Joules
(b) The Kinetic energy of the satellite can be found out by using the formula 1/2mv^2 = 1/2 * 50 * (3076)^2 = 236544400 J = 2.36544400 * 10^8 Joules
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