Question

How much energy is required to move an Earth satellite of mass m from a circular orbit of radius 2RE to one of radius 3RE? The constant RE is the radius of the Earth (just leave this as RE in your final answer)

Answer 1

If v is the velocity of the satellite,

Centripetal force = gravitational force.

mv^2 / (3Re) = G M m / (3Re) ^2
mv^2 = G M m / (3Re)

Kinetic energy = 1/2 mv^2 = G M m / (6Re)
Kinetic energy = 6.67e-11*5.98e24*2000 /(6*6.37e6)
2.09e+10 J
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Kinetic energy at 2Re = G M m / (4Re)
Difference in energy at 3Re and 2Re
G M m / (4Re) - G M m / (6Re) = G M m / (4Re) - G M m / (6Re)
1/4 - 1/6 = 0.083of
Hence work needed = 0.083of G M m / (Re) = 0.083*6*2.09e+10 J
= 1.04082e+10 J.