Question

# How much gravitational potential energy must a 3140-kg satellite acquire in order to attain a geosynchronous...

How much gravitational potential energy must a 3140-kg satellite acquire in order to attain a geosynchronous orbit?

The answer to the above question is 1.67x10^11 J.

How much kinetic energy must it gain? Note that because of the rotation of the Earth on its axis, the satellite had a velocity of 467 m/s relative to the center of the Earth just before launch.

Doing a 1/2mv^2 doesn't get the correct answer. The law of conservation of energy would state that KE = change in PE but KE = 1.67x10^11 J is not the correct answer either.

time period of satellite

T = 24 hr = 24*60*60

T = 2*pi*r^(3/2)/sqrt(GM)

24*60*60 = 2*pi*r^(3/2)/sqrt(6.67*10^-11*5.98*10^24)

r = 4.22*10^7 m

In orbit Fg = Fc

GMm/r^2 = mv0^2/r

v0 = sqrt(GM/r)

KE due to orbital motion k2 = (1/2)*m*v0^2 = (1/2)*m*GM/r

KE due to rotation of earth K1 = (1/2)*m*v^2

gain = k2 - k1

gain = (1/2)*m*GM/r - (1/2)*m*v^2

gain = (1/2)*m*( (GM/r) - v^2)

gain = (1/2)*3140*( (6.67*10^-11*5.98*10^24/(4.22*10^7)) - 467^2)

gain = 1.45*10^10 J

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