- #1
solarcat
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Homework Statement
Say you're launching a spacecraft of mass m from the surface of the Earth (mass Me and radius re) to a low height h (h is much smaller than r, so h is essentially negligible). How much work is required to move the spaceship from its low orbit to a great distance from earth? Ignore gravitational effects of other planets/moons/stars.
Homework Equations
Work = change in kinetic energy
Fc = mv^2/r
Fg = GmM/r^2
The Attempt at a Solution
When the spaceship is on the ground, PE=0. When the spaceship is in low orbit, PE=0 because h is essentially negligible. When the spaceship is very far away, my book says that PE will also be 0 (U = -GMem/r). So work depends only on kinetic energy.
I said the work required to get the spacecraft to height h would be GmMe/(2*re^2). This would also be its kinetic energy since kinetic energy on the ground is 0.
To find the change in kinetic energy:
KE when spaceship is in low orbit = GmMe/(2*re^2)
KE when spaceship is a great distance away from earth:
m*v^2/(re+d) = GmMe/(re+d)^2 (d=distance from earth)
Cancel m: v^2/(re+d) = GMe/(re+d)^2
v^2 = GMe/(re+d)
d is very large, so re+d approaches infinity, so v is zero. final kinetic energy = 0
Work = change in kinetic energy = - GmMe/(2*re^2).
But I don't think this can be correct because the next question essentially states that the energy of the spaceship at low orbit should be midway between the energy on Earth and the energy at a very great distance.