# Conservation of Energy and satellite

phy
Imagine a spherical, nonrotating planet of mass M, radius R, that has no atmosphere. A satellite is fired from the surface of the planet with speed vo at 30 degrees to the local vertical. In its subsequent orbit the satellite reaches a maximal distance of 5R/2 from the center of the planet. Using the principles of conservation of energy and angular momentum, show that vo - (5GM/4R)^1/2

This is what I've done so far and it's not right:

E=1/2 mv^2 - GMm/2R = GMm/4R
1/2mv^2-GMm/2(5R/2) = GMm/4(5R/2)
1/2mv^2-GMm/5R = 2GMm/20R
1/2v^2 = 3GM/10R
.:v = (3GM/5R)^1/2

Clearly, this isn't the answer I should be getting. Does anybody know where I'm going wrong?

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Doc Al
Mentor
phy said:
E=1/2 mv^2 - GMm/2R = GMm/4R
1/2mv^2-GMm/2(5R/2) = GMm/4(5R/2)
1/2mv^2-GMm/5R = 2GMm/20R
1/2v^2 = 3GM/10R
.:v = (3GM/5R)^1/2
I can't quite follow what you are doing. Apply conservation of energy to get one equation and conservation of angular momentum to get another. (I trust you realize that at the maximal distance, the speed is not zero.)

phy
So I'm just going to be solving a system of 2 linear equations for 2 unknowns?

Doc Al
Mentor
phy said:
So I'm just going to be solving a system of 2 linear equations for 2 unknowns?
I wouldn't call them linear... but, yes, you'll have two equations and two unknowns.

phy
Oh yeah my bad. I meant just two equations. Ok thanks for that. I guess I'll redo the question and get back to you on that one. Thanks again.