Conservation of Energy and satellite

In summary, the conversation discusses a spherical planet with no atmosphere and a satellite that is fired from its surface at a speed of vo and an angle of 30 degrees to the local vertical. The satellite reaches a maximum distance of 5R/2 from the center of the planet in its orbit. Using the principles of conservation of energy and angular momentum, it can be shown that the initial speed of the satellite, vo, is equal to (5GM/4R)^1/2. This is obtained by solving a system of two equations, one for conservation of energy and one for conservation of angular momentum.
  • #1
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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  • #2
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.)
 
  • #3
So I'm just going to be solving a system of 2 linear equations for 2 unknowns?
 
  • #4
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.
 
  • #5
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.
 

1. What is Conservation of Energy?

The Conservation of Energy is a fundamental law of physics that states that energy cannot be created or destroyed, but can only be transferred or transformed from one form to another.

2. How does Conservation of Energy apply to satellites?

Conservation of Energy applies to satellites in the sense that they must constantly maintain a balance between their potential energy (due to their height above the Earth) and their kinetic energy (due to their orbital velocity). Any changes in these energies must be accounted for to maintain a stable orbit.

3. Why is Conservation of Energy important in satellite design?

Conservation of Energy is crucial in satellite design because it ensures that the satellite will have enough energy to maintain its orbit for as long as possible. Any inefficiencies or losses in energy can lead to a shorter lifespan or even failure of the satellite.

4. Can satellites violate the law of Conservation of Energy?

No, satellites cannot violate the law of Conservation of Energy. While they may appear to be defying gravity or moving without a visible source of energy, they are actually following the laws of physics and constantly exchanging energy with their surroundings.

5. How does Conservation of Energy impact satellite missions?

The principle of Conservation of Energy plays a crucial role in the success of satellite missions. Engineers must carefully consider the energy needs of the satellite and design systems that efficiently manage and conserve energy to ensure the mission's objectives are met.

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