Satellite Energy and Orbit problem

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SUMMARY

The discussion centers on the energy dynamics of a spy satellite in a circular orbit around the Earth when an additional radial velocity is applied. The initial kinetic energy (K) is calculated as K = 1/2 GMm/r, and the potential energy (U) as U = -GMm/r. After the thruster fires, the new total energy of the satellite is determined to be zero, indicating that the satellite achieves an open orbit with infinite radius. The conclusion drawn is that the escape velocity must be equal to or greater than the tangential velocity for a satellite to escape gravitational influence.

PREREQUISITES
  • Understanding of gravitational potential energy (U = -GMm/r)
  • Knowledge of kinetic energy in orbital mechanics (K = 1/2 mv^2)
  • Familiarity with the concept of escape velocity
  • Basic principles of circular motion and orbits
NEXT STEPS
  • Study the derivation of escape velocity from gravitational potential energy
  • Explore the implications of adding velocity in different directions on orbital mechanics
  • Learn about different types of orbits, including open and closed orbits
  • Investigate the effects of mass and radius on satellite energy calculations
USEFUL FOR

Aerospace engineers, physics students, and anyone interested in orbital mechanics and satellite dynamics will benefit from this discussion.

Dextrine
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Homework Statement


A spy satellite of mass m is in a circular orbit with radius R and velocity v around
the earth. One of the satellites thrusters suddenly fires giving it an additional
velocity v in the outward radial direction (same v). What is the new total energy
of the satellite? What is the new orbit of the satellite?

Homework Equations


mv^2/r
GMm/r^2
-K=.5U
2K+U=0

The Attempt at a Solution


I honestly don't really have an idea how to even set up the problem. From what I understand mv^2=GMm/r^2 for circular orbit which should I could then use to find the energy, but I don't know how the velocity increasing radially will affect this. Any helpful nudge in the right direction would be greatly appreciated.
 
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Dextrine said:
mv^2=GMm/r^2
I think you mistyped the left hand side. As a result you have energy on the left and a force on the right.

What is the initial KE?
What is the initial PE?
How much energy was added?
 
haruspex said:
I think you mistyped the left hand side. As a result you have energy on the left and a force on the right.

What is the initial KE?
What is the initial PE?
How much energy was added?
So initially I got my K=1/2GMm/r and my U=-GMm/r

Since we are adding another V, which is Sqrt[GM/r], i get that my new total energy = -.5GMm/r+.5GMm/r=0

however, this doesn't seem right
 
Dextrine said:
So initially I got my K=1/2GMm/r and my U=-GMm/r

Since we are adding another V, which is Sqrt[GM/r], i get that my new total energy = -.5GMm/r+.5GMm/r=0

however, this doesn't seem right
It's right :). Remember, the total energy beforehand was negative.
So what do you get for the new orbit?
 
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haruspex said:
It's right :). Remember, the total energy beforehand was negative.
So what do you get for the new orbit?
AH, infinite radius? so we get an open orbit? Thanks a ton. Didn't think it would be so simple
 
Dextrine said:
AH, infinite radius? so we get an open orbit? Thanks a ton. Didn't think it would be so simple
That's it. So what general rule do you deduce for vertical escape velocity from a given orbit?
 
Hmm, it must be equal to or greater than tangential velocity?
 
Dextrine said:
Hmm, it must be equal to or greater than tangential velocity?
Equal. ("Escape velocity" means the minimum necessary to escape the gravitational field.)
 
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