Solve Orbital Velocity Equation: V=sqrt((g*R^2)/r)

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SUMMARY

The discussion centers on the derivation of the orbital velocity equation, V = sqrt((g * R^2) / r), and its relationship to other fundamental equations in orbital mechanics. Participants clarify that Vcirc = sqrt((GM) / r) and Vesc = sqrt((2GM) / r) are essential formulas, with the acceleration equations a = v^2 / r and a = GM / r^2 being pivotal in understanding satellite motion. The conversation emphasizes that for a satellite in a circular orbit, the normal component of acceleration is the only relevant factor, confirming that v^2 = GM / r can indeed be used to derive the orbital velocity.

PREREQUISITES
  • Understanding of gravitational force, specifically F = GMm / r^2
  • Familiarity with circular motion concepts, including a = v^2 / r
  • Knowledge of orbital mechanics, including escape velocity and circular velocity equations
  • Basic algebraic manipulation skills for equation derivation
NEXT STEPS
  • Study the derivation of the escape velocity formula, Vesc = sqrt((2GM) / r)
  • Learn about the implications of gravitational acceleration on satellite orbits
  • Explore the differences between circular and elliptical orbits in orbital mechanics
  • Investigate the role of centripetal force in maintaining satellite motion
USEFUL FOR

Students studying physics, particularly those focusing on mechanics and orbital dynamics, as well as educators looking to clarify concepts related to satellite motion and gravitational forces.

chris529911
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I have a homework problem that involves the equation v=sqrt((g*R^2)/r) I am confused as to where this equation comes from. I know that Vcirc = sqrt((GM)/r) and Vesc = sqrt((2GM)/r)
 
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Do you know the formulas a=v^2/r and a=GM/r^2? Equate the two and you get v^2/r=GM/r^2. With a bit of manipulation, you get that equation.
 
a=v^2/r is the equation for the normal component of acceleration

a=GM/r^2 derived from F= GMm/r^2

a satellite orbiting the Earth only has a normal component of acceleration correct ?
 
this is the problem. Why can't you just use v^2=GM/r?
 

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chris529911 said:
a=v^2/r is the equation for the normal component of acceleration

a=GM/r^2 derived from F= GMm/r^2

a satellite orbiting the Earth only has a normal component of acceleration correct ?

For a circular orbit, yes.
 
chris529911 said:
this is the problem. Why can't you just use v^2=GM/r?

You can. That comes from equating v^2/r with GM/r^2.
 

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