v = √(GM/r)
Where:
* v is the orbital velocity
* G is the gravitational constant (6.674 × 10-11 m3 kg-1 s-2)
* M is the mass of the central body (e.g., the Earth)
* r is the orbital radius (the distance between the center of the central body and the orbiting object)
Derivation:
This equation can be derived using the following steps:
1. Centripetal force: The orbiting object experiences a centripetal force that keeps it in its circular orbit. This force is provided by gravity.
2. Equating forces: The centripetal force (Fc) is equal to the gravitational force (Fg):
Fc = Fg
3. Formulas:
* Fc = mv²/r (where m is the mass of the orbiting object)
* Fg = GMm/r²
4. Substitution: Substituting the formulas for Fc and Fg into the equation from step 2:
mv²/r = GMm/r²
5. Simplifying: Canceling out 'm' and one 'r' from both sides, and rearranging:
v² = GM/r
6. Orbital velocity: Taking the square root of both sides:
v = √(GM/r)
Important Note: This equation assumes a circular orbit. For elliptical orbits, the velocity varies at different points in the orbit, and the equation becomes more complex.
