Question

Does anyone know or have seen a formula that relates the orbital velocity of a body around a primary and the orbital velocity of another body orbiting the body that orbits the primary?

Answer 1

Please see attachment.

Answer 2

if you are ok with vectors, then \(\vec {OB} = \vec{OA} + \vec {AC}\) ... and you can apply that to velocities as well

you could also try polar complex numbers as that might simplify computations


looks a bit like a cycloid to me.

Answer 3

this is a good related rates question

Answer 4

At first I thought about just trying simple geometry before using calculus, using the circumferences and radii. If the the speed of B around A is v2 and the speed of A around O is v1, then the two speeds are related in the following way \[v_{2} = \left( \frac{ r }{ R } k\right)v_{1}\]A very simplistic formula...

Answer 5

you need to start looking at the velocity as a vector......i reckon :-)

Answer 6

@isaiah.feynman , I don't think it's as simple as that. the speed of the "moon" will always be positive, but I am not sure it will be constant. In any case, as @IrishBoy123 has repeatedly established, speed and velocity are two VERY different things. One is directional, and one is a scalar.

Answer 7

I know all these already. The situation is very simple, no forces, only points going around with constant speeds. Its all just a small thought really. Of course the real situation will require rigorous mathematics.

Answer 8

Back to your original question:
Does anyone know or have seen a formula that relates the orbital velocity of a body around a primary and the orbital velocity of another body orbiting the body that orbits the primary?
Note that you stated velocities. You might think, oh, this is not a big deal, they probably knew what I was saying, etc.
Except it is a big deal. If you go on a grocery trip starting from home and returning home, you will have an average velocity of zero, but a non zero speed. My point? You wind up making no distance in progress but you definitely went on a trip... path dependence in a sense.
Because you asked for velocity we delivered regarding velocity. For speed, I would have to look into the derivation but intuition tells me that the speed is not constant either. Why? look at this

Answer 9

I see my error.

Answer 10

I put the moon on the right in both situations but when a planet orbits on one half the velocity of the moon and planet will be added and on the other half hte velocity of the moon and planet will be subtracted. I could derive a formula for this i guess but I am lazy so the exercise will be left to the reader :P (hint, use vectors and split components)

Answer 11

by the way i am not familiar with the study but the lagrangian of this system is probably a "better" way to look at it.

Answer 12

I could do those with the current physics. lol

Answer 13

actually I forgot about omega and omega makes it too much effort

Answer 14

to account for differences in angular velocity and phase... so much effort

Answer 15

also i have to apply kepler's laws... maybe after I wake up man

Answer 16

Its okay, I could figure out myself and show you later on.

Answer 17

I mean I will do it but not by hand... use mathematica lol

Answer 18

also there are different levels of consideration for this problem. General relativity? Three body problem? Center of mass approximation?