Two body problem, velocities of two bodies, relative velocity given

AI Thread Summary
The discussion revolves around the derivation of angular momentum and relative velocity between two bodies influenced by a central force. The key equations for angular momentum are presented, showing the relationship between angular velocity, tangential velocity, and distance from the origin. A clarification arises regarding the assumption of equal masses, with the consensus that while it simplifies calculations, it limits the generality of the model. The conversation highlights the importance of distinguishing between speeds and velocities, particularly in the context of relative motion. Ultimately, the participants agree on the validity of the derivation while acknowledging the complexities involved in simulations.
pobro44
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Homework Statement
I want to program a two body problem and wish to calculate velocities of bodies when relative velocity is known. Please let me know whether my derivation makes sense.
Relevant Equations
Angular momentum of system body acted upon with central force
When only force acting on body is a central force, angular momentum is constant and given by:

L = mr^2 * w

where r is distance from origin, and w is angular velocity.

Angular momentum can also be written as following:

L = r x mv = rmv * sin(theta) where v is tangential velocity, which is orbital velocity

so we can equate the two expressions above and

w = v * sin(theta) / r

since angular velocities of two bodies are the same, we can write

245343
where t1 stands for first body, and t2 for second

I suppose that angles that position vectors make with velocity vector are the same, so sines are equal and

245340


if we define relative velocity between two bodies as

245344


and plug that back into previous equation we get orbital velocities of each body from their relative velocity at that point

245345


Is this correct? I derived it by myself so any feedback would be appreciated :)
 
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pobro44 said:
L = r x mv = rmv * sin(theta) where v is tangential velocity
I assume you mean where v * sin(theta) is tangential velocity.
Did you mean to assume the masses are the same?
 
haruspex said:
I assume you mean where v * sin(theta) is tangential velocity.
Did you mean to assume the masses are the same?

Thank you for you reply, yes I assume the masses are the same because I equate two expressions for angular momentum for the same body at the same position. That way I get how angular velocity is related to tangential velocity and distance from origin.
 
pobro44 said:
because I equate two expressions for angular momentum for the same body at the same position
It sounds like you are saying that the masses came out to be the same as a result of the algebra. There is no reason they should be the same. If you want to make them the same, fine, but you will not then be able to model the more general behaviour.
 
haruspex said:
It sounds like you are saying that the masses came out to be the same as a result of the algebra. There is no reason they should be the same. If you want to make them the same, fine, but you will not then be able to model the more general behaviour.

I believe you think that I equate expressions for two different bodies of the same mass which orbit each other, but in derivation I was referring only to one of two bodies, with it's momentum expressed in two different ways, and then by equating the expressions for momentum masses cancel out.
 
pobro44 said:
I believe you think that I equate expressions for two different bodies of the same mass which orbit each other, but in derivation I was referring only to one of two bodies, with it's momentum expressed in two different ways, and then by equating the expressions for momentum masses cancel out.
Apologies - I see now what you are doing.
You refer to velocities but your equations are scalar, so you mean speeds. In particular, relative speed.
On that understanding your derivation is fine, but it might be awkward to use. When the masses happen to be the same you will get 0/0.
 
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haruspex said:
Apologies - I see now what you are doing.
You refer to velocities but your equations are scalar, so you mean speeds. In particular, relative speed.
On that understanding your derivation is fine, but it might be awkward to use. When the masses happen to be the same you will get 0/0.

yes, sorry, speeds as I use vis viva equation to calculate relative speeds at every point, and with those equations I derived at the end I calculate velocities of single bodies. However, I do not have any such problems in my simulation, this is what I get for relative speed in perihellion of 1, and eccentricity of 0:

245348


Bodies are on the same trajectory, a diameter of a circle apart, and their speeds are half of the initial relative speed on every point of trajectory.
 
I get it, in my sim I use vectors, and in the denominator I subtract vectors then calculate the magnitude. In final result of my derivation, magnitudes of r1 i r2 would be equal if masses were equal so that would yield 0 in the denominator.

If I argued that velocity vectors were in opposite directions, because cross product of their angular velocity and position vector is in opposite direction, so relative speed between two bodies must be equal to sum of their magnitudes instead of difference, I would get r1+r2 in the denominator and valid result, right?
 
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pobro44 said:
I get it, in my sim I use vectors, and in the denominator I subtract vectors than calculate the magnitude. In final result of my derivation, magnitudes of r1 i r2 would be equal if masses were equal so that would yield 0 in the denominator.

If I argued that velocity vectors were in opposite directions, because cross product of their angular velocity and position vector is in opposite direction, so relative speed between two bodies must be equal to sum of their magnitudes instead of difference, I would get r1+r2 in the denominator and valid result, right?
Yes, ##\frac{v_1}{r_1}=\frac{v_2}{r_2}=\frac{v_1+v_2}{r_1+r_2}##.
 
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Thank you Haruspex for taking the time to respond and assist, it is much appreciated:smile:
 
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