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

[pobro44]

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

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

if we define relative velocity between two bodies as

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

Is this correct? I derived it by myself so any feedback would be appreciated :)

 

[haruspex]

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?

 

[pobro44]

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.

 

[haruspex]

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.

 

[pobro44]

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.

 

[haruspex]

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.

 

[pobro44]

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:

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.

 

[pobro44]

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?

 

[haruspex]

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, $\frac{v_1}{r_1}=\frac{v_2}{r_2}=\frac{v_1+v_2}{r_1+r_2}$.

 

[pobro44]

Thank you Haruspex for taking the time to respond and assist, it is much appreciated