Generating position and velocities at different points on an ellipse

Is it possible to generate position and velocity vectors at different points on an ellipse? I have the red dot with a position vector of [28; 22] and velocity vector of [0; 2].

I know if a and b are given, we can get position vector at each point using x = a*cos(theta) and y =b*sin(theta) for theta values ranging from 0 to 2*pi radians. Is there a way to find the velocity vectors at those respective positions?

[John D'Errico]

Note that velocity has units of distance divided by time. I see no time involved with your question, so velocity has no meaning in this context.

You need to explain your problem more clearly. For example, is theta varying with time?

[Fiona Rae]

Sorry. Yes, theta must vary with time.

[James Tursa]

How does it vary with time? Is this 2-body motion of a satellite around a planet? Is this constant rate along the arclength? Or what?

[Fiona Rae]

Just a point moving in an elliptical path. I know for uniform circular motion, the magnitude of velocity (or speed) remains constant. The direction keeps changing and so the velocity keeps changing too.

Since speed cannot remain constant in an elliptical path, I'm guessing there is no such thing as uniform elliptical motion? And here too, direction keeps changing. So neither speed nor velocity remains constant.

[James Tursa]

".. Since speed cannot remain constant in an elliptical path ..."

I see no reason why not. The speed along the path can be whatever you want it to be. Are you telling me that a race car driving along an elliptical track can't drive at a constant speed? If there is a physical problem being modeled, then that physical model will determine the speed along the path. Is your question just a generic one with no physical system being modeled? If so, the speed can be whatever you define it to be. If there is a physical system being modeled, then you need to tell us what that is.

[Fiona Rae]

Yes, just a generic question. I don't have any specific application in mind. We were taught about uniform circular motion at school. So I was wondering if it still applies to an elliptical path.

Answer 1

[James Tursa]

For the elliptical equivalent to uniform circular motion, I suppose you could have three choices.

1) Pick uniform angular motion, in which case the speed and the area swept out along the arc will vary with time.

2) Pick uniform speed along the arc, in which case the angular rate and area swept out will vary with time.

3) Pick uniform area swept out by the arc, in which case the angular rate and the speed will vary with time. Incidentally, this last one matches planetary and satellite motion (assuming area is measured from attracting body and not from the center of the ellipse).

4) Maybe others ...?

But there is no "generic" translation from uniform circular motion to uniform elliptical motion.

[Fiona Rae]

Oh my. This is more complicated than I thought. So I browsed online and I came across this. For position,we have, x = a*cos(theta), y = b*sin(theta). So it's velocity would be, Vx = -a*omega*sin(theta), Vy = b*omega*cos(theta) where omega is angular velocity I suppose.

And then, it says omega = (Vo*ro)/(x^2+y^2) where Vo,ro are some intital velocity and radial distance from center of ellipse. Isn't angular velocity, omega = V/r? Which case are they talking about?

[James Tursa]

The first set of equations are just generic equations that apply to every ellipse relating position to angle:

x = a*cos(theta)
y = b*sin(theta)

IF you assume that the angle theta is uniform angular motion and has this simple form:

theta = omega * t

Then you can take simple derivatives of the position equations with respect to time and come up with the velocity equations you show:

Vx = -a*omega*sin(theta)
Vy = b*omega*cos(theta)

E.g.,

d(a*cos(theta))/dt =
d(a*cos(omega*t))/dt =
-a*sin(omega*t)*omega =
-a*omega*sin(theta)

Same for Vy. This is the constant angular rate case 1 above. But we had to assume something about the motion before we could derive velocity components.

If we made a different assumption about the motion, like the point was under the influence of a central attracting force that acted on an inverse distance squared law (e.g., gravity or electrical charge), then the velocity derivation we just did would not apply because the theta assumption we made does not apply. In this particular case, you end up with case 3 above. (The central attracting force is not at the center of the ellipse btw ... it is at one of the foci of the ellipse)

[Fiona Rae]

The first part is fine. It's the omega = (Vo*ro)/(x^2+y^2) which they used that confused me.

[John D'Errico]

James is doing quite well here. So I'll just add that in any problem like this, you need to decide what it is that you are modeling. I think you started out at the wrong end of things. You started out talking about velocity, without really focusing on what velocity means in this context. Is this a physical system that you are modeling, driven by what is equivalent to "gravitational" forces? Or is this a case of a "car" moving around a race track, at essentially a constant velocity? So now you need to deal with a constant distance per unit time, moving along an elliptic curve. Or is it uniform angular velocity, in which case you get the desired behavior by a simple application of differentiation.

So you need to first make the decision of what is the fundamental behavior of this system. Don't worry about equations YET. After you choose what systemic behavior you want, only then do you think about any equations.

[Fiona Rae]

Alright, I've thought about two 2D cases that interest me. One's where an unmanned aircraft is moving in an elliptical path (surveillance, perhaps?) and the other's where a satellite is moving in an elliptical orbit. Problem is, I'm not sure what quantity remains constant with time in real-life situations for both cases.

Is it speed or angular velocity that remains constant in the first case? Based on what James wrote, for the second case, I'm assuming the satellite sweeps out equal areas in equal intervals of time.

[James Tursa]

We showed earlier that if you assume theta = omega*t that you end up with the velocity equations. This of course carried the implicit assumption that omega is a constant. Now look at this equation that you have questions about:

 omega = (Vo*ro)/(x^2+y^2)

The left hand side is constant, but the right hand side is not constant. The numerator is constant, but the denominator varies with position on the ellipse. Therefore, the conclusion is that this particular equation does not represent the motion associated with the case we have been discussing, namely the theta = omega*t case.

Well, if it is not that, then what is it? To answer that, I would direct you attention to this link for planetary and satellite orbital motion:

https://en.wikipedia.org/wiki/Kepler_orbit

Scroll down to the "Mathematical solution of the differential equation (1) above" section and look at equation (3):

thetadot = H / r^2

Here, thetadot is not constant ... it can be thought of as an "instantaneous omega" that changes with time. I.e., we could rewrite this equation as follows with this understanding:

omega = H / r^2

That is, the instantaneous angular rate is a function of the orbital angular momentum magnitude and the current radius from the attracting body.

This explanation is getting long so I will cut to the chase. If we take the initial position to be at the "end-point" of the ellipse (i.e., close approach to the planet) so that the instantaneous velocity at that point is perpendicular to the current radial position line, then the angular momentum at that point will simply be:

H = Vo * ro

It turns out that angular momentum is constant for the 2-Body planetary problem. So that gives:

omega = Vo * ro / r^2

And the radius from the attracting body is simply r = sqrt(x^2 + y^2), which finally gives:

omega = Vo * ro / (x^2 + y^2)

What does all this mean?

It means that the omega = (Vo*ro)/(x^2+y^2) equation does NOT apply to the case (1) theta = omega*t case above. It doesn't even use the same definitions of what x and y are. It DOES apply to the case (3) I mentioned above, but only if the Vo and ro came from a specific point in the orbit where their simple product equals the angular momentum of the orbit, and only if you assume that x and y are measured from the attracting body and not from the center of the ellipse. If you found this omega equation on the same page as the velocity stuff, it is clearly in error and the author of the page should be contacted. Can you post the links you used to get these equations?

Unmanned aircraft flying an elliptical loop: Case (2) would probably be the closest match.

Satellite moving in elliptical orbit: Case (3).

[Fiona Rae]

Thank you, James!