Solving Central Force Motion: Sketch Equivalent Elliptic & Hyperbolic Orbits

[w3390]

Homework Statement

Sketch the elliptical and hyperbolic orbits two objects make around each other if their masses are equal. Next to each, sketch the equivalent one-body orbit.

Homework Equations

The Attempt at a Solution

For the elliptical orbit, I just drew two ellipses that overlap in a venn diagram sort of fashion where the center of mass is midway between the two objects.

I am confused about the equivalent one-body orbit. The system can be reduced to a single object of reduced mass \mu but I don't understand how to figure out its path.

For the hyperbolic orbit, wouldn't it just look the same as the elliptical orbit except with a higher eccentricity?

I have the same confusion when trying to deal with the path of the reduced mass in this one too.

If anyone can help clarify to me how I can visualize the sketches, I would greatly appreciate it.

 

[jambaugh]

Firstly go back and look at the details of the reduced mass business. The equivalence is to a single body in a central potential. It becomes essentially a two body problem with the central body fixed (i.e. as if it had infinite mass).

Secondly a hyperbolic orbit is not an ellipse it is a hyperbola.

I would start with the single body in the central well cases first. Then look again at what that reduced case means. You should have seen its derivation somewhere with the position and velocity derived in terms of the positions and velocities of the two bodies.

See how that relates the reduced cases (which are simpler) to the two body cases... especially the relationships of the position vectors.