Solving Central Force Motion: Sketch Equivalent Elliptic & Hyperbolic Orbits

In summary, the task is to sketch the elliptical and hyperbolic orbits of two objects with equal masses around each other, and then to sketch the equivalent one-body orbit for each scenario. For the elliptical orbit, the center of mass is located midway between the two objects, and the two ellipses overlap in a venn diagram fashion. The equivalent one-body orbit can be visualized by reducing the system to a single object with a reduced mass, and understanding its path in a central potential. For the hyperbolic orbit, the path would be similar to the elliptical orbit but with a higher eccentricity. It is important to first understand the single body in a central well cases, and then look at the reduced mass and
  • #1
w3390
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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 [tex]\mu[/tex] 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.
 
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  • #2
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.
 

Related to Solving Central Force Motion: Sketch Equivalent Elliptic & Hyperbolic Orbits

What is central force motion?

Central force motion is a type of motion in which a particle moves under the influence of a force that always points towards a fixed point, known as the center of force.

What is an equivalent elliptic orbit?

An equivalent elliptic orbit is a type of central force motion in which the particle moves in an elliptical path around the center of force, with the center of force located at one of the foci of the ellipse.

What is an equivalent hyperbolic orbit?

An equivalent hyperbolic orbit is a type of central force motion in which the particle moves in a hyperbolic path around the center of force, with the center of force located at one of the foci of the hyperbola.

How do you sketch an equivalent elliptic or hyperbolic orbit?

To sketch an equivalent elliptic or hyperbolic orbit, you will need to know the position of the center of force, the initial position and velocity of the particle, and the magnitude and direction of the force acting on the particle. Using this information, you can use mathematical equations to plot the path of the particle and create a sketch of the orbit.

What is the significance of equivalent orbits in central force motion?

Equivalent orbits are significant in central force motion because they allow us to understand and visualize the behavior of a particle under the influence of a central force. They also help us to predict the motion of planets and other celestial bodies in our solar system.

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