Quick Question on Kepler & angular momentum conservation

[RoboNerd]

Homework Statement

Homework Equations

I guess kepler's law but most importantly conservation of angular momentum are key here.

The Attempt at a Solution

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I put down E as the answer, but the solutions have D as the correct answer. Why is this the case?

Thanks in advance for the help!

 

[QuantumQuest]

I'll first ask why did you choose "E" as the correct answer. Through this you can clear out potential misundenstardings.

 

[RoboNerd]

My rocket exerts a thrust force on it backwards. This pushes it forward and increases its velocity. It must be a greater distance away from Earth in order to conserve angular momentum. Thus, E, which has a horizontal bulge, fits this description.

 

[gneill]

My rocket exerts a thrust force on it backwards. This pushes it forward and increases its velocity. It must be a greater distance away from Earth in order to conserve angular momentum. Thus, E, which has a horizontal bulge, fits this description.

Does the orbit depicted in E agree with Kepler's First Law?

 

[RoboNerd]

No. the Earth has to be at a focus

 

[QuantumQuest]

No. the Earth has to be at a focus

Exactly.The purpose of my question, was to direct you checking first with Kepler's laws, as gneiil points out, and see why "E" is wrong.

 

[RoboNerd]

OK. so why is D right?

 

[QuantumQuest]

Is "D" OK with Kepler's laws and conservation of energy?

 

[RoboNerd]

yes... it seems so

 

[gneill]

OK. so why is D right?

See if you can think of an argument that supports it. Consider what qualities of the orbit change when the maneuver is performed. What distinguishes C from D?

 

[RoboNerd]

C has a new orbit that is smaller than the first orbit. This does not work with conservation of angular momentum. It also has the Earth at the center, not of the focus.

D has a new orbit that is larger than the first. Earth is at the center. Angular momentum is thus conserved.

Right?

 

[gneill]

Because a force was applied when the rocket made its burn neither angular momentum nor energy will be conserved for the satellite. (you would have to include the rocket's exhaust material in the sum to conserve angular momentum, while the KE of the satellite increases because its speed is increased).

The important thing here is the increase in KE. Since the satellite is effectively at the same orbit radius immediately after the burn, the gravitational PE is the same but the KE increased. Thus the total energy of the orbit has increased. What do you know about orbits with larger total energy?

 

[RoboNerd]

The important thing here is the increase in KE. Since the satellite is effectively at the same orbit radius immediately after the burn, the gravitational PE is the same but the KE increased. Thus the total energy of the orbit has increased. What do you know about orbits with larger total energy?

I honestly do not know anything about orbits with larger total energy, or rather think I do not.

Sorry... what do I need to know?

 

[gneill]

I honestly do not know anything about orbits with larger total energy, or rather think I do not.

Sorry... what do I need to know?

The total mechanical energy of an orbit comprises its kinetic energy and its gravitational potential energy. Their sum is a constant for a given orbit. For bound orbits (circles, ellipses) the total energy is a negative value. As the energy increases the orbit becomes larger (the semimajor axis increases in size). When the energy value reaches zero the orbit is unbound, and the object will escape (parabolic trajectory for energy = 0, hyperbolic trajectory for energy > 0).

Do a web search on "specific mechanical energy of an orbit". Here's a wikipedia entry that's not too bad.

 

[RoboNerd]

So my energy has increased, and the object is moving away from the center of its orbit, the Earth that is pulling it towards itself. Thus orbit should be larger.

 

[gneill]

So my energy has increased, and the object is moving away from the center of its orbit, the Earth that is pulling it towards itself. Thus orbit should be larger.

Yes. The location where the KE was added (where the rocket fired) becomes the perigee of the new orbit.

 

[RoboNerd]

Great! Thanks so much for helping me understand this problem

 

[gneill]

You're very welcome.