Potential Energy of Nearly Hovering Rocket

[JerryR]

I have a question concerning gravitation potential energy and rockets under an unusual situation. Let a rocket be in a gravity well of a massive object such as a planet. For simplicity, assume that the rocket is in a vacuum. (we can add air effects later). The rocket engine thrust is dynamically adjusted to exactly counter the pull of gravity. Adjustment is needed as the mass of the rocket body will be changing. Three things could happen depending on initial velocity.

1. The rocket could hover at a fixed altitude. The potential energy of the rocket body will be decreasing as the altitude remains constant and the mass is decreasing. There would be no kinetic energy associated with the rocket body itself.
2. The rocket could be climbing at a constant rate as it coasts upward. The kinetic energy of the rocket body would be decreasing as the mass decreased and the velocity was constant. The potential energy could either increase or decrease depending on whether the altitude increased faster than the mass decreased.
3. The rocket could be falling at a constant rate as it coasts downward. The kinetic energy of the rocket body would be decreasing as the mass decreased and the velocity was constant. The potential energy would be decreasing as both the altitude and mass decreased.

I am interested in the change in potential energy of the rocket body. Assuming that the magnitude of the velocities are the same in cases 2 and 3 the kinetic energy/time profiles will be identical. The expenditure of energy from the rocket would appear to be identical. What is the source of sink of the potential energy change? I have tried to puzzle this our considering the mass ejected by the rocket motor but so far have come up blank.

I would appreciate any thoughts on this question.

 

[jbriggs444]

What is the sign of the work done on the descending rocket by its motor?

 

[A.T.]

I have tried to puzzle this our considering the mass ejected by the rocket motor

How does the movment of the rocket affect the speed of that ejected mass?

 

[JerryR]

jbriggs444 - in all three cases the thrust is in a direction to drive the rocket upward just enough to counter gravity. I think that this means that the sign of the work is identical. The descending rocket is coasting downward.

A.T. - I assumed that the speed of the ejected mass relative to the rocket was the same in all three cases. Relative to the planet the velocity of the rocket would need to be added.

 

[A.T.]

Relative to the planet the velocity of the rocket would need to be added.

Isn't that the reference frame where you do the energy calculations?

 

[jbriggs444]

jbriggs444 - in all three cases the thrust is in a direction to drive the rocket upward just enough to counter gravity. I think that this means that the sign of the work is identical.

What is the definition of work?

 

[JerryR]

A.T. - This is the reference frame I am using.

jbriggs444 - Work is force times distance. I think I miss-interpreted your question. If the potential energy is decreasing, as in the descending rocket, the work is negative. For the hovering rocket the potential energy goes down due to mass being ejected and the work is negative. For the ascending rocket the work could be either sign depending on the dominance of the potential energy reduction due to mass ejection versus potential energy increase due to altitude gain.

 

[Nugatory]

For the hovering rocket the potential energy goes down due to mass being ejected and the work is negative.

You've already said that work equals force time distance. What is the distance for the hovering rocket?

(A pretty good rule of thumb is: Any time that rocket calculation looks confusing or wrong... Go back and double check the way you're handling the momentum and energy that's being carried away by the exhaust).

 

[jbriggs444]

jbriggs444 - Work is force times distance.

More correctly, work is force times displacement. Distance is a scalar. Displacement is a vector. You determine whether work is positive or negative by whether the force is in the same direction as the displacement or in the opposite direction. If one is concerned specifically with the work done by the rocket motor on the rocket then one need not examine the exhaust stream. The work done is defined as the dot product of force and displacement. Nothing else enters in.

 

[A.T.]

This is the reference frame I am using.

Then consider the kinetic energy of the ejected mass in that frame.

 

[JerryR]

I think that I have this pulled together now using some of the suggestions above. I have redone the analysis using the potential and kinetic energies of both the rocket body and the ejected mass. The result is that the sum of these energies equals the energy supplied by the rocket engine. Energy balance is obtained. I still need to restructure this to make an argument I am pursuing in another area but that is outside the scope of this thread.

Thanks for the help and discussion!

I have written this out but have not found a way to attach a document.