Center of Mass of a jetpack-Earth System

[Physics Quandary]

It is known that the center of mass of an isolated system will assume the same location no matter what internal forces there are (as long as no net external force occurs).
My first question is if there exist two bodies of masses m1 and m2 in some space that has no friction, gravity, or external forces present, and one of the bodies move via some internal force (maybe a propulsion engine via fuel), will the other body move in the opposite direction to maintain the center of mass location or will it just stay there? I see two potential breaches regarding Newton’s First Law and the center of mass concept
My second question is a case of the previous question: if there is an automatic jet pack on Earth and one considers the Jetpack-Earth system (which is essentially isolated and closed), and the jet pack starts to take off from the surface of the Earth, won’t the Earth move a very, very small amount? If so, doesn’t this violate Newton’s First Law again? I know when you walk, you move the Earth a small amount to correct for the center of mass, but there is the internal force of your foot doing work on the Earth, thereby not violating Newton himself.

 

[Dale]

one of the bodies move via some internal force (maybe a propulsion engine via fuel)

I am not sure if you mean an internal force as in a force between m1 and m2 which is internal to the m1 m2 system or an internal force which is internal to m1 itself.

if there is an automatic jet pack on Earth and one considers the Jetpack-Earth system (which is essentially isolated and closed), and the jet pack starts to take off from the surface of the Earth, won’t the Earth move a very, very small amount?

Yes.

If so, doesn’t this violate Newton’s First Law again?

No. The jet pack exerts a force on the Earth so Newton’s first law doesn’t apply.

 

[Janus]

It is known that the center of mass of an isolated system will assume the same location no matter what internal forces there are (as long as no net external force occurs).
My first question is if there exist two bodies of masses m1 and m2 in some space that has no friction, gravity, or external forces present, and one of the bodies move via some internal force (maybe a propulsion engine via fuel), will the other body move in the opposite direction to maintain the center of mass location or will it just stay there? I see two potential breaches regarding Newton’s First Law and the center of mass concept

The only way to move that body via propulsion is for it to throw mass in the opposite direction. The center of mass of expelled fuel and object will remain fixed. Since the center of mass of this system doesn't move, the center of mass of the whole system doesn't move,and the other body doesn't move.

My second question is a case of the previous question: if there is an automatic jet pack on Earth and one considers the Jetpack-Earth system (which is essentially isolated and closed), and the jet pack starts to take off from the surface of the Earth, won’t the Earth move a very, very small amount? If so, doesn’t this violate Newton’s First Law again? I know when you walk, you move the Earth a small amount to correct for the center of mass, but there is the internal force of your foot doing work on the Earth, thereby not violating Newton himself.

Again, the jet pack has to throw mass downwards in order to move upwards.

 

[A.T.]

The only way to move that body via propulsion is for it to throw mass in the opposite direction.

You can also use mass-less photons. Then the question becomes if the center of mass concept can be extended such that the rule mentioned by the OP still applies in that case.

 

[sophiecentaur]

You can also use mass-less photons.

In which case, the system would need to be analysed in terms of Momentum and everything would be OK.
The 'propellant' in both cases would be intercepted by the Earth (or at least some of it) and that would affect the overall result (no / very little net acceleration)

 

[A.T.]

In which case, the system would need to be analysed in terms of Momentum and everything would be OK.

Sure, but the OP specifically asks about the center of mass concept.

 

[jbriggs444]

You can also use mass-less photons. Then the question becomes if the center of mass concept can be extended such that the rule mentioned by the OP still applies in that case.

In which case, the system would need to be analysed in terms of Momentum

Sure, but the OP specifically asks about the center of mass concept.

So let's use the momentum concept and find the center of mass that way. The object will be to see whether [massless] photons enter into the computation. I am spit-balling here, never having thought much about the problem of defining the center of mass of a system containing massless components.

Start by finding a reference frame where the total momentum of the object $m_1$ plus its expelled photon plume is zero. Clearly there is such a frame. It is equally clear that it will be identical to the frame where $m_1$ started out stationary before it expelled any photons. This is the center-of-momentum frame.

[For our purposes we will be pretending that photons are classical particles (little massless bullets moving at c) and not weird quantum thingies]

Define the "center of mass" for this system. We will define it using angular momentum in the following way.

In its center of momentum frame, the system has an intrinsic angular momentum that is independent of the point we choose to use as the origin/reference point for computing angular momentum.

1. Pick a candidate center of mass. Use it as the origin of a coordinate system anchored to the center of momentum frame.
2. Perform an infinitesimal boost by $\Delta v$ in any direction.
3. If the angular momentum of the system has changed, that candidate is not the center of mass.

Repeat until you find the unique point where an infinitesimal boost in any direction results in no change to angular momentum.

Note that massless photons have linear momentum and can contribute to the angular momentum of the system.

 

[sophiecentaur]

Sure, but the OP specifically asks about the center of mass concept.

In which case, with massless photons, it's not soluble in those terms.