Exploring Rocket Momentum Conservation: A Followers Tale

[willoughby]

I would like to preface this by saying that I do not in any way resemble a physicist - and I'm sure the crudeness of my work will confirm that, but I thought that this was so cool, I wanted to share it. I'm just a follower of physics.

I understand that some of the concepts I'm going to be mentioning here are probably second-hand knowledge to a lot of you, but I still wanted to share. This originally started with me verifying for myself that the momentum of a rocket-fuel system indeed remains conserved. I'm certain that there are better ways to do what I have done, but I am unaware of them. Here's what I did.

Using Excel, I tracked the instantaneous variables for a hypothetical rocket that instead of firing continuously, it fired "bursts" of thrust at even (but arbitrary) time intervals - ejecting equal amounts of mass with each burst at equal velocities. I tracked the cumulative momentum and mass of all the fuel that had been ejected as well as momentum and mass of the "remaining" mass (the rocket). To get the most simplistic results, I ejected 1 kg of mass at 1 m/s at equal intervals. The longest a row can be in Excel is 16K and some change, so basically, my beginning mass was 16,383 kg and 1 kg of mass was ejected at 1 m/s every arbitrary time interval until the "remaining" mass was 1kg. Each column calculated the "new" momentum, mass, velocities, etc. based on the previous column - which is why I was limited to 16,383 "steps".

I did this several different ways, but the above is what was graphed and saved an image and uploaded here. By a "few different ways", I mean that I varied the initial mass, the velocity of the bursts and the mass of the bursts.

I wanted to talk about a few things -

The momentum didn't continually rise. This surprised me at first. I thought I had made a mistake, but once I started thinking about it, I came to this conclusion : At some point, the velocity of the ejected mass will be pointing the same direction as the rocket relative to the initial reference frame - in fact, once the rocket reaches the velocity the mass is being ejected at, all subsequent "bursts" will have negative velocity (assuming the positive direction is the direction of the first burst of mass),. so the combined momentum of all the ejected mass will start to decrease once that happens. The flip side of that is that the momentum of the rocket must decrease as well. This confused me. I know it MUST for the momentum to remain conserved, but I can't find a reason that satisfies me. The reason the momentum starts to decrease for the fuel became obvious, but not so much for the rocket. What is happening to make the momentum of the rocket begin to decrease - other than it MUST because the momentum of the ejected mass is decreasing?

(I am talking about magnitude of the momentum. Technically the momentum of the ejected mass is getting bigger, but it's max is really a min since it's negative. I hope that doesn't confuse anyone.)

I also noticed that no matter what, the point at which the momentum was at its greatest was the point that the rocket's mass was about 36.78% of its initial mass. I found this pretty remarkable. I did some research into this and found where this is actually a well-documented thing derived from the rocket equation (which I don't understand fully). Come to find out that the maximum/minimum momentum would be when the mass of the rocket is 1/e*initial mass - or about 36.78%. Again - I understand that this is common knowledge to some of you out there (if not most), but I just wanted to share my excitement.

Thanks for reading. I have attached a graph, and I apologize for how convoluted it is. The 'x' axis are in units of arbitrary time intervals. The 'y' axis is just the units of either kg for mass or kg m/s for momentum.

The arrow pointing to where the rocket mass and fuel momentum (same value as rocket momentum) is mass = 6,026 kg - which is 36.78% of the initial mass of 16,383 kg.

 

[Drakkith]

The momentum didn't continually rise. This surprised me at first. I thought I had made a mistake, but once I started thinking about it, I came to this conclusion : At some point, the velocity of the ejected mass will be pointing the same direction as the rocket relative to the initial reference frame - in fact, once the rocket reaches the velocity the mass is being ejected at, all subsequent "bursts" will have negative velocity (assuming the positive direction is the direction of the first burst of mass),. so the combined momentum of all the ejected mass will start to decrease once that happens. The flip side of that is that the momentum of the rocket must decrease as well. This confused me. I know it MUST for the momentum to remain conserved, but I can't find a reason that satisfies me. The reason the momentum starts to decrease for the fuel became obvious, but not so much for the rocket. What is happening to make the momentum of the rocket begin to decrease - other than it MUST because the momentum of the ejected mass is decreasing?

Have you tried tracking the momentum of the remaining fuel and the momentum of the rocket itself separately?

 

[willoughby]

Yes, that's what I did. There are two lines in the graph that make like a fish shape. These are the two independent momentum values for each the rocket and fuel. Obviously, the sum of which is zero the entire time. Arbitrarily, the rocket's momentum is the negative value; fuel is positive.

Like I said before, there is an intuitive reason that the fuel's momentum will decrease - being that it will eventually switch from positive velocity to negative velocity - thus beginning to decrease its momentum. I cannot manage a similarly intuitive reason for the rocket's momentum to switch directions - although I know it must to mirror the fuel's momentum. That's what I'm having trouble with. With the fuel, the velocity is the culprit - and it's obvious. With the rocket, the mass is decreasing, but there is no intuitive reason why it starts falling at the point it does (that I can come up with).

EDIT : You mean the fuel that hasn't been ejected. I misunderstood. No, I haven't, but isn't that arbitrary? I ultimately ended up with a mass of 1 kg, so the momentum would simply be the velocity - which I did track, but is not on this graph.

 

[Drakkith]

EDIT : You mean the fuel that hasn't been ejected. I misunderstood. No, I haven't, but isn't that arbitrary? I ultimately ended up with a mass of 1 kg, so the momentum would simply be the velocity - which I did track, but is not on this graph.

I don't know if it's arbitrary. I'd say make a graph of the momentum of the remaining fuel and a graph of the momentum of the rocket itself and see if that helps.

 

[willoughby]

I added it, but it doesn't do much. The "dry mass" of just 1 kg is virtually zero almost the entire time (until the very end long after the issue I have a question about) and the "remaining fuel" line is virtually identical to the total remaining since it's just the total minus the line that is pretty much zero the whole time.

By the way, thanks for chiming in and giving your input.

 

[A.T.]

but there is no intuitive reason why it starts falling at the point it does

What counts as an "intuitive reason" is always highly subjective. It might help you to decompose the firing of the 1kg fuel packets into 2 steps:

STEP 1:
Just detaching 1kg from the rocket, so it it continues to fly along the rocket as the same speed v_rocket. It merely no longer counts as part of the rocket. That step is always a loss in momentum for the rocket of 1kg * v_rocket.

STEP 2:
Push off from the detached 1kg.

Now, it should be obvious that if you push back that 1kg so fast that is stops, then it loses all momentum it had, so the rocket regains exactly what it lost in STEP 1. This is the transition point between gaining and losing momentum.