I really wanted to fully understand that, but some of it went right over my head.
It's relatively easy to break down.
Rocket thrust in a vacuum comes from two sources: the momentum of the exhaust and the static pressure of the exhaust. The first term in the equation looks at momentum, which has been written extensively about since Newton first discovered it. The momentum of the exhaust leaving in one direction matches the momentum of the rocket moving in the opposite direction, per Newton's third law of motion. The mass part of the momentum formula is the mass of the propellants, here described as the mass applied per unit time, because the exhaust leaves the system and must be replaced over time by new propellant.
The key here is that the mass enters the thrust chamber as a liquid -- often a dense cryogenic liquid. But it leaves the engine as a gas of much greater volume and much less density than before. To be sure, the mass flow rate into the thrust chamber is the same as the mass flow rate out of the thrust chamber, but the exhaust mass is in a different form that must move much faster in order to sustain that flow rate.
The propellant is first converted to gas in the top of the thrust chamber. It's sprayed together in such a way that it mixes thoroughly, and then the radiant heat from the reaction downstream vaporizes it. Those thoroughly mixed gases are then ignited, creating vast amounts of thermal energy in the working fluid. Every gas responds to an increase in thermal energy by trying to increase its volume and/or pressure. The urge to do so in a rocket engine is extremely powerful. A wonderful Victorian-era gadget called a de Laval nozzle -- a convergent-divergent nozzle -- lets the gas escape from the only remaining hole in the thrust chamber in a way that collimates the flow. All the gas molecules are going in the same direction, maximizing the momentum. Otherwise, pressurized gas escaping from a plain hole in a pressure vessel will expand in a cone-shaped pattern.
The conversion of chemical energy thermodynamically to gas pressure, and from there to gas velocity, is what the poster's proof is missing.
That's a more nuts-and-bolts explanation of where the momentum thrust comes from. Pressure thrust comes from the
static pressure of the exhaust gas. The gas streaming in linear fashion out of the de Laval nozzle has momentum. But it's still a gas with measurable static pressure. It doesn't have zero density. As such, it pushes against the walls of the nozzle just like the contained air in a balloon pushes against the balloon walls, even though the balloon air isn't hot and isn't moving. It's ordinary gas pressure. The term for this effect is "adiabatic," and it's the same principle by which steam locomotives conserved water by opening the steam valve only a little bit at the beginning of the power stroke.
If the static pressure of the exhaust is greater than the ambient into which it is exhausted, it will continue to expand in static fashion irrespective of its velocity. That urge to expand into a relatively unpressurized space is the ability to do what engineers call "pressure and volume work," in this case, to continue pushing in all directions. "All directions" in this case includes the direction of the rocket nozzle, which results in thrust. That's the second term of the equation -- the static pressure of the exhaust per unit area, minus the static pressure of the ambient (i.e., the pressure difference) times the area of the exit plane of the nozzle -- sort of like the area of the piston face in a steam cylinder.
Ironically, the poster here thinks he has cleverly discarded momentum thrust. But he hasn't dealt with the notion that in a vacuum, the ambient pressure is zero so the pressure component of the rocket equation actually contributes more. The notion of "having something to push against" is actually the opposite of what makes rockets more efficient as they climb.
Science is hard, but you don't see me making stuff up to compensate, like some other people.
That's just it. Science is hard, but if overturning well-established, well-used principles were as easy as scribbling some poorly-remembered physics onto a sheet of notebook paper, it wouldn't come across as so inaccessible.
NASA didn't invent rocketry. NASA isn't the only state-funded space program. NASA isn't even the biggest consumer of rocketry. NASA relies on private industry to supply its rockets, the same private industry that sells access to space to other private industries. Spacefaring is a multibillion-dollar
industry. It doesn't give a rat's patootie about some ideological spat.
