The fuel in an airplane is also exhausted but I'm not sure how much of momentum is gained due to this rocket fuel effect, in the traditional working regime.
For lower stages, you can inject water into the nozzle and ISP (exhaust velocity) goes down, but thrust goes up, more than compensating for the weight of the water tanks.
But you wouldn't want to do that for an upper stage, as all the rocket below that would need to carry the water tank to a higher velocity.
Saturn V second stage had variable mixture ratio, using a leaner mixture with lower ISP (but better density and thrust) at lower altitude and richer mixture with higher ISP at higher altitudes. https://yarchive.net/space/launchers/saturn_variable_mixture...
So, ISP is clearly not everything. At low altitudes, you're looking at basically maximum thrust and thrust to weight. At some level you're also looking at volume and thus density, to minimize cost.
My upper point was more about asking about the regime in which we can consider an airplane a constant mass system, and could totally neglect the mass of the consumed fuel compared to the momentum of the air moved.
One of the main difference between a jet engine and a rocket engine is that the rocket engine carry its own oxygen, but I'm also not quite sure how much the momentum of the surrounding air is relevant, in helping push the plane forward.
There is probably some kind of transition where either when air density get lower, or when speed get higher, than your jet engine behave more like a rocket motor than a blowing-fan (like the turbo-prop vs turbo-fan in the other comments).
In outer space, the rocket equation says that the delta-v is only a function of propellant mass and exhaust velocity. It is always better to have higher exhaust velocity, no exceptions.
But until you get to orbit, you experience the “gravity drag”. Gravity acts in the opposite direction of your motion, and reduces your speed, so you end up needing overall more delta-v.
In order to reduce the effect of the gravity drag, you want to accelerate as quickly as possible. That’s why astronauts have to be able to withstand high Gs for a few minutes. Ballistic missiles, which are not limited by the human limits can (and do) acceletate much faster. A Trident II gets to about 50 Gs.
To get to these insane accelerations, you need huge thrust. You achieve that at the cost of lower exhaust velocity (like in the case of the Space Shuttle). Lower exhaust velocity means lower delta-v, but higher thrust means lower delta-v losses due to the gravity drag, so in many cases this tradeoff makes sense.
The problem for first stages is that some high ISP propellants are very non-dense. Liquid hydrogen is 70 kg per cubic meter. Kerosene is about ten times as dense. So the tanks need to be a lot bigger for the same payload. The same holds for pumps and turbines too. So the engines need to be bigger for the same thrust. Hydrogen is also annoying to deal with as it's much colder than liquid oxygen, the molecule is also much smaller and leaks much more easily.
Technically the fueled all-up hydrogen rocket might be lighter, so the engines should be smaller, but the density disadvantage eats back all that advantage. The dry mass, which is the actual cost driver, is about the same.
For upper stages, it makes sense because the cost driver is not the dry mass but the wet mass. Also they need less thrust so engine mass is a smaller proportion of stage mass.
The other day I saw the TechIngredients video about afterburner https://www.youtube.com/watch?v=EP4Hf63zR6w that I found interesting.
Edit: The wikipedia page about specific impulse https://en.wikipedia.org/wiki/Specific_impulse and in particular its graph https://upload.wikimedia.org/wikipedia/commons/thumb/4/4f/Sp... helped me quantify the importance of air in the propulsion efficiency of a jet engine with respect to a rocket engine.
For rockets, there's Rocket Lab's Electron that uses electric pumps to pump the regular propellants to the combustion chamber. This was much easier to develop than a turbine for the turbopump. It drops some spent batteries along its flight.
Could an electric airplane drop batteries at certain points? Like after climbing to target altitude. A lot of operational headaches there, so not likely. There have been some military planes that have used takeoff trolleys or takeoff rockets so there is some precedence there.
Rocket engines are extremely inefficient, they just have to be used because of the very wide speed range they support. So it wouldn't really make sense to eject water with an electric pump. Batteries have such low energy density and the pumps are heavy too. It's the worst combination to carry your own reaction mass and propel it with battery stored energy!
A rocket engine has tremendous trust to weight and power density, it can be gigawatts in a car sized package. This is enabled by the very energy dense liquid fuel and liquid oxidizer. Even a jet engine pales in comparison, as it has to use gaseous air as oxidizer which is not dense, and which is 80% "fluff" nitrogen. Hence the jet engine compressors have to be large, and the compression ratios are meager 10 to 20 bars. A rocket engine can run at 200 bars main chamber pressure. The compressors (pumps) can operate at hundreds of megawatts (this is actually not far from big jet engines). The thrust to weight ratio can be 100 to 1, while for jets it's 10 to 1.
Until then we have to stay tethered to the sea with something like Zapata's Flyride / Flyboard https://www.youtube.com/watch?v=PCAvIHGylKo
Edit : (We just have to put an electric jetski inside a fire-fighting plane, and have the Flyride pull the plane forward until it reach lift-off speed, (no expensive fuel needed :) )
(This ignores potential energy from changes in altitude.)
This is one reason why it's better to use a lower Isp propellant on the first stage of a multistage launcher (also, those propellants are much denser.)
This is a complicated area, but in general basically: Supersonic flow is bad. At best you get a ton of noise (like the Russian Tu-95). Efficiency also goes to hell.
This is also why the SR-71 had those crazy moving brake cones… they manipulated the shockwave so the air actually entering the engine was subsonic.
Now, for a given energy, you can push some mass of air with speed v or 4 times as much with speed v/2. The momentum of the air in the latter case is double the first case: 4 m v/2 = 2mv. You always want to move more air at lower speed.
You achieve that with higher and higher bypass ratios. Of course, with a higher bypass ratio the engine becomes larger and heavier (and draggier), so there are limits. But if you could increase the bypass ratio without increasing the engine weight and drag, you would always do it. As a bonus, with lower air speed come lower vibrations and noise.