In Portland's OMSI there is a hands-on water bottle rocket station. (https://www.youtube.com/watch?v=cdtmVY76_PQ). The rockets are normal PET bottlers. The visitors fill their bottle with an amount of water and then fill the remaining volume with compressed air at a given pressure.
The challenge is to find the best water-to-air ratio so that the rocket flies highest. Too much water is bad, not only because it makes the rocket heavy. As I explained to my son, the compressed air is also the energy store for this rocket (not the water, since water is almost incompressible). To have less compressed air means to have less energy available.
But then I got stuck because by inversion this means that an "all air" configuration should be best: Most available energy, highest kinetic energy, highest speed of empty bottle. This is obviously wrong. It was clear experimentally that the best ratio is somewhere in the middle. Also it makes intuitive sense that some mass in form of water is needed to produce thrust, since actio = reactio. In order to produce momentum, mass is needed to "push off of".
I'm aware of the fairly complex rocket flight physics. (For example, https://www.ohio.edu/mechanical/programming/rocket/analysis1.html gives an accessible overview.) But because I am not interested in an exact result much of it can be neglected. The basics are fairly simple: Energy stored in the compressed air is transformed into kinetic energy of the expelled water, rocket and earth, plus "losses" through heat from turbulences.
My question is on a more general, abstract level. Momentum or not, we have a given energy in the air which must go somewhere.
Where does the energy go which is stored in the compressed air in a "compressed air only" configuration? It should be more energy than with a partly water-filled bottle; but the rocket's final velocity (and hence kinetic energy) is much lower. Did we produce that much heat? I don't think so. Did we accelerate the earth? No, the "burn phase" was short.
I am missing something. What is it?