We all have seen basic water rockets, where pressurized air pushes out water creating downward force. Is it theoretically possible that a water rocket could create enough force, for a long enough duration, to reach orbit? If not is it possible that it could assist a solid or liquid booster engine?
A water bottle rocket is similar to this question about using a pump to expel a liquid at high speed. We concluded that the highest possible exhaust velocity with this strategy is very low (1.3 km/s, about 1/3 of the Shuttle main engines).
A water bottle that stores water plus air at high pressure is even less efficient, because to contain the pressure you need a thick, heavy tank (my estimate: 60 cm thick steel walls for 6000 bar).
So how about using ordinary Coke bottles (chosen because they're common and among the strongest bottles available in the supermarket)? We'd have to use a lot of them in parallel. Randall Munroe did the physics on a similar concept: using model rocket engines to launch into space.
If you try to produce an orbital rocket using the same design math we used for the suborbital rocket, it spits out a description of a pancake-shaped mountain of model rocket engines over a mile wide. It would taper to a 10-meter-high spire in the center and would weigh about as much as the Great Pyramid.
Not only would this vehicle never get out of the atmosphere, it would probably not stand up under its own weight.
So, with enough bottles to provide the thrust we need, we have an unstable structure. This is not a feasible way to get to orbit.
There have been hot water rockets used for crash tests of cars to accelerate on a short distance. The water in the tank was heated electrically to a very high temperature and pressure. When the valve to the nozzle was opened, the water turned into steam and left the nozzle with high speed. This principle is far more effective than just water and air under high pressure in a tank.
These type of rocket is very simple, only one tank, no turbo pumps, only one valve, no combustion chamber and no need to cool the nozzle walls.
But the tank must be build for very high pressure at high temperature and will be very heavy. Some tank isolation might be necessary to limit the heat loss on the start pad and during the flight. The temperature in the nozzle will be much lower compared with a LH2/LOX rocket and therefore also the specific impulse too.
We have a rocket with a very bad ratio of structural mass to fuel and a low exhaust velocity. To calculate the necessary number of stages, we need a good guess of structural mass and exhaust velocity, I would think about 30 % and some 300 m/s, but we need an educated guess I can't give. Here https://en.wikipedia.org/wiki/Steam_rocket is a value for the Isp, 195 s, not much compared with 448 s of a modern hydrogen rocket engine. Here http://www.tecaeromex.com/ingles/vapori.html I found an exhaust velocity of 457 m/s. If somebody may confirm my guess of 30 % structural mass, we may calculate the number of stages.
In the meantime I calculated the number of stages for 8 km/s, 10 % payload, 30 % structural mass and 60 % fuel (hot water), but the result is absurd: 19 to 20 stages. Even with 20 % structural mass and 70 % water we need 15 stages.
The start weight of a Saturn V was 2,935 t, the fourth stage of the hot water rocket for a payload of 1 t only (counted from top, not from bottom as usual) would have a weight of 10,000 t with fuel.