How does fire create thrust in rocket?

All big rockets are burning either gas or fluid to create thrust. While this is so, I have filled up a plastic bottle with air at high pressure, and it can go long distances by blowing the pressurised air at reverse direction. If my bottle can do this without using any fire, why don't rockets just use air? How is the effect of combustion in thrust?

What the rocket does is holds the propellant in a lower-pressure, easier to store form, with higher energy density. Basically you have a lot of that kinetic energy you'll need in the form of chemical energy. Then you can pipe that fuel to the combustion chamber and ignite it there. At that point all that chemical energy is released in the form of expanding gases and heat.

So, instead of having a very heavy container holding all that hot pressurized gas, you just have a smaller container holding your fuel.

Another way to think about this, is to realize that your bottle is basically a combustion-chamber-only rocket: You have a hollow body with hole in the bottom, which is under some high pressure.

The differences between your bottle and a real rocket are:

• The combustion chamber contains a much, much higher pressure than your bottle ever could.

• The combustion chamber is constantly being refilled by the turbopumps.

These two factors alone mean, that a real rocket can reach much higher speeds than your bottle ever could, and that it can do so with a combustion chamber that's only a tiny part of the rocket. This small combustion chamber can be built as robust as necessary to withstand the highest possible pressures, without contributing significantly to the weight of the rocket.

Now, why do we still need the fire inside the combustion chamber? After all, we could just put the output of the turbopumps through a nozzle, and drive the rocket with that, no?

The fire turns the cold fuel into very hot gases. The fuel that is pumped into the combustion chamber is rather cool, so it's rather dense. The fire that leaves the nozzle, is the exact opposite. That means, that the pumps have a lot less work to do. They pump a volume $$V_{in}$$ into the combustion chamber, overcoming its internal pressure $$p$$, and a much larger volume $$V_{out}$$ leaves through the nozzle, driven by that same pressure $$p$$. Work is $$W = pV$$, and the difference $$W_{out} - W_{in} = p(V_{out} - V_{in})$$ is the work that is contributed by the burning fuel, and which the turbopumps do not have to deliver.

So, yes, any rocket is similar to your bottle rocket in principle, but it's the differences that allow real rockets to accelerate to over 7km/s which your bottle won't manage.