The Science of Water Rockets

There has been a lot of publications lately about water rockets. These are rockets which are usually constructed of empty plastic soda bottles, pressurized by a bicycle pump and launched into the air. I haven’t done any of this, but it sounds like great fun. I even picked up a copy of Soda Pop Rockets by Paul Jarvis, which is a colorful if rather arts-n-craftsy book on the subject. While reading it, I couldn’t help but muse about the physics involved. How do things like diameter of the rocket, diameter of the exhaust nozzle, and the amount of pressure affect the height that the rocket might achieve?

I mentioned this to Loren over lunch the other day, since I know that he has done his fair share of water bottle rocket launches, usually using liquid nitrogen as the propellant. He had a few interesting insights, namely that the nozzle design which is common in ordinary rockets isn’t really useful in water rockets, as water is essentially incompressible. This means that you can use Bernoulli’s equation to compute the force generated. Of course, the mass of the rocket is continuously dropping as water is expelled, but that’s not too hard to deal with in the simulation.

What wasn’t clear to me was that after the water is exhausted, there is still residual air pressure in the rocket, and this pressure is significant and must be dealt with differently since the air is compressible. A bit of research led me to this rather nice website:

Rocket Science.

He has an interesting example of a 2 liter bottle which is pressurized to 100 psi and filled 20% with water. It’s fascinating: the “water burnout” (when all water has left the rocket) occurs only 0.042 seconds after launch, when the rocket has an altitude of only 3.2 feet (!). It continues to accelerate though as the air pressure equalizes. In this example, one third of the velocity of the rocket is obtained after water burnout.

It might be fun to make an implementation of this.

Addendum: Here’s another link to a water rocket simulator.
Addendum2: Another simulator.