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What is the Math behind bottle rocketry?

Posted: Tue Mar 01, 2016 10:00 am
by Astrofox
I have a good question for the forums - what are the mathematics behind water rocketry?
Well, more specifically, how do I calculate the best water levels without having to guess and check with simulators?

It's just something I want to know more about. Also, since this is related to this topic, how do you calculate the force of drag without the drag coefficient? I need to know this so that I can build better, more aerodynamic rockets using simple materials.


Re: What is the Math behind bottle rocketry?

Posted: Thu Mar 03, 2016 8:21 am
by anoymous
Well, you can't calculate the drag force without the coefficient. But you could try to measure it indirect:
If you know the initial speed (after the thrust phase), and you know the height of apogee, you can calculate the deceleration of your rocket. When comparing this to g, I think you can calculate the drag (and maybe the drag coëfficient?

Re: What is the Math behind bottle rocketry?

Posted: Sun Mar 13, 2016 4:44 pm
by Astrofox
So, I've come up with some more questions involving the Math behind bottle rocketry - How does one determine the Specific Impulse of a bottle rocket by using the volume of the pressure vessel, the pressure of the air inside the pressure chamber, and the volume water takes up?

Re: What is the Math behind bottle rocketry?

Posted: Thu May 26, 2016 3:31 am
by david2665
Out of my own personal interest, I began looking into this.

Searching the web, brought up a few, quite insightful resources, mostly from educational institutions: ... ysis1.html which contains a link to an interesting .pdf file (nielson equations) ... t_eqns.pdf ... %20430.pdf

and the most useful I found…. ... CTIONS.doc page 21 onwards.

I plugged the equations into excel, iterating line by line over time, to come up with a mathematical simulator:

However, whilst the output looks good, I checked it against the Clifford Heath simulator, and the results were not the same….


Re: What is the Math behind bottle rocketry?

Posted: Sat Nov 04, 2017 6:28 pm
by anachronist
I just found this thread. Earlier this year I spent a few months developing my own simulator, which accounts for just about everything that can possibly be accounted for. I documented the calculations in 5 posts of my blog starting here:

To answer Astrofox's original question: For a given initial pressure (like 100 psi), there's an optimal combination of empty weight and total fill volume that achieves the maximum altitude, for a rocket with a given drag coefficient. Because all the mathematical relationships are nonlinear, the only realistic way to calculate this is by numerical analysis. So yes, you need a simulator.