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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.

Thanks!

### 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:

https://www.ohio.edu/mechanical/program ... ysis1.html which contains a link to an interesting .pdf file (nielson equations)

http://www.et.byu.edu/~wheeler/benchtop ... t_eqns.pdf
http://stemstates.org/assets/files/430_ ... %20430.pdf
and the most useful I found….

http://research.che.tamu.edu/groups/Sem ... 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….

Dave

### 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:

http://www.nablu.com/2017/07/the-water-rocket.html
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.

-Alex