# Optimization of Bottle Rocket Water Level

My (entry-level) physics class is building bottle rockets, and we are competing to build the longest-flying bottle rocket. The rockets are filled partway (we get to decide how much to fill them) with water, placed on the launcher, and filled with compressed air to a certain arbitrary pressure, the value of which will be the same for each rocket, before it is launched.

I am trying to maximize the impulse given to the rocket when it is launched so as to maximize its initial velocity (and thereby its height).

I am not very familiar with fluid dynamics, but I did find this equation after some digging: $v = \sqrt{2q \over \rho}$. However, I am struggling to get beyond this point. I want to express everything in terms of the volume of water added to the bottle (so I can take the derivative and find the maximum altitude/flight time), but I have issues with the density in this equation; it seems like it will change depending upon the water volume as well, and I am struggling to model how.

I have a thorough understanding of single-variable calculus and a decent understanding of multi-variable calculus. Please help me solve this, whatever it takes. I don't even care about the rocket – I just think this is really cool.

• This is a related question you might find useful. – fibonatic May 16 '15 at 3:28
• You normally assume an incompressible fluid, so the density of the propellant does not change. – Rob Jeffries May 16 '15 at 7:58
• Please show us your working and ask a question about a specific physics concept. – Rob Jeffries May 16 '15 at 8:01
• @fibonatic the answer in that question used a formula which is valid for constant exhaust velocity under the assumption that unused fuel has the same velocity as rocket. both of which are wrong here. – Azad May 16 '15 at 14:07
• I've seen many different formulas on the internet but all of them had used steady form of Bernoulli equation which is blatantly wrong in the case at hand. It's not even quasi steady (since ${\partial V \over \partial t}\gg V{\partial V \over \partial s}$) – Azad May 16 '15 at 14:12

Your analysis will need to be a combination of theory and experiment.

You are not maximizing impulse alone: you need to think about drag as well. So you need to think about the shape of your rocket as well as the volume of water and initial pressure. The drag on the rocket is probably well modelled by the ram pressure equation $F_d=\frac{1}{2}\,\rho_{air}\,C_d\,v^2$ where $\rho$ is the air density and $C_d$ the drag co-efficient of the rocket's cross section. This latter you will need to find experimentally.

The equation you cite is the application of Bernoulli's principle to the calculation of the exhaust speed. It seems that the rocketteers at this link:

Water Rocket Analysis at Ohio University

also found Bernoulli's equation effective for this analysis. This link also gives you the differential equations that describe the decrease in exhaust velocity as the rocket empties itself, which they derive by assuming an adiabatic expansion of the compressed air.

Lastly, you will need to adapt the Tsiolkovsky Rocket Equation to the variable exhaust velocity situation you have (by dint of the decreasing velocity as the rocket empties and thus lowers its pressure) to work out the change in velocity the water will beget (the "delta V" for your rocket.

I outline a derivation of the rocket equation for two slightly different situations in my answer to the Physics SE question "Variable mass dynamics: Particle and Rigid Body".