Honors Differential Equations, Dr. John McCuan
Fall 2002

Vertical Motion of Water Bottle Rockets
Kevin Guthrie, Laurence Lindsey, Kathryn Sisson, Mashruba Tasneem 
http://backwarde.resnet.gatech.edu/rocket/

Abstract. A discussion of the vertical flight of water bottle rockets as built, measured, analyzed and derived by undergraduate honors differential equations students at The Georgia Institute of Technology.

Problem.  As supplies and facilities for this initial stage of experimentation were slightly limited, we chose to model only the vertical motion of the rocket's trajectory when launched from a self-constructed, pressurized launcher at ground level.  During the formulation of the differential equation of the rocket's trajectory, we found that it's flight must be modeled in three parts: the rocket’s launch with thrust, its continuing upward travel after propulsion has ceased, and free fall after the rocket has reached the apex of it's trajectory. The following is a picture of the vertical motion which we are attempting to model:

We devised formulas for the first two phases of flight only, as the third phase was beyond our knowledge of both mathematics and physics due to its arbitrary rotation in freefall.  Using Newton’s Law, F=(mv)’, and it’s influence to the conservation of energy, we applied Bernoulli’s equation, the ideal gas laws, and the equation for thrust to produce our ordinary differential equation for the velocity. 
	
Mathematics. Beginning with Newton’s law of motion, we researched and revised this equation to incorporate all of the relevant forces (thrust, gravity, and air resistance) to obtain 

   *

where u is the exit velocity of the water leaving the nozzle and v is the velocity of the rocket. To find u, we used Bernoulli’s Equation for Fluids, neglecting the compressibility of water.

P1 + rv12 + rgh1 = P2 + rv22 + rgh2

(We assume that the rocket is in freefall, so gravity plays no part in this equation. Hence, it may be removed)

Next we find the time derivative of m, and solve for m itself.

The pressure difference between the inside and outside of the bottle (P) was found using an incarnation of the ideal gas law.

PV = P0V0

V= 

.

Differentiating both sides yields

and using the Separation of Variables method to solve for P gives

This equation for P may then be substituted into (*) to find the fully derived equation for the vertical motion of the rocket. We then solved the equation numerically, a process which is described in the Results section.
	We encountered some logistic difficulties in this derivation: our first obstacle surfaced in the equation for thrust. The u-term is dependent on the nozzle pressure, which depends, in turn, on the exit velocity of the water. With some advice and academic inspiration we used Bernoulli’s equation for fluids (which is relevant to the conservation of energy), which relates this pressure and velocity coherently in the thrust equation.  Secondly, our original equation for pressure included an integral (obtained from the ideal gas law), which was unsolvable in that form. But, after some more thought and derivation, we were able to substitute it with a more appropriate and easily-dealt with equation, which then was solvable with the Separation of Variables technique,. 

Materials. The apparatus with which we experimented was built as follows:
	The rocket was constructed from a simple 2-liter soda bottle, which we modified by removing the cap and attaching 3 fins at equal distances and angles from the nozzle and each other. These fins were made from 1"x5" pieces of foam board and secured to the rocket with duct tape. 
	Our launching apparatus was more complicated. We constructed a three sided base from 4"x6" pieces of plywood with pieces of 2" wooden dowels as legs. This base had a smaller rectangular piece cut out in which we placed the actual launcher. The launcher was made from a similar 4"x6" piece of wood on which we screwed a double switch electrical housing. We drilled a hole with approximately a 1" diameter through which we inserted a jointed piece of PVC plumbing pipe. We mounted a rubber beaker stopper with two holes in it onto the PVC pipe in the electrical house as a seal for the rocket nozzle. On the opposite end of the pipe we epoxied a  valve- stem from a car tire for water proof pressurizing. This launcher was then fitted into the larger base and fastened with metal shelf reinforcements which were screwed down with wing nuts.  
	Our apparatus is pictured below:

Methodology. For launching, the above-mentioned rocket was filled with a standard 500mL of water measured with a graduated cylinder. The removable launcher was fitted stopper-to-nozzle on the rocket, and then a metal fork with a twine leash was inserted through two holes in one side of the electrical housing, under either side of the lip of the rocket's nozzle and finally in two identical holes on the opposite side of the housing. The launcher was fastened into the base which had been secured to the ground by tent stakes and heavy twine.  Then, using an electric compressor on the stem of the PVC pipe, the rocket was pressurized to approximately 70psi, and hence prepared for launch. 
	For each launch, one member of the group would pressurize the rocket while another member would concentrate on pulling the fork consistently, and horizontally out. The trials were recorded with a digital video camera operated by a member of the ACE lab. The rocket launcher was placed in front of a 40-foot length of twine, which had been flagged at 3 foot intervals and suspended from a fourth story balcony. 

Results and Analysis. Experimentation was conducted on Wednesday, December 4th from 3pm to 5pm in the courtyard of the William Vernon Skiles Building on the Georgia Institute of Technology campus. The temperature was 36° F  with a 30mph Easternly wind.
	We launched multiple trials but because of the weather conditions and the low resolution of the video camera (30 frames per second), we were unable to gather data from these trials.  The videos of these trials may be found on our project's website under The Experiment page (http://backwarde.resnet.gatech.edu/rocket/experiment.html). 
	We attempted re-launching in following days, but without a more advanced camera, like that which was on loan to the ACE lab earlier this year, or some sort of (at this point) unattainable sensor which would measure the time and angle of the rocket's flight, data collection in this manner is impossible. 
	Although we were unable to obtain data from our experimentation, we used the same initial conditions for our calculations, and then compared the results to data we found on various websites.

Revision. For future trials there are many factors that could be improved upon.  Primarily, we need to devise an apparatus for data collection as described in the previous section. We would also like to experiment with different nose cones to add mass.
	Mathematically, our model needs to consider turbulence (as we assume laminar flow for all time), and the friction between the water and nozzle. Plus, after all the water has left the bottle, there still remains some pressurized air, which leaves immediately but, consequently, adds some thrust which is unaccounted for in our model. This thrust is very difficult to represent because air is compressible and turbulent.  We would also like to develop a model for the third phase of the flight, which, as stated earlier is extremely complex due to its variable rotation. As we began formulation of our model with a revised equation, these additional issues are even more complicated. They are topics which may not be considered in a single semester’s time, but instead to be understood properly, should and must be studied over a course of an academic career.