How a Water Rocket Flies
After reading this you will know why a soda-bottle rocket flies, how to compute its thrust and apogee from the water fill and the pump pressure, and why about a third of a bottle of water beats a full one.
What the simulator does and one launch to anchor it
A water rocket is a plastic bottle holding some water and a lot of compressed air. You seal the neck with a nozzle, pump air in through a valve, and release. The air pushes the water out of the nozzle in a short, hard jet. That jet carries momentum down, so the bottle is pushed up. When the water runs out, a brief blast of air follows, and then the rocket coasts upward, slowing under gravity and air drag until it stops at apogee and falls back.
The Bottle Rocket Simulator runs this whole flight in your browser. You choose the bottle size, the water fill, the pump pressure, the nozzle, the nose cone, the fins and the empty mass, and you can tilt the launcher or fit a parachute. It integrates the equations of motion and draws the thrust curve, the altitude and speed over time, and the apogee for every possible fill. One launch to keep in mind is the simulator's classroom preset: a 2-litre bottle one third full of water (0.66 litre), pumped to 60 psi, the plain 21.5 mm bottle neck as the nozzle, a rounded nose cone, three small fins and an empty mass of 150 g. It climbs to 46 m. The sections below show where that number comes from.
When this model is the right tool
Use the simulator to plan a school or club launch, to compare two fills before you waste a pump, or to see how fins and nose shape change apogee. The physics is the same textbook model used to plan school competitions, so the trends it shows (more pressure helps, too much water hurts) are the trends you will see on the field.
Do not treat the exact altitude as a measurement. The model leaves out the water-hammer spike at the very start, the effect of the rocket's acceleration on the water column, and launch-tube effects, so real rockets of the same design can fly 10 to 20% differently. If you need orbital mechanics or multiple stages, that is a different problem; see the Rocket Designer & Staging Lab.
The pressure setting is gauge pressure, the pressure above the surrounding air, which is what a pump gauge shows. The classroom 60 psi is 4.14 bar above the air, or about 5.15 bar absolute. Thrust depends on the difference, so gauge pressure is the number that matters.
The physics of the water phase
While water leaves, treat the air above it as an ideal gas expanding adiabatically. No heat crosses the wall in the fraction of a second the burn lasts, so pressure and volume follow:
Here p is the absolute air pressure, V is the air volume, and the exponent 1.4 is the ratio of specific heats for air. As water leaves, V grows and p falls.
The water leaves the nozzle at a speed set by the pressure difference across it. Treat the water as incompressible and apply Bernoulli between the inside and the outside:
where \Delta p is the gauge pressure (inside minus outside) and \rho = 1000 kg/m³ is the density of water. The thrust is the momentum carried away each second. Because a pressurized jet gives twice the naive momentum flux, the thrust works out to:
with A the nozzle area. A real nozzle passes a little less water than the ideal formula, so the simulator multiplies the flow and the thrust by a discharge coefficient of 0.97. A bigger nozzle or a higher pressure both raise the thrust, but a bigger nozzle also empties the bottle faster, so the burn is shorter.
A worked example: the classroom rocket
2 L bottle, one third water, 60 psi
Take the classroom preset: a 2 L bottle 10.4 cm across, 0.66 L of water, 60 psi, the 21.5 mm bottle neck as the nozzle and 150 g of empty mass. Work in SI units.
- Nozzle area: A = \pi (0.01075)^2 = 3.63 \times 10^{-4} m².
- Gauge pressure: \Delta p = 60 \times 6895 = 4.14 \times 10^{5} Pa.
- Initial thrust: F = 2 \times 0.97 \times 3.63 \times 10^{-4} \times 4.14 \times 10^{5} = 291 N. That is the peak, before the pressure drops.
- Initial exhaust speed: v = \sqrt{2 \times 4.14 \times 10^{5} / 1000} = 28.8 m/s.
- Initial water flow: \dot m = 0.97 \times 1000 \times 3.63 \times 10^{-4} \times 28.8 = 10.1 kg/s.
- Launch mass: 0.150 kg empty, 0.660 kg of water and 0.008 kg of compressed air make 0.818 kg. Its weight is 0.818 \times 9.81 = 8.02 N, so the thrust starts at 36 times the weight.
At 10.1 kg/s the water would be gone in 0.065 s; because the flow slows as the pressure falls, it actually lasts 0.080 s. The air grows from 1.34 L to the full 2 L, so at water-out its pressure is 5.15 \times (1.34/2.00)^{1.4} = 2.94 bar absolute. That leaves 1.93 bar (28 psi) above the outside air for the air blast, which lasts another 0.03 s. The whole push is over after 0.11 s, and the rocket is moving at 42 m/s. Without drag it would coast to 42.3^2 / (2 \times 9.81) = 91 m. Drag on a 10.4 cm bottle at that speed is strong and halves that: the simulator's apogee is 46 m, reached 2.8 s after launch.
Reading the thrust and apogee charts
The thrust curve is a spike that decays. It starts at the 291 N you computed and falls as the air expands. When the water runs out at 0.080 s, the thrust drops to the air blast, which fades to nothing by 0.11 s. All of the speed is gained in that first tenth of a second; the remaining 2.7 s of the climb are coasting.
The apogee-versus-fill chart is the one worth studying. It plots the peak height for every water fill from empty to nearly full, with everything else held fixed. It rises from 12.7 m with air alone, peaks at 46.1 m with 30% water, and falls again as the bottle fills. The explanation is a trade: water is the reaction mass, so too little water means too little to throw, while too much water leaves too little air to throw it. Above about 69% the air is spent before the water is out, and the leftover water rides along as dead weight.
Explore the fill trade-off
Common mistakes
The most frequent error is overfilling. A full bottle feels like it should fly farthest because it holds the most water, but with almost no air there is nothing to expel the water, and apogee collapses. Fill to roughly a third.
The second is expecting the nozzle to change the height much. With the same water and air, the total push hardly changes: the classroom rocket reaches 45.9 m with the 21.5 mm bottle neck, 45.9 m with a 15 mm reducer and 45.6 m with a 9 mm garden-hose fitting. What changes is how the push is delivered. The bottle neck gives a 291 N kick that is over in 0.11 s, and the rocket feels up to 85 g; the 9 mm fitting pushes with at most 51 N for 0.64 s, and the peak load drops to 14 g. A narrow nozzle is kinder to a payload, not a way to fly higher.
Do not read the printed apogee as a prediction good to the metre. The model omits water-hammer, column acceleration and launch-tube boost, so expect the real flight to differ by 10 to 20%. Use it to compare designs, not to certify a record.
A third mistake is leaving off the fins. Without fins the center of pressure moves ahead of the center of mass and the rocket tumbles, turning sideways to the air and thrusting in the wrong direction. A nose cone and three or four fins keep it pointed up. In the simulator, taking the fins off the classroom rocket cuts its apogee from 46 m to 26 m.
Related tools
If you want the drag side of the problem in detail, draw the bottle profile into the Wind Tunnel (CFD Sandbox) and read its drag, or shape a fin section in the Airfoil Designer & Analyzer. For the much harder problem of reaching space with staging and real propellant, use the Rocket Designer & Staging Lab, and for planning a transfer between planets see the Orbital Mission Planner.
Frequently asked questions
How much water should I put in a 2-litre bottle?
About 0.6 litre, 30% of the bottle. For the classroom rocket at 60 psi that is the simulator's best fill (46.1 m), and anything from about 0.5 to 0.7 litre stays within a metre of it. Less water leaves little to throw; more water starves the air.
Why is the thrust twice the pressure times the nozzle area?
Momentum flux through the nozzle is \dot m v = \rho A v^2. With v^2 = 2\Delta p / \rho this becomes 2 A \Delta p. The factor of two comes from the pressurized jet, not from any pressure term added by hand.
Does higher pressure always fly higher?
In the model, yes: the classroom rocket reaches 32 m at 40 psi, 46 m at 60 psi and 62 m at 90 psi. In practice the bottle sets the ceiling. PET bottles burst at around 150 psi, most school rules stop at 90 psi, and the simulator warns above that.
What does the empty mass change?
There is a sweet spot. A heavier rocket gains less speed from the same push, but it carries its momentum through the air better, because drag slows a light rocket more. The classroom rocket reaches 41 m at 60 g, 46 m at 100 to 150 g, 38 m at 250 g and 26 m at 400 g.
Why does a tilted launch go sideways but not much higher?
Tilting trades height for distance. In a vacuum 45 degrees would give the longest flight, but drag pulls the best angle lower: the simulator's distance preset lands farthest at about 40 degrees (79 m), a little short of that at 45 degrees (78 m). Straight up gives the most altitude.
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