# A more thorough analysis of water rockets: Moist adiabats, transient flows, and inertial forces in a soda bottle

Abstract: The water rocket 1 is a popular toy that is often used in first year physics courses to illustrate Newton’s laws of motion and rocket propulsion. In its simplest version, a water rocket is made of a soda bottle, a bicycle pump, a rubber stopper, and some piping see Fig. 1. The bottle is half-filled with water, turned upside-down, and air is pushed inside the bottle via a flexible pipe that runs through the stopper. When the pressure builds up, the stopper eventually pops out of the neck. The water is then ejected and the rocket takes off. Witnesses of the launch of a water rocket cannot but be amazed that such a simple device can reach a height of tens of meters in a fraction of a second. The popularity of water rockets extends beyond physics classrooms, with many existing associations and competitions organized worldwide. 1 The more than 5000 videos posted on YouTube with the words “water rocket” in their title testify to their popularity. Some of these videos involve elaborate technical developments such as multistage water rockets, nozzles that adapt to the pressure, the replacement of water by foam or flour, underwater rocket launches, and even a water-propelled human flight. The public’s passionate explorations with water rockets contrast with the small number of articles devoted to their analysis. I found only two papers 2,3 that treat the simplest possible rocket, similar to

## Summary (2 min read)

### A more thorough analysis of water rockets: Moist adiabats, transient flows, and inertial forces in a soda bottle

- Accurate measurements show that they outperform the usual textbook analysis at the beginning of the thrust phase.
- The bottle is half-filled with water, turned upside-down, and air is pushed inside the bottle via a flexible pipe that runs through the stopper.
- In Sec. III numerical solutions are compared with published experimental data.2.

### A. Moist air expansion

- Air expansion is the only source of energy of the rocket.
- Because this distance is much smaller than the radius of the rocket, the gas expansion has to be modeled as an adiabatic process.
- The pressure-volume relation is derived, assuming that the total entropy resulting from dry air, water vapor, and condensed water the fog is constant during the adiabatic expansion.
- The total number of water molecules NL in the liquid is the difference between the initial and the current values of NV.

### B. Water ejection

- The starting point of the analysis is to assess the importance of viscous forces.
- Water can therefore be assumed to be inviscid in this context, and ejection can be analyzed with Euler’s equation.
- The ratio in the square brackets in Eq. 10 ensures that the total flow of water is the same over any section of the rocket.
- The notation highlights that this term accounts for the inertia of the accelerated water.

### C. Rocket acceleration

- For a conventional rocket, MU is generally estimated by assuming that the fuel moves upward at the same speed as the rocket.
- This term accounts for the fact that the velocity of water in the rocket is smaller than the velocity of the rocket itself.
- The three forces on the right-hand side of Eq. 14 are estimated in the usual way.
- For the case of a uniform velocity profile, the first contribution is the product of the rate of mass loss with the exhaust velocity; the second term accounts for the fact that the thrust is not estimated in the reference frame of the rocket.
- The last two forces exerted on the rocket are the weight and the aerodynamic drag.

### A. Numerical solutions and experimental data

- These authors used a 2-l soda bottle as a rocket, the takeoff of which they measured with a high-speed camera and a “smart-pulley” system.10.
- The authors have accounted for several effects which were previously not considered and which may have an important role during the burnout.
- From dimensional analysis, the acceleration is expected to be as significant as the pressure difference between the inside and the outside of the rocket.
- For both pressures, curve 1 is slightly above the analysis of Ref. 2, but the theoretical prediction remains below experiment.

### B. Analysis of simplified equations

- A real rocket still outperforms the theoretical prediction, also known as The improvements are small.
- And thus the authors shall here assume g0=0 and CA=0.
- The time derivatives dh /dt and dZ /dt are plotted in Figs.
- For large values of / 0, the two numbers are very close to each other.
- In general, applying Bernoulli’s equation leads to an underestimation of the initial ejection velocity.

### IV. CONCLUSIONS AND PROSPECTS

- This paper was motivated by the observation2 that water rockets outperform the usual textbook analysis at the beginning of the thrust phase.
- The present analysis does answer some questions raised in Ref. 2.
- The authors have shown that this energy can be accounted for using a polytropic exponent smaller than =1.4.
- The movement of water inside the rocket contributes significantly to the total momentum of the rocket, an effect that is more pronounced for a larger hole.
- Maximum energy efficiency would require that water be ejected with a constant velocity with respect to the ground.

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