Water rocket enthusiasts have created all sorts of amazing water rocket designs, including this one shown in the video below which is a two-stage water rocket.

Next I will get into some of the physics of water rockets. The analysis will be somewhat advanced, but it's a means to an end in which the end result will help you build and set up a water rocket that will reach a maximum height in the air.

The figure below shows a schematic for this analysis.

Where:

The nose cone (shown in the above figure) reduces air resistance as the rocket flies through the air. Also note that we are assuming a thin-walled rocket body in which the dimensions

When the rocket nozzle is opened the water starts to exit at high speed due to the pressure inside the rocket body forcing the water out. As the water exits it accelerates downward due to a large downward force (caused by the internal pressure). By Newton's third law there is also an equal and opposite force pushing upward on the rocket, causing it to accelerate upward. As the water exits, the water height

As the rocket speed increases it encounters air resistance (drag) which can cause the rocket to tumble end over end. To prevent this fins must be placed on the rocket (shown in the above figure) which cause the resultant drag force from air resistance to act at a point underneath

We will now develop the equations to model the flight of a water rocket.

The thrust

where

We can reasonably assume that, when the water is exiting, the volume of air inside the rocket body expands quickly enough so that it has no time to experience either heat gain from the external environment or heat loss to the external environment. In thermodynamics this is called an adiabatic expansion (or an isentropic process), and it can be represented mathematically by the following equation:

Where:

From Bernoulli's equation for constant density fluid flow,

Where:

Note that

This equation assumes the following:

• The flow velocity of the water at the air-water interface inside the rocket body is negligible compared to the outlet flow velocity

• The gravitational change in energy of the water as it changes elevation while flowing from inside the rocket body to outside has negligible contribution.

• Quasi-steady flow conditions for the water, with negligible fluid friction.

Now,

From equation (1)

Substitute this equation into equation (3) and we get

Solve for

Substitute equation (2) into equation (3). This gives us

We now seek an equation in which

where

Substitute this equation into equation (8). We get

Substitute for

From equation (5),

We can substitute for

An interesting and practical problem is to determine the maximum height reached by a water rocket, taking into account air resistance. By Newton's second law the (one-dimensional) force balance for the water rocket is

where

Where:

Therefore the force balance for the water rocket becomes

Substitute equation (5) into equation (11) and we can solve for

Equation (11) is only valid up until the point the peak height is reached. This is because (in this equation) the drag term does not reverse sign to oppose motion when the rocket velocity

A nice optimization problem is to determine how much water to put in the rocket body to obtain the maximum height, given an initial pressure.

The equations derived here have been incorporated into an Excel spreadsheet which you can easily use to help you design a water rocket that reaches the maximum height possible. To download the Excel spreadsheet right-click on this link. This Excel file is in compressed "zip" format and you have to uncompress it before you can use it.

Filling the rocket with pressurized air only will not result in nearly as great a height reached as you would by filling the rocket with pressurized air

Ideally the rocket is pressurized so that, as the last of the water exits, the pressure inside the rocket is at atmospheric pressure. This represents the most efficient design, even though it won't necessarily result in the greatest height reached for a given initial pressure. If the air pressure inside the rocket is greater than atmospheric pressure, as the last of the water exits, then the air will of course shoot out the bottom and some additional thrust (and speed) will be given to the rocket. However, this is not accounted for here given that its contribution to peak height reached is almost negligible, for reasons given in the previous paragraph.

Completely filling the water rocket with water will prevent it from being pressurized since water is incompressible for the typical pressures used in water rockets.

If the mass of the rocket body (water excluded) is too small, the drag force will be excessively high relative to the gravity force, during the part of the flight stage where there is no thrust. As a result, the height reached by the rocket will actually be less than for a heavier rocket body. As an analogy, if you launch a Styrofoam ball and a metal ball of the same size straight up into the air, both with the same initial kinetic energy, the metal ball will reach a greater height even though the Styrofoam ball has to be launched with much greater speed in order to have the same kinetic energy as the metal ball (due to its much lower mass).

The typical water thrust stage for a bottle size rocket is very short, typically lasting a tenth of a second or so. This is how long it takes for all the water to exit the rocket. During this time period the rocket accelerates very quickly.

A long and narrow rocket body (long

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