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Water Rocket Physics
Flight profile

This section explains what happens during the flight of a water rocket. The diagram below shows a graphical representation of a typical water rocket flight profile. From launch to landing (in this case: crashdown).

Flight profile

Flight phases

Thrust Phase

The pink line is thrust. The water is pushed out of the rocket by the expansion of the compressed air inside it. As the air expands, the pressure drops, and thus the thrust wears off. When all the water has been expelled, the remaining air bursts from the rocket at close to the speed of sound, and the rocket is depressurerized in milliseconds. After this, there is no more thrust for the rest of the flight.
As thrust is produced, the rocket accelerates upwards (blue line) due to Newton's 3rd law of motion. Even though the thrust decreases, acceleration actually increases. This is because the rocket looses weight as the water is lost, and according to Newton's 2nd law, acceleration equals force divided by mass.
This acceleration obviously causes the rocket to pick up speed (green line). Water rockets can reach very high speeds very fast. In this example, about 45 m/s (162 Km/h) in 0.7 seconds - and this is a pretty slow rocket compared to so many others!
As the speed increases, the rocket climbs off the launchpad and gains altitude into the air (red line).

Burnout

"Burnout" is a term from pyrotechnic rockets that actually burns their fuel. In water rockets, the term is used for the point in time where the rocket is emptied, and thrust goes to zero. In the diagram, this happens at 0.7 seconds.
At this point the rocket is at it's fastest upwards speed, and biggest upwards acceleration, but it has only climbed about 10 meters. As the burnout happens, there is a discontinuity in the acceleration (blue line, dashed). The acceleration does not go to zero, but becomes negative. This negative acceleration is the combined effect of gravity(-1g; -9.8m/s2; 32ft/s2) and drag caused by air-resistance on the rocket. Therefore, the rocket begins to slow down (green line).

Upwards Coast Phase

In this phase, there is only two forces acting on the rocket: drag and gravity. Gravity is constant and acts downwards from the center of mass of the rocket. Drag is proportional to the square of the airpeed of the rocket, and acts towards the rear from the aerodynamic center of the rocket. See the stability section for in-depth explanations of these terms.
Apart from the square of the airspeed, drag is also proportional to the size of the rocket - namely its cross-sectional area - and its coefficient of drag (Cd), which is a measure of the aerodynamic efficiency of the rocket. The better the streamlining, the smaller the Cd, and the less speed it will loose.

Apogee

As the rocket's vertical velocity (green line) goes to zero, the rocket reaches its highest point - the apogee. Now, if the rocket was going perfectly vertical, it would come to a stop before falling down again. But no rocket i have ever seen has been going vertically - it is almost impossible that some small gust of wind, assymmetry or other factor has not caused a slight veer in the rocket's flightpath. Therefore, the rocket will still have some forward speed (dark green line).
At this point, the rocket's speed is at its minimum, and and acceleration crosses the 1g-mark because the drag now begins to slow the rocket's descent, not its ascent.

Downwards Coast Phase

This is very similar to the upwards coast phase: The only forces are drag and gravity. But now, drag acts upwards, and not downwards, because the rocket is travelling downwards. As the rocket picks up speed in its fall towards the ground, the drag-force increases, and starts offsetting the force of gravity. (blue line goes towards zero, green line curves). If allowed to fall for enough time, these forces would equalize and the rocket would reach its terminal velocity.
In this case, it does not reach terminal velocity, because it touches down before it has the chance.

Touchdown

The downwards coast phase ends when the rocket reaches the ground. In the rocket used in the diagram, there is nothing to slow it, so it crashes into the ground at about 34m/s (122Km/h) *AUCH*
Usually some contraption - mainly a parachute - is used to slow down the rocket, so it does not destroy itself and anything it hits during touchdown.

Aerodynamics

Drag

Drag is a force that counters the movement of the rocket through the air. The drag on a rocket can be described by the following equation:

Drag equation

Fd
Is the drag-force: This works to slow down the rocket
Cd
Is the coefficient of drag for the rocket Depends mainly upon the shape of the rocket - usually it is between 0.1 and 0.3 for water rockets.
pair
Is the density of the air. For dry air this is between 1.2 and 1.3 Kg/m^3.
A
Is the cross-sectional area of the rocket. A skinnier rocket has lower drag than a fat one...
vair
Is the rocket,s forward velocity through the surrounding air. The rocket's true air speed. This does not matter if the rocket is going up or down. It is just the speed as felt by the rocket itself. It is important to note that vair is squared; a fast rocket looses its speed four times as fast as a rocket, that is half as fast.

Stability

Aerodynamic stability is needed to keep a rocket pointing in the direction of its flight.
When a rocket is flying there is only 3 kinds of forces acting upon it: thrust, gravity and aerodynamic forces. Thrust always acts axially along the length of the rocket, so it can never change the direction of the rocket by itself.
Gravity always acts downwards, and the rocket is drawn towards the ground by gravity. Gravity acts upon the rocket in its center-of-mass - its "balancing point". This point is often referred to as the center-of-gravity (CG).
If the rocket was a totally homogeneous solid (had the same density throughout), aerodynamic forces would also act in the cg. But our rockets are not homogeneous. This infers that there is another point where the aerodynamics on the rocket: the center-of-aerodynamics. When the rocket is stationary that point is where the combined air-pressure from the surrounding air would converge. This is why the point is usually referred to as the center-of-pressure (CP)
The most prominent aerodynamic force on a flying rocket is drag. Drag acts in the CP, but the rocket will rotate around its CG, so if these are not perfectly aligned, the rocket will rotate untill they are. This may lead to the rocket flying through the air on its side, and this causes a huge amount of drag that slows the fast rocket down. To make sure that this rotation will be to our benefit, we must ensure that the rocket is pointing directly in the direction it is travelling.
This can be done by moving the CG forwards with ballast in the nose - or we can move the CP back by adding fins to the rear.
The distance between the CG in front, and the CP in the back, is called the stability margin. The stability margin determines how strong a force is created to bring the rocket back on course when it has veered from it. As long as there is a margin, there will be a correcting force, but you don't want it to be too little, as the rocket might not be able to correct large deviations. You do not want it to be larger than neccesary. Large fins causes more drag, and even a short gust of side-wind will make the rocket "weathercock" and head into the wind.
A rule of thumb is to have a magin og between one and two rocket-diameters.

Adding things up

So: there are three forces that acts upon the rocket: Thrust, gravity and aerodynamics.
These are what makes it fly the way it flies. I have made some small movies that shows what happens from the rocket's point of view. They are based on the same rocket as the one used in the diagrams on this page

External forces

The movies require some explanation. The semi-transparant rocket takes-off from a checkerboard ground, and the camera follows the rocket as it follows its flightpath, while staying level with the ground at all times.

The red arrow represents gravity
The pink is thrust
The green is drag

The arrows are not fully to scale: Thrust should be more than 10 times the size of the gravity and drag arrows. But it shows the basics:
The huge spike of thrust at takeoff. Drag force peaking at burnout and with minimum at apogee. Gravity constantly pulling towards the ground. Drag increasing in the downwards phase, and almost off-setting gravity just before landing.

Internal forces

This next movie shows what forces affects an object inside the frame of the rocket itself.
Because gravity pulls the same in both the rocket and the object inside it, there is not apparant effect of mass. Therefore, there is only thrust and drag left. The arrow inside the rocket in the movie is the result of thrust and drag added together, and this time, it is to scale.
As you can see, there is a huge shift in the force at launch and at burnout. Then there is a sloooow dip untill apogee, and a slow increase untill crashdown.
These two movies also highlights the common misconception that you can use a gravity-based apogee-detector for deploying a parachute. In the first movie, it looks like the red arrow swings around, when compared to the rocket, and that this would be the perfect indication that the rocket has reached apogee.
In reality, the gravity acts the same on the rocket as on the object that should sense it - nullifying its usefulness. The point at which such a device would trigger, is at burnout - when forces shift from acting down, to acting up inside the rocket

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