A rocket water or hydropneumatic rocket is a wheel gear which comprises a bottle in PET propelled by reaction , using the water and the air under pressure . These machines can exceed the speed of 400 km / h and 100 metersof altitude ^{1} . Their launch requires the use of a launch pad (or no shot) manufactured for this purpose. For its educational virtues and its spectacular side, the construction of water rockets is an activity often practiced in schools and holiday centers.
Principle
The propulsion of a water rocket is based on the wellknown principle of actionreaction : when a certain mass (the mass of water, here) is ejected violently from a container, a reaction force is created. in the opposite direction. It is by this same principle that rockets such as Ariane are propelled, except that instead of a body of water, these machines eject a mass of ignited gas.
In water rockets, the mass to be ejected is water, fluid perfectly neutral and devoid of energy. The means for ejecting this water is not a chemical reaction but the pressurization of the air in the container. This container is a simple PET bottle . For safety reasons, it is important that this bottle is PET, to withstand the strong internal pressure.
The higher the air pressure in the rocket, the faster the water will be ejected, and therefore the greater the reaction force, even if it lasts less.
In practice, the propulsion phase of a water rocket is usually very short (of the order of a tenth of a second for a standard rocket of 1.5 L). However, after this propulsive phase, the water rocket continues its rise thanks to the acquired kinetic energy , and this in spite of the gravity and the resistance of the air.
The protocol for launching a typical rocket is as follows:
 The rocket filled with 30% to 40% water is placed on its firing point .
 The air in the bottle is then pressurized (at 5,000 hPa in the school environment, ie 5 bars or 5 atmospheres in old units, and up to 8,000 hPa ), generally using a pneumatic pump. manual.
 Once the correct pressure is reached, the rocket is released and begins its flight.

Launch of water rocket H + 30ms.
Security measures
Water rockets are devices that consume a lot of energy and can cause serious injury if simple safety measures are not followed.
It is therefore recommended that no person be within 10 meters of the firing point during pressurization and takeoff of a rocket ^{2}^{ , }^{3} .
It is strongly recommended to test the strength of the PET bottle before mounting the rocket. To do this, fill the bottle to the brim and inflate to the expected pressure. Thus, in case of explosion, as there is very little compressed air, the explosion will be weaker.
You must use a manometer pump and a long hose.
The components of a rocket
A water rocket consists of 4 distinct parts:
 The warhead; the tip of our rocket consists of a cap or warhead (definition more military), whose utility is the improvement of aesthetics and especially aerodynamics. The warhead will preferably contain a charge to stabilize the trajectory of the rocket, under the basic principle of ballistics.
A sharp profile like that of a fighter plane or the Concorde, which seems to us, at first sight, the best is useless for our FHP (hydropneumatic rocket) whose speed remains largely below that of the sound. But this cap can also contain a device that will allow the rocket to return to the ground smoothly, a parachute for example.  The fuselage is the main element of the FHP because it will serve as a reservoir and chamber for the pressure to increase and expels the water through the nozzle.
 The nozzle is the neck of the bottle; it controls the flow of water during propulsion.
 Ailerons make it possible to control the trajectory of our FHP. They must be aerodynamic to offer low resistance to air. At our level it is almost impossible to set up a system of fins constantly driven by an electronic system because it would be heavy, bulky and complex.
Minimum fuse
A simple PET bottle is in itself a water rocket to highlight the existence of a very large propulsive force. However this simple bottle will be aerodynamically unstable: it will start to spin, so that it will not gain much altitude.
To make it stable , you have to add fins to the rocket. A 1.5liter rocket with fins can easily exceed 60 meters .
The tank
The tank of the water rocket is the internal volume of the bottle (s).
It is possible to increase the volume of the tank of the rocket by assembling between them several elements of bottles. This requires a collage prohibiting leaks and disembowelings under the strong launch pressure. The polyurethane glue , generally used for this kind of bonding, provides a good seal, good resistance to pressure, and the elasticity necessary to withstand the violence of some ground returns.
Some advanced water flares have multiple stages, each of which is triggered by various mechanisms ^{4} .
The fins
Ailerons have a lot of influence on the performance of the rocket. They allow a stabilization of the trajectory.
They must be well dimensioned, located below the center of gravity and firmly attached to the body of the rocket. Too big fins will cause a problem of “overstability” and fins too small will not stabilize the rocket enough.
The payload
The payload is the set of things that the rocket embarks and that do not serve directly to its operation. It can be a small camera or a camera, data capture instruments (acceleration, speed, altitude, …) or anything else.
The recovery system
When a water rocket is sufficiently advanced, the rocketist wants to recover without damage the rocket and / or the onboard equipment. For this we use a recovery system whose role is to slow down the speed of fall of the rocket. The lower the fall speed, the less the shock will be violent.
The most used system is the parachute although there are other rather innovative techniques. It is often made from a nylon fabric or a garbage bag. Strings connect the parachute to the rocket.
The tricky thing is to create a reliable parachute trigger. The parachute must be enclosed in the rocket during the ascension phase of the flight and deploy during the fall.
There are various popular methods of simplicity and variable reliability:
 the “sylapus” method, an acronym for “Ultra Simple Parachute LArgage System”, uses inertia when the rocket is turned upside down so that the cap separates from the body of the rocket, thus freeing the parachute. This method is simple to perform. It has the disadvantage of being effective only for vertical shots.
 with a mechanical timer “tomy timer” (clockwork mechanism used in small toys).
 with a shutter pressed against the rocket that will open when the rocket loses its speed (by pulling a rubber band), thus freeing the parachute.
 radio controlled system
 or using “glider” fins: they are tackled to the rocket during takeoff and make it hover during the descent
The theft of a water rocket
The stages of the flight of a singlestage water rocket are as follows:
 Propulsion using the launch tube (if there is one)
 Propulsion by ejection of water
 Propulsion by ejection of air
 Ascended ascent (deceleration due to gravity and drag )
 Achievement of the climax (maximum altitude, zero vertical speed)
 Fall to the ground (braked or not by a parachute)
Propulsion is only a very small part of Flight ^{5} .
Launch pad
The launch pad, or no firing, is essential for sending rockets. It is used to pressurize and put in the trajectory at takeoff. There are two main families of launches: those called “full mouth” and those with “garden hose connection”. Each of these systems has its share of advantages and disadvantages. The simplest design consists of a cork provided with a valve of air chamber .
Base “garden connection”
The most common method for making a step type “garden hose connection” shot is to set firmly on a stable one female connector pipe sprinkler, connector type quick release for easy pull , or better, by rotation . The male part is adapted to the neck of the bottle in a sealed manner . The easiest way is to pierce the plug to insert the reduced male connector by filing. A pipe of a few meters is attached to the female connector, and closed at its end by a bicycle valve that allows to connect the pump and maintain the pressure. The release of the rocket is done by pulling a string adapted to the connection.
A much more reliable and powerful launching base can be made with gutter bends (to support), a garden hose casing mechanism (to hold the bottle in place during pressurization), and a brake bike (to trigger the shot).
Fullmouth basis
On this type of launcher, the neck of the bottle is used directly as a nozzle . The thrust is then stronger, but also shorter than with the “garden fitting” systems, because of the larger diameter.
On this type of launching base, it is possible and often desirable to create a restraint system which holds the rocket on the firing point during pressurization and until the user actuates the release mechanism of the rocket. This allows to reach high pressures (higher than 4 or 5 bars) and gives a control of the moment of takeoff.
Launch tube
A launch tube is a tube added to the launch pad, which is located above the neck when the rocket is positioned. Thus the tube enters the rocket.
This system offers two main advantages:
 Guiding the rocket at the beginning of the trajectory.
 Propulsion of the rocket during takeoff avoiding to eject too much water.
The diameter and the length of the tube condition its effectiveness. The larger the dimensions, the more effective the tube will be.
The water rocket in equations
To completely describe the flight of the rocket it is necessary to resort to complex equations of fluid mechanics but here the goal is to get as close as possible to reality with a basic theoretical model, with physical laws in a form as simple as possible.
We define :
{\ displaystyle m_ {1} (t)} : mass of water in the rocket
{\ displaystyle m_ {0}} : mass of the empty rocket
{\ displaystyle v (t)} : speed of the rocket
{\ displaystyle v_ {e}} : speed of ejection of water (in what follows, admitted constant during the propulsion phase by ejection of water)
{\ displaystyle Q} : water flow that we consider constant to simplify
{\ displaystyle s} : small section of the rocket
{\ displaystyle S} : large section of the rocket
{\ displaystyle r} : radius of the small section of the rocket
{\ displaystyle R} : radius of the large section of the rocket
{\ displaystyle \ rho} : density of water
Conservation of the momentum:
At the time {\ displaystyle t} : {\ displaystyle p (t) = \ left (m_ {1} (t) + m_ {0} \ right) \ cdot v (t)}
At the time {\ displaystyle t + dt} : {\ displaystyle p (t + dt) = (m_ {0} + m_ {1} (t + dt)) \ cdot v (t + dt) + (m_ {1} (t + dt) m_ {1} (t)) \ cdot (v (t) v_ {e})}
{\ displaystyle dp = m_ {0} \ cdot dv + m_ {1} \ cdot dvdm_ {1} \ cdot v_ {e}}
Finally {\ displaystyle dp = (m_ {0} + m_ {1}) \ cdot dvdm_ {1} \ cdot v_ {e}}: (1)
Preservation of the material for an incompressible fluid
{\ displaystyle Q = v_ {e} \ cdot s}
{\ displaystyle m_ {1} (t) = m_ {1} (0) Q \ cdot \ rho \ cdot t}
{\ displaystyle m_ {1} (t) = m_ {1} (0) v_ {e} \ cdot s \ cdot \ rho \ cdot t} : (2)
and {\ displaystyle {\ frac {dm_ {1}} {dt}} = – v_ {e} \ cdot s \ cdot \ rho}
Flow calculation with Bernoulli’s theorem
{\ displaystyle P_ {a} + \ rho \ cdot \ left (g + {\ frac {dv} {dt}} \ right) \ cdot z (a) + {\ frac {1} {2}} \ cdot \ rho \ cdot v (a) ^ {2} = P_ {b} + \ rho \ cdot \ left (g + {\ frac {dv} {dt}} \ right) \ cdot z (b) + {\ frac {1} {2}} \ cdot \ rho \ cdot v (b) ^ {2}}
This law only applies in a laminar regime, which is not the case here, but we do not have many other choices.
Knowing that {\ displaystyle z (b) = 20cm} about, we can neglect the term in {\ displaystyle \ rho \ cdot \ left (g + {\ frac {dv} {dt}} \ right) \ cdot z (b)} in front of {\ displaystyle P} which is on the order of {\ displaystyle 2 \ cdot 10 ^ {5} Pa}. To be convinced we will take{\ textstyle g + {\ frac {dv} {dt}} = 100} : we obtain {\ displaystyle 100 \ times 1000 \ times 0.2 = 2 \ cdot 10 ^ {4} Pa}
So this term is negligible, but narrowly.
We measure {\ displaystyle R} = 5.03e2m to 3e4m and {\ displaystyle r}= 4.5e3m to 1e4m. As{\ displaystyle s \ times v (a) = S \ times v (b) = Q}, we deduce that {\ textstyle v (b) ^ {2}} is negligible in front of {\ textstyle v (a) ^ {2}}
He stays : {\ displaystyle P_ {b} P_ {a} = {\ frac {1} {2}} \ cdot \ rho \ cdot v (a) ^ {2}} equals {\ displaystyle PP_ {o} = {\ frac {1} {2}} \ cdot \ rho \ cdot v_ {e} ^ {2}} or {\ displaystyle P} and {\ displaystyle P_ {o}}are respectively the internal pressure and the pressure at the outlet. So{\ displaystyle v_ {e} = {\ sqrt {2 \ cdot {\ frac {PP_ {o}} {\ rho}}}}} : (3)
Fundamental principle of dynamics:
{\ displaystyle {\ frac {dp} {dt}} = (m_ {0} + m_ {1} (t)) \ cdot g}
When projecting on the Oz axis, pointing upwards:
{\ displaystyle (m_ {0} + m_ {1} (t)) \ cdot {\ frac {dv} {dt}} + {\ frac {dm_ {1}} {dt}} \ cdot v_ {e} = – (m_ {0} + m_ {1} (t)) \ cdot g}
{\ displaystyle (m_ {0} + m_ {1} (t)) \ cdot {\ frac {dv} {dt}} = – (m_ {0} + m_ {1} (t)) \ cdot g { \ frac {dm_ {1}} {dt}} \ cdot v_ {e}} with {\ textstyle {\ frac {dm_ {1}} {dt}} <0}
{\ displaystyle {\ frac {dv} {dt}} = – g – {\ frac {dm_ {1}} {dt}} \ cdot {\ frac {v_ {e}} {m_ {0} + m_ {1 } (t)}}}
We use (2) :{\ displaystyle m_ {1} (t) = m_ {1} (0) v_ {e} \ cdot s \ cdot \ rho \ cdot t} : (2)
and {\ displaystyle {\ frac {dm_ {1}} {dt}} = – v_ {e} \ cdot s \ cdot \ rho}
{\ displaystyle {\ frac {dv} {dt}} = – g + {\ frac {v_ {e} ^ {2} \ cdot \ rho} {m_ {0} + m_ {1} (0) – v_ {e} \ cdot s \ cdot \ rho \ cdot t}}}
If we take {\ displaystyle M_ {0} = m_ {0} + m_ {1} (0)} the initial mass of the filled rocket:
{\ displaystyle {\ frac {dv} {dt}} = {\ frac {v_ {e}} {{\ frac {M_ {0}} {v_ {e} \ cdot s \ cdot \ rho}} – t} }} g
We obtain v (t) by integrating:
{\ displaystyle v (t) = – v_ {e} \ cdot ln \ left (1 – {\ frac {\ rho \ cdot s \ cdot v_ {e} \ cdot t} {M_ {0}}} \ right) g \ cdot t}
The integration of v (t) gives, a priori:
{\ displaystyle z (t) = – v_ {e} \ cdot t \ cdot \ left (1ln \ left (\ left ({\ frac {M_ {0}} {\ rho \ cdot s \ cdot v_ {e }}} \ right) ^ {2} – {\ frac {M_ {0} \ cdot t} {\ rho \ cdot s \ cdot v_ {e}}} \ right) \ right) – {\ frac {1}} {2}} g \ cdot t ^ {2}}
These equations concern the phase of ejection of water and can be pushed further by taking into account the neglected elements but this requires a more advanced level of knowledge.
Depth study of the influence of pressure and initial volume of water
By studying the equations proposed above on the page, we use: {\ displaystyle v (t) = – v_ {e} \ cdot ln \ left (1 – {\ frac {\ rho \ cdot s \ cdot v_ {e} \ cdot t} {M_ {0}}} \ right) g \ cdot t}
From this, it is enough to study the speed of ejection of the water to deduce the speed of the rocket. By fixing the rocket, we can observe the speed of ejection and thus to study more finely what is the influence of initial pressure or initial water volume on takeoff.
The study was conducted, it appears that the speed of the rocket does not vary linearly with the pressure.
NB: this study only takes into account the ejection phase of the water and not the contribution of the air ejection at the end.
See also
Internal Links
 Water engine . When water is an additive, or a source of energy.
 Rocket with Sodium Bicarbonate or Rocket VBS, or Rocket Chemical
External links
On other Wikimedia projects:
 Water Rocket , on Wikimedia Commons
 French Forum of Rocket Fans [ archive ]
 very complete site intended especially for high school students [ archive ]
 Video of making a water rocket [ archive ]
 Videos tutorials on water rocket [ archive ]
 Blog dedicated to water rockets [ archive ]
 An article on the physics of the water rocket [ archive ]
 Tutorial to quickly build a basic but effective water rocket [ archive ]
Bibliography
 Ivan Lanoë, Build and Launch Water Rockets , Dunod , Paris, 2003 ( ISBN 2100069888 )
 Cedric J. Gums, A more thorough analysis of water rockets: moist adiabats, transient flows, and inertial forces in a soda bottle [ Archive ] , American Journal of Physics 78, 236 (2010).
Notes and references
 ↑ The altitude world record on June 14, 2007 is 625,1557 meters . See on WRN.com [ archive ] ( en )
 ↑ “Always stay more than 5 m or 10 m from a water rocket and behind the axis of launch or frankly on the side of the planned trajectory (at least 10m). “
 ↑ See also the Security folder on the site planetesciences.org [ archive ]
 ↑ http://fuzeao.free.fr/fus_example_bietage.htm [ archive ]
 ↑ To visualize quantitatively the flight of a water rocket, go to the Dean Wheeler simulator [ archive ] ( en )