Water rockets have been fascinating children and adults alike throughout the world for some time now, possibly because they are conceptually simple, requiring an ingenious combination of "exotic" components such as plastic soda pop bottles, water, a bicycle pump - and their behavior is totally unpredictable. A great website to visit is the Water Rockets page of the University of Leeds in England, which includes a number of fascinating video clips of various rocket launches and tests. Their video clip of the first documented manned water rocket launch made us realize the great potential of the water rocket as the possible transportation system of the future.
The cross section of a typical water rocket illustrating the principle of operation is shown in the following diagram:
Thus the compressed air in the bottle forces the water through a nozzle (bottle neck) which produces the thrust required to accelerate the bottle (hopefully) vertically upwards. We determine the time derivative of its vertical velocity by Newton's second law of motion:
where:
m is the instantaneous total mass of the rocket [kg]
u is the upwards velocity [m/s]
Fthrust is the thrust forsce (due to the expelled water) [N]
Fdrag is the drag force from the surrounding air [N]
g is the acceleration due to gravity [9.81 m/s2]
The thrust force is proportional to the exhaust mass flow through the nozzle times the velocity of the exhaust relative to the rocket.
where:
is the rate of mass flow of the expelled water [kg/s]
uex is the exhaust velocity of the expelled water through the nozzle [m/s]
is the density of water [1000 kg/m3]
AN is the area of the nozzle [m2]
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Bernoulli's equation can be derived from the energy equation applied to the water flowing through the nozzle. It relates the kinetic energy of the exhaust water to the compressed air pressure applied at the water surface. Neglecting potential energy terms, we have: |
Combining equations (2) and (3) above we obtain:
We now continue with Page 2 of the water rocket analysis, leading to the compressed air volume variation differential equation. Solving this equation will allow us to evaluate the rocket performance, leading ultimately to the altitude attained by the rocket.
