Les faits

"Ce sont des hypothèses honnêtes qui rendent compte des faits: mais je sens bien qu'elles viennent de moi, qu'elles sont tout simplement une manière d'unifier mes connaissances (...) Lents, paresseux, maussades, les faits s'accommodent à la rigueur de l'ordre que je veux leur donner; mais il leur reste extérieur." (Jean-Paul Sartre, La Nausée)

The language of Nature

Every practitioner of physics knows that the laws of physics, in their mathematical form, do not enable to make predictions about the real world. Quite often, solving the equations of mathematical physics provides two solutions, of which one is rejected on the basis of non-mathematical arguments.

A first example is that of the retarded and advanced potentials of classical electromagnetism. Only the former are considered, the latter being in contradiction with causality. A second example is the common situation in which one finds two possible functions: one that diverges at infinity and another one that converges to zero. Only the latter is kept, the former being termed “unphysical”.

What is interesting in the two examples is that mathematics predicts one thing as well as the opposite: causality and anti-causality, divergence and convergence. We choose the solution on the basis of non-mathematical arguments.

Mathematics may be the language of Nature, but we take some liberty in its interpretation.

Soda-bottle water rockets

To build a water rocket [e.g. 1, 2], you need:
(i) a plastic soda-like bottle, (ii) a rubber tap with a plastic flexible pipe through it, and (iii) a bike pump.

Fill half the bottle with water, close it with the tap, turn it upside down, and pump air through the pipe until the tap plops off . The water is pushed out of the bottle by the air pressure, and there you go!

Below are some pictures of a few launches with a 2 l mineral water bottle. The pressure in the bottle is about 4 atmospheres when the tap plops (my bike pump has a manometer). The camera we had did not enable us to take more than 5 pictures per second. In each case, the water expulsion (i.e. the burnout) is shorter one fifth of a second!

On the first picture I am hiding behind the cabin and pumping. Less than 0.2 seconds later, the bottle has expelled all its water and it is 5 meters high. With some rough kinematics (velocity = acceleration*time, distance = 1/2 acceleration*time squared), one finds that the average acceleration of the bottle is more than 25 g’s, and its velocity after 0.2 s is more than 50 m/s!

The inset shows the nicely even bouncing of the water jet on the ground at the rocket’s takeoff. The cloud in the second picture (indicated by an arrow) is formed when the bottle empties entirely from all its water. From the second and third pictures, it is interesting to note that, although the water is being expelled downwards, the water of the cloud is moving upwards. The reason for this is that the water is expelled downwards relative to the bottle. As the bottle itself is climbing faster than the water is descending compared to it, the absolute velocity of the water is still upwards.

Incidentally, this is what multistage rocket systems aim at avoiding [3]. The first stage uses low exhaust velocity fuels, and high exhaust velocity fuels are used in the upper stages. Under these conditions, the exhaust velocity increases when the rocket gains speed: the fuel is always expelled backwards, which makes the propulsion very efficient.

It would be interesting to check whether the same can be obtained with a water rocket by optimizing the shape of the bottle.

Here is another launch that was pretty successful, as for the height reached by the bottle.

There is something disappointing with all these launches though: the bottle rotates. This is clear from the second picture of the first launch, and from the third picture of the second. The rotation clearly increases the aerodynamic drag on the rocket, which reduces its maximum height.

An additional issue related to the rotation of the bottle is that it is not possible to put a parachute on the rocket (which prospect made my daughters hectic). The parachute opens as soon as the bottle starts rotating, i.e. immediately after the burnout, and the launch is a fiasco.

Perhaps an appropriate shape of the bottle can make the rocket more stable, and prevent it from rotating.

How about side wings?
Surely my daughters will love it!

[1] D. Kagan, L. Buchholtz, L. Klein, “Soda-bottle water rockets”, The Physics Teacher, 33 (1995) 150-157.
[2] http://en.wikipedia.org/wiki/Water_rocket.

[3] R.H. Gowdy, “The physics of perfect rockets”, American Journal of Physics, 63 (1995) 229-232.