Theoretical Computer Science 
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I think I was around 12 when I read about the famous thought experiment by Galileo that disproved the old notion of Aristotle that heavier bodies fall faster than lighter ones.
Suppose we have two stones, the first being lighter than the second. Release the two stones from a height to fall to Earth. Stone 2, being heavier than stone 1, falls more rapidly. If they are joined together, argues Galileo, then the combined object should fall at a speed somewhere between that of the light stone and that of the heavy stone since the light stone by falling more slowly will retard the speed of the heavier. But if we think of the two stones tied together as a single object, then Aristotle says it falls more rapidly than the heavy stone. How do the stones know if they are one object or two?
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I remember being stunned by the simplicity and elegance of the argument. No tower of Pisa needed, no pendulum, no inclined plane, nothing! Just a clever way of arranging thoughts.
In any case, I was telling my dad about this argument a couple of weeks ago. He asked me why then do pieces of paper fall to ground more slowly than rocks. It took me a few minutes to come up with a satisfactory answer.
Two questions then for you dear readers.
Where exactly does Galileo’s reasoning break down in the presence of air resistance?
What are your favorite examples of arguments in theoretical CS or math which are too clever for their own good?
For anyone actually wondering where the argument breaks down, it is in the false assumption that weight and weight alone determines the rate at which an object falls.
To expound on this a bit more:
Galileo’s argument asks how the stones know, when they are one object. However, that is not even the problem.
For clarity think about two extremes: a stone and a parachute tied together will clearly behave as predicted by Galileo. Yet, gluing a short stone cylinder (light object) onto the end of a long stone cylinder with the same radius will likely result in a body that orients in the same way when falling (“tip” down) but whose weight is increased while its drag stays roughly the same (or decreases according to Wikipedia), thus accelerating faster and having higher terminal velocity, being in line with Aristotle’s prediction.
In both cases the objects “know” that they are joined. Yet, they behave differently.
So Galileo’s theory breaks down for example when the air resistance of the joined object does not change while it gets heavier.
A number of points. I put these together not as rigorously as I should; so there may be fallacies.
(1) Neither make the distinction between acceleration and speed. Indeed, the argument above does not prove much (it posits a dilemma and then prematurely jumps onto one possible resolution), until Galileo followed up with an experiment of measuring acceleration (and total time-of-flight) of a ball rolling down an inclined track.
Therefore, I think modern thinkers put too much importance into his thought experiment. It might have inspired him to conduct experiments physically; but without the latter, the former was at best a dilemma.
(2) Neither mentioned the role of tension, the manner in which the speeds of two objects can be affected by being joined or tied together with a string.
When the string is loose, the two objects could be traveling on diverging speeds and trajectory, until the point the string is in tension (stretched). At that point, the two objects start exchanging force (via action and reaction), which then affects momentums, which then affects the speeds.
When two objects are tightly tied together to begin with, they are not allowed to travel with any divergence in position or speed (disregarding rotation for now). Neither argument describes how force would be exchanged in this case.
However, Galileo’s argument drew from everyday experience that the faster object would have to slow down, and the slower object would have to speed up, in order for their speeds and trajectory to converge. (Consider a herder pulling a horse.) Thus, Galileo pointed out that Aristotle’s prediction would have contradicted everyday experience.
When this argument was made, Galileo might not have been aware of that the speed of two falling stones are actually continuously changing – in acceleration – until the inclined plane experiment was conducted. Does the intuition from everyday experience of tension (a herder pulling a horse) still apply when both objects are accelerating?
(3) How does air resistance fit into the picture. It is an example of configuration (the shape and positioning of objects) affecting the force and therefore the accelerations experienced by the objects.
Since the effects of configuration (beyond the tying of strings) were never discussed, no falsifiable theory about configuration effects were put forward in these arguments, and these arguments would not have been able to account for it. This discrepancy could only be found by quantitatively comparing predictions with experimental results.
(4) Could Aristotle have been right? Well, in a simulated environment (like a computer game of FPS (first-person shooting)), it is possible to simulate Aristotle’s version of falling speed (non-realistically and in violation of conservations of momentum and energy). This is why experimentation is necessary to find out the laws of the nature.
1) Galileo’s reasoning breaks down due to the need to conserve momentum, also known as the conservation of momentum principle. If you look at any of the formulas for air resistance (there are many approximations), they all indicate that air resistance is proportional to velocity and area of the falling object. Intuitively, using conservation of momentum principle, a heavier object is able to keep more of its velocity during collisions with small air particles. Wide objects (like paper) collide with more air particles. If Galileo had better instruments to measure he would have see the heavier object hit slightly sooner than the lighter one. In a vaccume they would hit at the same time. 2)
More specifically, the force due to air resistance (drag force) experienced by an object is dependent on the shape of an object, its velocity relative to the air, and some other properties of the air, but NOT the mass of the object. That is, a higher-massed object would experience the same drag force as an identically shaped lighter object traveling at the same speed. However, since F=ma (that is, a=F/m), the “slowing” effects on acceleration would be much greater for the lighter object.
@Joe Just to be clear, I’m not correcting you, just adding more info in case of curiosity 🙂
Incredible argument. I’ve just recently had some interesting conversation about Galileo, the Church, and all that history. I also recommend Paul Feyerabend’s “Against method” for an interesting, if controversial, perspective on Galileo’s story.
Favorite arguments? The argument that there are infinitely many prime numbers is of the same kind. Assume there are finitely many of them, multiply all together, and add 1. You are supposed to get a prime number, but you had already listed all of them. We could also recall Cantor’s diagonal argument. For example, there are uncountable many numbers between 0 and 1. Why? Well, assume we could enumerate all of them. Then take the i-th number in this enumeration, look at its i-th digit after the decimal point, and pick some other digit. Proceeding in this way, we would arrive at some other number, which had not been enumerated. A contradiction.
Notice that in these cases, as in the Galileo’s case, we are using reduction ad absurdum (or tertium non datur). Perhaps this is what you found so interesting. Then perhaps read about intuitionistic logic.
Now that I think about it, we could simplify Galileo’s argument even further. Let’s just take two identical stones and through them side by side. Then they are supposed to be falling at an identical speed, because everything about them is identical, regardless of whether you accept Aristotle’s theory or not. Then join the two stones by a little “bridge”. You get a heavier stone that is still supposed to be falling at the same speed…
The simplest and most beautiful proof techniques use the same idea as Galileo used. The same reality can be observed from different points of view. Double counting (http://en.wikipedia.org/wiki/Double_counting_%28proof_technique%29) is an instance of this idea, many proofs from algebra use it too.
Galileo was demonstrating the inconsistency of Aristotle’s argument, he certainly didn’t agree with it.
In a vacuum, all objects fall at the same rate regardless of weight. If only Galileo could have seen this: http://www.youtube.com/watch?v=XtvESpQipcw&t=70
Personally I find Zeno’s paradox interesting, and Godel’s incompleteness theorem hurts my brain!