One of the more entertaining and yet illustrative examples of the principle:
(credit:
http://www.physicsforums.com/showthread.php?p=2063418)
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This is the story of Atom Man, a miniature sized superhero from the planet Zogg. Planet Zogg is much like Earth, the laws of physics are the same except that their "air" molecules are heavier and slower than ours, so when the wind blows hard the motion of these molecules is predominately just downwind with not very much random thermal motion.
Now Atom Man and his arch rival Mini-Schroder both have downwind carts that they can ride in, and they like to race each other on windy days. After much racing and tweaking both Atom Man and Mini-Schroder have tuned their carts so that they travel downwind with very little friction. They both attain almost the same speed as the wind, so their finishes are generally pretty closely contested.
Atom Man is very small and nimble, he can even see individual molecules of gas in the air. When he gets into his downwind cart to race he can see the molecules of air colliding with the back of his cart and bouncing off, providing the change in momentum to accelerate the cart forward. As his cart increases to closer to the wind speed he sees these molecules colliding with ever less relative speed and imparting ever less momentum, so he never can quite reach the speed of the wind, let alone exceed it.
One day while Atom Man is racing Mini-Schroder downwind he gets a great idea. The race is neck a neck, with both carts doing almost exactly the wind speed, when Atom Man notices that the air molecules are now almost stationary relative to him and he can easily reach out and grab them. So he starts grabbing these molecules as fast as he can and hurling them backwards to create a forward thrust (in much the same way as a spacewalking astronaut could throw a spanner to create thrust if he had no other propulsion available). This idea works and Atom Man actually gets his cart to faster than wind speed, resoundingly beating Mini-Schroder in the process.
The victory is short lived however. Mini-Schroder immediately protests to the DWC Academy about Atom Mans cheating tactics. The complaint is upheld, and rightly so. After all Atom Man was using up much of his own stored energy (from his internal batteries, you see Atom Man is battery powered) in the process of hurling those atoms. The Academy's rules strictly state that the only allowed source of power in these races is the wind.
Atom Man is not so keen to give up on the idea of DWFTTW just yet, so he asks the Academy to rule on whether he can continue to use his air molecule throwing trick if he obtains the power to throw those molecules from a generator running off the wheels of the cart instead of from his internal batteries. After some deliberation the Academy decides that this is ok and violates no rules. Meanwhile Mini-Schroder is not the least bit concerned about the ruling. He knows full well that Atoms Mans generator will create drag at the wheel and he "knows" (without any calculation or anything, he just knows ok) that this drag will exceed any thrust that Atom Man is able to produce by hurling air molecules.
Atom Man is not so sure about that, he knows a little about basic Physics so he does the analysis. He figures out that if he throws "n" molecules per second, with each molecule having a mass "m" and leaving at velocity "v" then the average forward thrust will be equal to the average rate of change of momentum. That is, atom mans produced thrust is :
F = nmv
Now Atom Man knows that if his strategy is to be successful he must be sure that the drag from his generator is less than this amount, so he decides to use only half of this extra thrust in powering the generator. The theoretical maximum power that his generator can deliver is therefore :
P = 1/2 Fw = 1/2 nmvw
Where "w" is the speed of the cart (and approximately the speed of the wind when atom man starts his hurling).
Finally Atom man calculates how much power he requires to hurl those molecules. The KE given to each molecule is 1/2 mv2, and this is done "n" times per second, so the average power required is :
P = 1/2 nmv2
Since 1/2 nmvw > 1/2nmv2 whenever w > v Atom Man sees that he can easily exceed the wind speed whenever the wind speed is greater than his throwing speed. The result is successful and Atom Man now can always beat Mini Schroder in these races.
Mini Schroder again complains to the Academy, this time he even accuses Atom Man of witchcraft, but nobody will take him seriously. Eventually he storms off and falls into a Bessel null, never to be heard of again. Everyone else lives happily ever after.
