Each second a rabbit moves half the remaining distance from his nose to a head of lettuce 10m away. When does he get to the lettuce?
by 
kewelll07
ANSWERS: 7
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He'll never get there. If you continue to cut the distance in half, it can never be zero, but it will be infinitely small.
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Your rabbit and lettuce have to be theoretical constructs to make this a meaningful exercise, but... 0 10m 1 5m (hmmm... never seen a rabbit do 5m/s) 2 2.5m (he's braking hard!) 3 1.25 4 .625 5 .3125 6 .15625 7 .078125 8 .0390625 9 .01953125 10 about .01 11 about .005 12 about .0025 13 about .0012 14 about.0006 at 14 seconds your rabbit is well with a tooth length of the lettuce and probably would have just streched out his snout and grabbed the thing at about 10 seconds. It would be very difficult for the hypothetical rabbit to take those last few steps at 5mm or less per step. At 14 seconds that .6mm step might be hard to measure since real rabbits change shape as they move and breathe... In a few more seconds your rabbit's nose will be within molecules of the lettuce and both things in real life are too dynamic to measure to that tolerance.
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This reminds me of the one about the engineer and the technician who are given that challenge to reach a prize. The engineer says, "We'll never get there." The technician says, "I'll get close enough." That illustrates the issue. At some point the rabbit will get close enough to the lettuce that he'll just reach out and take a bite. Mathematics has a way of dealing with distances that are subatomic. Rabbits don't. In the absolute, mathematical sense, the distance becomes infinitesimally small, but never reaches zero. Rabbits, and people, don't live in a quantum world though, but in a Newtonian one, so this is a question with illustrative power, but no meanigful answer.
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I am adding an additional answer to what I have already said, because it takes another tack. I have recently learned that what you are describing is known as "Zeno's (or Xeno's) Paradox. It was originally a philosophical argument against motion as anything but illusion. There's a fairly thorough treatment at Wikipedia: http://en.wikipedia.org/wiki/Zeno's_paradoxes I can see that the apostrophe is giving the Answerbag system some trouble in displaying the link. It should end with .../Zeno's_paradoxes
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To put it simply... In mathematical terms, the rabbit will never reach the lettuce. He will forever keep getting half as close to it as his last hop. In mechanical terms, some subatomic particle of an atom at the tip of the rabbit's nose will eventually reach a single unit of quantum distance from some subatomic particle of an atom on the surface of the lettuce. His next movement will cover that quantum distance, and he will have dinner.
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First off you need to know the distance between the rabbits head and it's nose. second of all, as long as you keep dissecting the distance, in actuallity the rabbit will never reach the lettuce...
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This won't address the question directly, but it will address the answers that say he will never. He will. It is true, it is called Zeno's paradox, but has been solved in calculus through the use of converging series. Basically, converging series are a series of numbers that go on infinitely but can have a finite answer.
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