There is a wonderful story circulated among physicists that illustrates the answer.
Johnny von Neumann was a great mathematician and a great physicist. So someone (I don't remember who) challenged him with this problem:
Two trains are 100 meters apart, moving towards each other. Each train is moving at 10 meters per second. A bee is flying back and forth between those two trains at 20 meters per second. Eventually the bee will be squished when the two trains crash against each other. When they do, what will be the total distance flown by the bee?
According to legend, von Neumann thought for a moment, and then said "100 meters".
The questioner said, "Correct. But now I know, from the quickness of your response, that you are really a physicist, not a mathematician. The mathematician would have calculated when each bee-train encounter takes place, calculated the position, taken the sum of those distances as an infinite series, and then summed the series. But a physicist would have taken a short-cut; the physicists would have realized that the bee was flying for 5 seconds at 20 meters per second, so it would have traveled 100 meters total."
Von Neumann's purported response: "Oh, that's a clever way to solve it! No, I summed the series."
(For the series solution, and a graphic showing the story, look at Two Trains Puzzle.)
-Richard Muller, Former Professor of Physics,UC Berkeley
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