Two trains 150 miles apart are traveling toward each other along the same track. The first train goes 60 miles per hour; the second train rushes along at 90 miles per hour. A fly is hovering just above the nose of the first train. It buzzes from the first train to the second train, turns around immediately, flies back to the first train, and turns around again. It goes on flying back and forth between the two trains until they collide. If the fly's speed is 120 miles per hour, how far will it travel?
We want to know the total distance that the fly covers, so let's use Distance = Rate * Time to solve the problem. We already know the fly's rate of flight. If we can find the time that the fly spends in the air, we can figure out how far it travels.
Ignore the fly for a minute, and concentrate on the trains. The first train is traveling at 60 miles/ hour and the second train is going 90 miles/ hour, so they are approaching each other at 60 miles/ hour + 90 miles/ hour = 150 miles/ hour. Now we know the rate at which the trains are closing in on each other and their distance apart (150 miles), so we can find the time until they crash:
Time = Distance / Rate
= (150 miles) / (150 miles/ hour)
= 1 hour.
The fly spends the same amount of time traveling as the trains. It goes 120 miles/ hour, so in the one hour the trains take to collide, the fly will go 120 miles.
The solution above ignores the shape of the fly's path. To picture this shape, think of the fly as a point made out of rubber. It's bouncing between the trains at a very high speed. As the trains get closer and closer the bounces get shorter and shorter, until they are microscopic. Even then, if you had a strong enough magnifying glass, you could still see the bounces getting shorter. No matter how much you magnify, there will always be a tinier bounce that you can't see.
You can analyze this path by combining the bounces into a series of round trips, from the first train to the second and back again. It turns out that the length of each trip is a fraction of the trip before. No matter how many times you multiply by a fraction, you will never reach zero. The fly makes an infinite number of round trips, each one smaller than the last.
We could use this information about the shape of the fly's path to solve this problem. We might add the length of each of the fly's round trips, from the first train to the second and back again. It's also possible to add up the time that each round trip requires, then use Distance = Rate * Time to finish the problem. Of course, adding every term directly would take forever. There are shorter methods for summing an infinite number of terms in calculus and other branches of advanced mathematics. To use them, we would need to find a pattern in the trip lengths. (Finding the pattern is the messy part!)
There's a famous story about John von Neumann, a physicist and mathematician:
Another mathematician knew the quick solution to the fly problem, and wanted to see von Neumann struggle with it. He posed the question, and von Neumann responded with the right answer in a few seconds.
"Interesting," said the first mathematician. "Most people try to sum the infinite series."
"What do you mean?" von Neumann replied. "That's how I did it."
See Lee Bradley's presentation: the "von Neumann" way.
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