Due to an extremely unfortunate error involving a train that was behind schedule and a watch that may have been four minutes slow, there was a head-on collision of two trains in the town of Kipton in 1891. The plan was that the slow westbound passenger train, traveling at 8 miles per hour, would be able to get onto a siding in time to get out of the way of the fast eastbound mail train, traveling at 45 miles per hour. But it didn’t work out that way. (You can Google Kipton Train Wreck to find out the details.) Our concern here is with a fly. When the two trains were exactly 10 miles apart, the fly, sitting on the front of the westbound engine, flew down the tracks to the eastbound train at 60 miles per hour. When it got there, it turned around immediately and flew back to the westbound train, thence to the eastbound, thence to the westbound, back and forth, over and over, going a shorter distance each time. The question, of course, is how far did the fly fly before being crushed between the two engines as they collided?
The statement of the problem implies we should be summing an infinite series, but there is an easier way to get the answer. If the westbound train is going 8 mph and the eastbound train is going 45 mph, they are closing the distance between them at 53 mph. To traverse the 10 miles between them will thus take 10/53 of an hour. The fly is flying at a rate of 60 mph and will fly 60(10/53) = 11.321 miles before the trains collide.