Passing ShipsDate: 05/12/99 at 08:56:19 From: Mary Covington Subject: "Till We Meet Again" Every day at noon a ship leaves San Francisco for Tokyo, and at the same instant a ship leaves Tokyo for San Francisco. Each trip lasts exactly eight days. How many Tokyo ships will each San Francisco ship meet? This is a problem we have been working on. We have drawn it, acted it out, and come up with either 8 or 9. The answer that was given at a workshop last summer was 17. We are frustrated. Can you help us? Thanks, The 3rd Grade of The Raleigh School Raleigh, NC Date: 05/12/99 at 11:47:17 From: Doctor Peterson Subject: Re: "Till We Meet Again" Hi, Mary (and the rest of the class). I like to approach this sort of problem with a time-distance graph, at least to get an idea of what's happening. The distance left to right is the position of the ship from San Francisco to Tokyo; each * on the vertical lines marks noon on one day, when a ship leaves from each end. The 8-day trip from SF to Tokyo is a diagonal line from top left to bottom right; the ships from Tokyo to SF that it meets are the diagonal lines that cross it at the X's: Day |/ / / / / / / / | 0 X / / / / / / / * |\ / / / / / / / /| | X / / / / / / / | |/ \ / / / / / / / | 1 * X / / / / / / * | / \ / / / / / / /| | / X / / / / / / | |/ / \ / / / / / / | 2 * / X / / / / / * | / / \ / / / / / /| | / / X / / / / / | |/ / / \ / / / / / | 3 * / / X / / / / * | / / / \ / / / / /| | / / / X / / / / | |/ / / / \ / / / / | 4 * / / / X / / / * | / / / / \ / / / /| | / / / / X / / / | |/ / / / / \ / / / | 5 * / / / / X / / * | / / / / / \ / / /| | / / / / / X / / | |/ / / / / / \ / / | 6 * / / / / / X / * | / / / / / / \ / /| | / / / / / / X / | |/ / / / / / / \ / | 7 * / / / / / / X * | / / / / / / / \ /| | / / / / / / / X | |/ / / / / / / / \| 8 + / / / / / / / X | / / / / / / / /| SF Tokyo I count 17 X's, though I'd question whether to call the first and last "meetings," since they represent a boat that arrives just as they depart, and one that departs just as they arrive. They meet each of the ships that left Tokyo in the last 8 days, as well as each ship that leaves Tokyo while they are on the way, making 8 + 8 + 1 = 17 because we're including both ends of the 16 day interval. You don't need the picture to see this, but it helps a lot to convince me I'm right. Perhaps what you missed, and what I might have missed if I hadn't drawn it, was the ships that were already on the way when our ship left port. Why is this more than the 8 or 9 you expect? Shouldn't they meet one ship per day? No; because they are going toward those ships as fast as the ships are coming toward them, it's as if the ships were coming twice as fast and twice as frequently. We meet all the ships that will arrive in San Francisco in the next 16 days, all in 8 days. This is related to the Doppler effect you may have heard of: if I'm moving toward a source of sound, it will seem as if the sound has a higher frequency. Here, I'm going at the speed of "sound," and I meet a "wave" twice as often as I would if I stayed at San Francisco, so the "sound" has twice the frequency. - Doctor Peterson, The Math Forum http://mathforum.org/dr.math/ |
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