bug on band

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This problem was contributed to The Feynman Lectures website by Lev B. Okun. The following story is translated from Lev's 1990 article "Three Encounters," published in the Russian journal Nature shortly after Sakharov's death.

21 July 1976, Restaurant Aragvi in Tbilisi, Georgia, where a dinner party was held for the International Conference on High Energy Physics (XVIII in a series of so-called Rochester Conferences). There were lots of long tables, and at one of them I sat near Sakharov. The conversation meandered randomly, we started talking about new problems, and then I suggested to Andrei Dmitrievich the problem of a bug on an ideal rubber band: You have a 1 km long rubber band with one end attached to the wall, and the other in your hand. The bug begins to crawl towards you on the rubber band, starting from the wall, at a rate of 1 cm/sec. As he crawls the first centimeter you extend the rubber band 1 km; when he crawls the second centimeter you extend the rubber band another 1 km, and so on, every second. The question is: Does the bug ever reach you, and if so, in how much time? Both before and after that evening I gave the problem to different people. One demanded about an hour to solve it, another demanded a day, the third remained firmly convinced that the bug does not reach you and the question of how much time is given to send you barking up the wrong tree. Sakharov asked for the conditions of the problem and a piece of paper. I gave him my invitation to the banquet, and immediately, without any comment, he wrote the solution on the back. All together it took about a minute.

An infinitely stretchable rubber band has one end nailed to a wall, while the other end is pulled away from the wall at the rate of 1 m/s;  initially the band is 1 meter long. A bug on the rubber band, initially near the wall end, is crawling toward the other end at the rate of 0.001 cm/s. Will the bug ever reach the other end? If so, when?


Solutions (listed by author)

Andrei Sakharov (pdf, 94K)  (translated into English - html, 2k)

Michael A. Gottlieb (approximate stepwise solution html, 2k)

Michael A. Gottlieb (exact general solution html, 2k)

Rudi Pfeiffer (html, 2k)

Twan Mennink (html, 2k)

Adrian Cozac (pdf, 86k)


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