This is a problem that i found while on this website: http://feynmanlectures.info/ (all credit goes there). The problem is this (and there is a funny story behind it):
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?
Lev Okun gave this problem to Andrei Sakharov to pass the time while they were being driven from Moscow to JINR in Dubna. However, it didn't work (to pass the time) because immediately after being told the problem, Sakharov pulled out a pen, took Lev's magazine and wrote down the solution, without any hesitation whatsoever.
So it actually has a pretty nice history, and much respect should be given to Sakharov for his quick solution! Although, i see how, after having done the problem, that once you do one, you've done 'em all. I also want to mention another similar problem that i came across some time ago, and just remembered when i read this one. It is from a book called Professor Stewart's Cabinet of Mathematical Curiosities by Ian Stewart; the problem is as follows and is made slightly simpler than the one above (but trivially so, as it turns out. But it is actually kind of nice that way! - I'll explain more what I mean after I present my solutions below...):
The spaceship Indefensible is at the center of a spherical galaxy with radius 1000 lightyears (lyrs). The spaceship travels at a rate of one lightyear every year - the speed of light (assume it's made of photons...). At exactly one year intervals after the Indefensible starts its voyage, the universe expands instantly by 1000 lyrs, and the ship is carried along with the space in which it sits.
So these two problems are really one and the same, except that the bug is being carried along as the band expands constantly while the spaceship is only effected by the expanding space at discrete intervals.
We'll see how this effects the solution later.
At this point I kind of want to leave these problems up to you guys to solve, without giving away too many hints. One hint I will give for Stewart's problem (which he gives as well) is that the following link may be helpful: http://en.wikipedia.org/wiki/Harmonic_series_(mathematics) (particularly the section on Rate of Divergence...)
I'll upload the solutions sometime later, but don't peek! It's (They're) a quite rewarding problem(s) to work!
I'll note that the entire time I was working the Bug on Band problem I thought that there was absolutely no way that my solution was even remotely correct, until I got the right answer! Just goes to show you sometimes...
Hello, Nick. I was amused to find you copied the "bug on band" problem from my website, so I sent the URL of your blog to my friend Lev Okun, in Moscow. Lev was amused to find that I had completely confused his story about Sakharov! Here is the correct story, translated (as best I could) from Lev's article, "Three Encounters," published in the Russian journal, "Nature" in 1990, shortly after Sakharov's death.
ReplyDelete21 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.
I will be updating this on my website presently!