## Wednesday, October 19, 2011

### The ant and the rubber band

In a draft I'm writing, I'm including a classic puzzle as a demonstrative example.  But I should let my readers have a try first!

There is an ant on a rubber band.  The ant is crawling from one end of the rubber band towards the other at 1 inch per second.  The rubber band is one foot long, but getting longer.  After each second, the rubber band's length grows by one foot.

Question: Will the ant ever reach the other side of the rubber band?

(It's an infinitely stretchy rubber band, and the ant lives infinitely long.  Since I am a physicist, I will not worry about whether the rubber band is stretching instantaneously every second or stretching continuously.  Just assume whichever you prefer.)

You have less than a week to solve this one.  Bonus points if you can predict what my draft is about.

(solution contained here)

Larry, The Barefoot Bum said...

Does the ant grow along with the rubber band?

Larry, The Barefoot Bum said...
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Larry, The Barefoot Bum said...

I'm assuming that the ant doesn't grow, and that the increase in the rubber band's size is evenly distributed over the length. Therefore, as the ant moves along the rubber band, more of the rubber band's increase is added behind the ant, and therefore the ant will eventually reach the other end of the rubber band.

I don't yet have the maths to figure out how long it will take the ant to reach the end.

miller said...

You may assume the ant is a point, and that the rubber band gets stretched uniformly.

Larry, The Barefoot Bum said...

OK, so yes, the ant will eventually reach the other end.

Eduard said...

If the band is only one inch long the discretification is more important. Crawl then stretch: ant reaches the end after 1 second. Stretch then crawl: ant reaches the end. Stretch and crawl simultaneously: ant reaches the end.

miller said...

The rubber band's length begins at twelve inches before the ant starts crawling.

Larry, The Barefoot Bum said...

I'm guessing the post is about light traveling across an expanding universe.

Using brute force (I should have paid more attention in Calculus), I'm guessing around 80,850 seconds for the ant to reach the other end.

miller said...

That's correct, and also a correct guess as to the subject of my draft!

However, the length of time you got is different from mine, because I assumed continuous stretching. It's a factor of two. I'm surprised it's that different, actually.

Larry, The Barefoot Bum said...
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Larry, The Barefoot Bum said...
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Larry, The Barefoot Bum said...

I used Excel; you probably used calculus. Also, I could easily have an error in my calculations. In any event, I count coming in within a factor of two (and getting the general answer without doing any math or calculation) as a win.

I think, however, that the ant will never reach the end if its travel and the expansion are geometric instead of arithmetic.

Larry, The Barefoot Bum said...

The integral for acceleration puts the 1/2 in the formula. Since I indeed did instantaneous increases every 1, 10, 100, and 1000 seconds (to keep the problem tractable in Excel), I suspect the factor of 2 is indeed due to the difference between your continuous and and my instantaneous solutions.

miller said...

The factor of two is due to the fact that sum(1/n) is not quite the same as ln(n), even for large n. The difference is called the Euler-Mascheroni constant. The correction is a factor of e^(-gamma), which is about one half.

But yeah, I would definitely count getting within a factor of two as a win.

Peter Jeffery said...