My intention with this blog is to share some of the more interesting curiosities of math. Often in the form of mathematical puzzles, riddles or paradoxes. Some can be applied to the real world, showing that all is not always what it seems. I'll include a calculation now and then but will keep it light — heck I'm far from a math wizz myself.
Saturday, July 2, 2011
The Ant On An Elastic Rope
An ant starts to walk along an elastic rope which is 1 km long, at a speed of 1 cm per second (relative to the rope it is crawling on). While this happens the rope is uniformly stretched by 1 km per second (i.e. after 1 second it is 2 km long, after 2 seconds it is 3 km long, etc.). Assume the rope will never break and we have an infinite amount of time on our hands. Will the ant ever reach the end of the rope?
Here's a hint: Yes, yes it will.
If it were crawling along, say, an iron bar being lengthened, then no. The ant could obviously never reach the end. A speed of 1 cm/s is not enough to reach a target that is moving away at 1 km/s.
However, this is a an elastic rope. The intro said “relative to the rope it is crawling on”: that's your clue right there. As the rope stretches, the ant is carried along. To visualize this, imagine the ant not crawling but sitting still — at 10% the length of the rope, thus 100 meters from the beginning. Initially the rope is 1 km, a second later it's 2 km. Because of the rope's elastic properties, the ant is still at 10% except this is now 200 meters from the beginning.
Now visualize it has been crawling during this time. In that case, after the second the ant would be at 200 m plus 1 cm: 10.0005%. Hey, some small progress has been made! Keep this up long enough, and eventually 100% (the end of the rope) is reached. And yeah, that will get tougher the further it crawls. It's a very discouraging walk if you ask me.
In fact, people much smarter than me calculated how long it would take. It's not that bad, a mere 8.9 × 1043421 years ought to do it...
(Disclaimer: unfortunately for the ant it's unlikely the universe is still around by then. But points for trying.)
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I remember this riddle, classic! I would always mess up the answer though and nothing has changed lol. Mind if I share another rope riddle on your blog here: Rope Burn
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