A rubber band is 1 meter long. An ant starts at one end, crawling at 1 millimeter per second. At the end of each second, the rubber band is instantaneously stretched by an additional meter. (So, at the end of the nth second, the rubber band becomes n+1 meters long.)

Does the ant ever reach the far end of the band? If so, when?

Each second the ant moves 0.001m towards the destination (the end of the band).
Each second the destination moves 1m away from the ant.
So the destination is moving away from the ant faster than the ant is moving towards the destination, and therefore, the ant will never get there.
In this case, the fact that the movement of the destination is non-linear is a red-herring.
You could plot a graph of the position of the end of the rubber band and the position of the ant both relative to the start of the band against time (x vs t). The end of the band would be a series of unit steps every second (starting at t=0,x=1), and the ant would be a diagonal line with a gradient of 0.001. The lines would diverge with increasing time and not cross for t>0.
So the solution given to this problem is incorrect, so what is the flaw?
The sum terms become incorrect with time. Following the logic of the solution presented, in the first second, the ant moves 1/1000 of the _original_ length of the band, and in the second, it moves 1/2000 of the _new_ length of the band. The solution proposes that in two seconds, the ant has traversed (1/1000 + 1/2000) * the length of the band which is assumed to be 2m, so the ant has traversed a total of 3/2000 * 2 = 3/1000 or 0.003m. We know from its speed that the ant has actually traversed 0.002m in 2s, and we can also get that from L0/1000 + L1/2000 = 1/1000 + 2/2000 =0.002.

Posted by rich harran on 2004-03-23 04:31:27