Can an ant reach the free end of a stretching rubber band?

[Tide]

Nate,

That ant's speed relative to the rubber band is u, a constant. Also, the speed of the point on the band upon which the ant is standing at any instant depends on where that point started. Since the ant is moving relative to the band, the speed of the point it is standing on varies so the numerator in your expression cannot be a constant. (I.e., the instantaneous point upon which the ant stands changes from one step to the next and each of those points started off at a different location on the initial band.) Finally, you are assuming the (incorrect) answer by setting the upper limit of your integral to infinity. The ant reaches the end of the band in a finite time.

 

[NateTG]

Nate,

That ant's speed relative to the rubber band is u, a constant. Also, the speed of the point on the band upon which the ant is standing at any instant depends on where that point started. Since the ant is moving relative to the band, the speed of the point it is standing on varies so the numerator in your expression cannot be a constant. (I.e., the instantaneous point upon which the ant stands changes from one step to the next and each of those points started off at a different location on the initial band.) Finally, you are assuming the (incorrect) answer by setting the upper limit of your integral to infinity. The ant reaches the end of the band in a finite time.

It's probably my fault, but it's clear that you didn't understand what I wrote.

I was taking the (unorthodox) approach of changing from an absolute unit of length to using the rubber band's length as a unit of length. This has the disadvantage that some quantities - like the ant's speed relative to the rubber band - which were constant, are now variable, but has the advantage that other quantities - the length of the rubber band - become constants.

Regarding the use of the improper (only in the sense of limits of integration) integral:
The question whether the ant reaches the end of the rubber band is quite similar to asking whether, for:
$$f(x)=\int_{0}^{x}\frac{1}{y}dy$$
it is ever true that
[tex]f(x)>a[/itex]<br /> for some arbitrary $a$<br /> This is equivalent to asking wether:<br /> $$\int_{0}^{\infty}\frac{1}{y}dy$$<br /> tends to positive infinity.[/tex]

 

[Tide]

Nate,

Yes, I did miss that the first time around. I like the approach - nice!