I was thinking about this last night (how sad is that!). I think it's a case of proportional growth of the worm's speed, which is why the continuous model increases so much faster than the discrete model.
Basically, the worm travels at a constant rate under its own power, and gets a speed boost from the rope being stretched. At the beginning of its trip the worm hardly gets any boost from the stretch, because the fraction of the rope that it has already traveled only gets a fractional increase in length. As the worm travels further down the rope, it gets more of a boost, which makes it travel down the rope faster, which increase the boost, etc... which ends up making the worm's speed increase exponentially.
This is analagous to proportional growth in a population (where the growth rate is proportional to the population size), or continuously compounded interest (where the monetary increase is proportional to the size of the bank account).
In the discrete model, the worm only gets a boost at certain intervals, so the boost is only contributing to the factor which controls the change in speed with time (the worm's position) part of the time.
Interesting, though. I think the continuously stretched rope gives the minimum time to traverse the rope.
Originally posted by PBE6Yeah I saw that, but it doesn't change anything that follows, so I didn't see the need to be picky. With the actual substance of the analysis, I don't see anything wrong...
Oh, I found a typo in my solution. The very first equation (dL/dt = k) describes the movement of the end of the rope, not the worm's position.
Originally posted by PBE6Okay, I am beginning to think that there may not be a discrepancy between the continuous and discrete cases:
What do you think about my other post? The one before the typo correction?
Consider the following (sort of) general case:
Suppose the rope is initially k cm long, and we stretch it at a rate of k cm/sec. The worm speed is w cm/sec.
Then, by your continuous model, we can find the time it takes the worm to traverse the rope by setting W/L = 1, and by noting that in this case, L = L0 + kt, and L0 = k.
Doing this, I get the traverse time (given as N number of seconds) to be
N = exp(k/w) - 1.
Now consider the discrete model: according to it, the fraction of rope traversed after the Nth second is given by
f(N) = (w/k)[S(N)]
where S(N) is just the Harmonic series taken out to the Nth term. A cool little fact is that for large N, the following approximation applies
S(N) ~ ln(N) + &,
where & is the "Euler Constant" and is approximately 0.5772156649...
Using this handy approximation, we can solve for the time it takes the worm to traverse by simply setting f = 1: Upon doing this, I get that the discrete model predicts
N = exp[(k/w) - &].
Comparing the two, you find there is virtually no difference between them for large k/w, and I imagine that the discrepancy between the two can be explained in the nature of the approximation we used for S(N).
Thus I don't think the discrete and continuous models necessarily conflict.
Does any of this make sense?
FYI, you can find more information regarding the approximation I used here:
http://www.jimloy.com/algebra/hseries.htm
or here:
http://mercury.easternct.edu/s/shamaj/web/projects/harmonicSeries.pdf