Re: The worm problem

[rgrandy@ruf.rice.edu (Richard E. Grandy)]

> Ludwik and Donald objected to my statement of the rubber worm problem as
> being ill formed. I went over Ludwik's complaints, but I easily recognized
> that he was being ironic. Donald, however, makes me wonder. The problem I
> posed is included below. I would appreciate a serious comment on just how
> this problem is ill formed.
>
> > I call this (inappropriately) "The Rubber Worm Problem": You are provided
> > with an elastic rope exactly one kilometer in length. At time t=0 a worm
> > starts to crawl from one end of the rope to the other at a speed of one
> > centimeter per second. The rope itself is attached to a tractor which
> > is stretching the rope at a rate of one kilometer per second. The problem
> > as it was originally stated specifies that the rope lengthens by one
> > kilometer at the end of each second, and it is this discrete version of
> > the problem I will continue with here. At the end of one second the worm
> > had marched one centimeter toward the other end of the rope. At that
> > instant he is instantaneously transported to the 2 centimeter mark on a
> > rope which is now two kilometers in length. By calculation you will see
> > that at the end of the second second he will be three centimeters along
> > his way on the two kilometer rope, which will then lengthen to three
> > kilometers, taking the worm to the 4.5 cm point, etc.
> >
> > First question: Has our worm embarked on a Sisyphusian journey; will he
> > ever reach the end of his rope? If not, prove that.
> >
> > Evidently the answer to the first question is "No" as the sophisticate
> > will recognize. Otherwise, what's the point of the question? The second
> > question is: How long will it take the worm to reach the end of his rope?
>
> A physicist will assume the worm needs no more than one coordinate to
> specify position. The sample calculation certainly makes that clear. How
> could this problem possibly be misinterpreted? In my view it comes
> nearer being overspecific.
>
> Leigh


I can't help with your question because it seems quite well specified to
me.  I would prefer to reformulate it into an equivalent form.  Since the
effect of the stretching is not to move the worm backward but to give it a
longer rope to go along, the effect is the same as if instead of stretching
the rope we slowed the worm, so it would be centimetering 1,1/2,1/3,1/4,
....  in successive seconds. So it turns into the question when that series
= 10^5.

Richard Grandy
Philosophy & Cognitive Sciences
Rice University