Chronology Current Month Current Thread Current Date
[Year List] [Month List (current year)] [Date Index] [Thread Index] [Thread Prev] [Thread Next] [Date Prev] [Date Next]

Re: worm problem/PEDAGOGY



Hi!
This is the CONTINUOUS MODEL of the worm on a rubber band. Since most
people wrote about the discrete model, let's analyze the continuous one.
ASSUMPTIONS: The rubber band extends from A to B, and its length at
time t is L = L_0 (1+kt). The worm is crawling from B to A, with velocity
v (as measured by an non-streching observer).
COORDINATES: Let x be the distance from A in the direction towards B.
Then for point A, x_A=0, and for point B, at time t=), x_B=L_0.
Let z be streching coordinate (marks on the rubber band), such that
z_A=0 and, at time t=0, z=x everywhere.
Then x = (1+kt)z and, for fixed z, dx/dt = kz = (k/(1+kt))x.
Let x(t) and z(t) be the location of the worm at time t (I beg your
paedon for the sloppiness of the notation, identifying physical magnitudes
with mathematical functions). Then, the velocity of the worm is obtained
by superposition: dx/dt = kx/(1+kt)-v. This is a linear o.d.e. of the
type x'=p(t)x+q(t), whose solution is (here Int = integral, and
P(t) is any indefinite integral of p(t)
x(t) = exp(P(t)) (C - Int exp(-P(s)) q(s) ds) =
= (1+kt)(C - (v/k) ln(1+kt)).
The arbitrary constant C turns out to be C = x(0).
CONCLUSIONS: a) If v=k=0, x(t) = C = constant.
b) For all the other v,k >=0, x(t) tends to - infinity as
t tends to infinity, and the worm starting at x_B = L_0 > 0
passes the point A where x = 0.
Isn't that nice? Emilio