gabee
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Homework Statement
http://www.feynmanlectures.info/exercises/bug_on_band.html
An infinitely stretchable rubber band has one end nailed to a wall, while the other end is pulled away from the wall at the rate of 1 m/s; initially the band is 1 meter long. A bug on the rubber band, initially near the wall end, is crawling toward the other end at the rate of 0.001 cm/s. Will the bug ever reach the other end? If so, when?
Homework Equations
Differential ones!
The Attempt at a Solution
I solved this one in what I think is a sort of novel way. I imagine that we are viewing the situation in a "stretching" frame--as if we view the stretching band with a camera and we continuously zoom out to keep the image of the band on the film exactly 1 meter wide. Then, the velocity of the bug on the film is described by
v_{bug\,in\,frame} = \frac{l_0}{l} v_{bug\,0}
where l_0 is the initial length of the band, l is the length of the band as a function of time, and v_{bug\,0} is the initial velocity of the bug as seen on the film (which is the same as the real velocity, 1E-5 m/s).
In effect, the 'image on the film' becomes a representation for the fraction of the band traversed by the bug.
So, using the information given in the problem, this equation becomes
v_{bug\,in\,frame} = \frac{1}{1+t} 1 \times 10^{-5}
which we can integrate with respect to time to find the x position of the bug on the film:
\int v\,dt = \int \frac{1}{1+t} 1 \times 10^{-5}\,dt
x = 1 \times 10^{-5} \ln(1+t)
When x = 1, the bug has reached the end of the band:
1 = 1 \times 10^{-5} ln(1+t), and with a little algebra,
t = e^{100000} - 1.
My original plan, however, was to use the following differential equation to describe the actual distance of the bug from the wall:
\frac{dx}{dt} = \frac{x}{l} v_{end} + v_{bug}
where l (which is 1+t) is a function of t that describes the length of the band, v_{end} is the velocity of the end of the band (dl/dt=1 m/s) and v_{bug} is the velocity of the bug by itself (1E-5 m/s). But I didn't know how to solve this differential equation (I haven't taken a diff eq course yet and separation of variables won't work). I'm wondering--is there a general solution to this form of DE?
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