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Motion along a stretching band

  1. Jul 23, 2015 #1
    This was given as a problem in a calculus textbook I'm working through (apologies if this should have gone in the physics forum)

    1. The problem statement, all variables and given/known data

    An ant crawls at 1foot/second along a rubber band whose
    original length is 2 feet. The band is being stretched at 1
    foot/second by pulling the other end. At what time T, if ever,
    does the ant reach the other end?
    One approach: The band's length at time t is t + 2. Let y(t)
    be the fraction of that length which the ant has covered, and
    explain
    (a) y' = 1/(t + 2) (b)y =ln(t + 2) -ln 2 (c) T = 2e -2.

    2. Relevant equations
    ∫1/x dx = ln(x)

    3. The attempt at a solution
    Given a, I can get to b by integrating and finding the constant, and then to c by solving for y=1, but I'm stumped on how to get to explain a. y' seems to be the ant's speed over the length of the band, by I don't understand why that is so.
     
  2. jcsd
  3. Jul 23, 2015 #2

    Nathanael

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    Homework Helper

    Calculus by Gilbert Strang? :woot: I read that one too.

    y' is the rate of change of the fraction which has been traversed. Suppose the ant was not crawling: when the rubber band is being stretched, the fraction of the rubber band behind the ant would not change (because it also stretches).
     
  4. Jul 23, 2015 #3
    Yup.

    I see that y' is the rate of change of the fraction, but I'm still not sure why why it equals what it does.
     
  5. Jul 23, 2015 #4

    Nathanael

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    Homework Helper

    Consider two special cases to try to get an intuition:

    The case where the ant is not crawling (but starts out at some initial fraction y0).
    The case where the rubber band is a fixed length.

    What would the equation for y' be in each of these (separate) cases?
     
  6. Jul 23, 2015 #5
    Got it, Thanks.
     
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