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Math Help - the ant on a rubber rope

  1. #1
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    the ant on a rubber rope

    I cannot solve the problem of the ant on a rubber rope (Ant on a rubber rope - Wikipedia, the free encyclopedia)
    in the case of a rope stretching exponentially:

    the target end of the rope y(t) goes away exponentially

    y(t)= D*exp(H*t)

    where D and H are positive constants. D is the initial length of the rope to be crossed by the ant.

    Given that the rope stretches uniformly, I believe that the position of the ant on the rope x(t) should evolve according to


    dx(t)/dt=c+x(t)*exp(H*t)

    where c is the constant proper speed of the ant irrespective of rubber band stretching
    initial condition x(0)=0

    Can the ant reach the end of the rope depending on the relative values of c, D and H?
    is it possible to prove that the ant will never reach the target, whatever the values of the constants?
    or is it a critical point on the rope beyond which the target cannot be reached

    many thanks in advance

    Denis
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  2. #2
    mfb
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    Re: the ant on a rubber rope

    Consider the problem in coordinates which get stretched together with the rubber band:
    dy(t)/dt = c/exp(H t) = c exp(-H t)
    This is easy to solve, and the total distance of the ant approaches c/H. For c>H, it reaches the other side, for c<=H, it does not.
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  3. #3
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    Re: the ant on a rubber rope

    Quote Originally Posted by mfb View Post
    Consider the problem in coordinates which get stretched together with the rubber band:
    dy(t)/dt = c/exp(H t) = c exp(-H t)
    This is easy to solve,
    .

    thank you for your answer; I don't see precisely the equation to be solved, would you be kind enough writing it

    Quote Originally Posted by mfb View Post
    and the total distance of the ant approaches c/H. For c>H, it reaches the other side, for c<=H, it does not.
    for any value of c and H, does the ant win the race if D<c/H ?
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  4. #4
    mfb
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    Re: the ant on a rubber rope

    dy(t)/dt = c exp(-H t)
    This can be solved by integration of both sides from t=0 to t=T (or infinity).

    I missed the additional D in your formula, but this is just a scaling of c:
    dy(t)/dt = c/D exp(-H t)

    The ant reaches the other side for c/(DH)>1. In other words, the initial velocity of the ant has to be larger than the initial extension velocity of the rubber band.
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  5. #5
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    Re: the ant on a rubber rope

    thanks!
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