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Homework Help: Ant on an expanding balloon

  1. Nov 16, 2011 #1
    This isnt homework but nobody is going to believe that so here it is:

    1. The problem statement, all variables and given/known data
    at t=1 (or any nonzero value of t)
    an ant starts moving across a balloon at velocity
    v=c=1 (c is the speed of light)​
    the radius of the balloon as a function of time is given by
    r(t)=nct (n is a constant much larger than 1)​

    how far does the ant move as a function of time?

    2. Relevant equations

    3. The attempt at a solution
    the angular velocity of the ant is
    ω(t) = v/r(t) = c/nct = 1/nt​
    integrating to get angular displacement I get
    multiplying by radius to get distance I get
    d(t) = r(t)*θ(t) = nct*log(t)/n = ct*log(t)​

    The trouble is that this cant be right.
    if n>>1 then the ant can never get all the way around the balloon
    therefore the angular displacement θ must approach a limit as time goes to infinity.
    But log(t) increases without limit.

    I've gone over and over it
    it seems too simple to be wrong but it must be.
    Last edited: Nov 16, 2011
  2. jcsd
  3. Nov 16, 2011 #2


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    Can the balloon physically expand as fast as nct? Or is that an assumption?
  4. Nov 16, 2011 #3
    that is assumed
  5. Nov 16, 2011 #4


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    I am not sure you can assume this. If the balloon's expansion is limited then your paradox might never occur.
  6. Nov 16, 2011 #5
    the point is that the balloon is much expanding faster than the ant can walk
    so the ant cant walk all the way around it.

    there is no paradox. My math must be wrong somehow

    the trouble still remains even if the radius of the balloon were as low as r(t) = ct/2π
    the circumference would then be c(t) = 2πct/2π = ct
    Last edited: Nov 16, 2011
  7. Nov 16, 2011 #6


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    Part of the problem seems to be the initial singularity (the fact that the balloon has 0 radius) at the beginning of the expansion. Your angular velocity blows up at t = 0 if you assume that the ant has always been moving at c "around" the surface of the balloon.

    [tex] \theta(t) = \frac{1}{n}\int_0^t \frac{d\tau}{\tau} [/tex]

    [tex] = \frac{1}{n} [\log(t) - \log(0)] [/tex].

    The logarithm of 0 is not defined. This integral is not defined.

    I'm curious what happens if you stipulate that the ant has to start at rest and start moving at c only after a finite time. OR you could stipulate that the balloon has a finite radius at t = 0.
  8. Nov 16, 2011 #7
    that was stipulated but I assumed that the equation would be the same.
  9. Nov 16, 2011 #8
    I'm more concerned by the fact that log(infinity)=infinity

    for large values of n it should by finite
  10. Nov 16, 2011 #9
    log does have the nice property of equaling zero at t=1
    so I originally tried to make that the point where the ant starts moving.

    actually I dont see any trouble with doing that,
    you just have to adjust the distance scale to match it so that c=1
    Last edited: Nov 16, 2011
  11. Nov 16, 2011 #10


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    Sorry, my bad. So if we just consider things from t = 1 onward, then it is indeed true that θ(t) = log(t)/n.

    So it seems like the answer is that the ant will make it all the way around in a finite time equal to t = exp(2πn). So the time required to make a full circle increases exponentially with n, but it is still finite.

    Why are you so convinced that this answer cannot be right?
  12. Nov 16, 2011 #11
    because some points on teh balloon will be receding from the point of origin of the ant faster than c (much faster)

    consider the rate at which the circumference is increasing.

    c(t) = 2πr(t)=2πnct

    a point on the opposite side of the balloon will be receding at half that speed

    space is opening up between the ant and that opposite point much faster than the ant can move forward.
    therefore it should be receding from the ant.

    now the total distance the ant travels will increase without limit
    but the angle θ(t) will not.
    Last edited: Nov 16, 2011
  13. Nov 16, 2011 #12


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    Your expression for the distance travelled s(t) is wrong. It's NOT true that s(t) = θ(t)r(t). Instead, we must write:

    ds = r(t) dθ = r(t) (dθ/dt) dt = r(t)ω(t) dt = c dt

    s(t) = ∫ds = ∫cdt = ct

    So, that's progress. We found one mistake.
  14. Nov 16, 2011 #13
    thats just the velocity of teh ant.

    you also have to consider taht the space already traveled by the ant is expanding.
  15. Nov 16, 2011 #14
    according to someone on another forum

    ds/dt = r dθ/dt + θ dr/dt.

    the first part is the velocity of the ant
    the second part is the expansion of space.

    but the second part just seems to reduce to what I already have above.

    I cant believe that such a simple looking problem has not only baffled me but even you guys arent sure about it.
    Last edited: Nov 16, 2011
  16. Nov 16, 2011 #15
    θ(t) is teh total angular distance traveled.

    it seems to me that if you know the radius then it should be trivial to figure out teh total distance
  17. Nov 16, 2011 #16


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    That 'correction' is dead wrong. The original analysis is correct, even it SEEMS paradoxical. s(t) is supposed to be the displacement from the original point at time t. Remember the space behind the ant is growing as well as the space in front of the ant. The ant will circle the balloon. Very slowly if n is large, but it will.
  18. Nov 16, 2011 #17
    so even though a point opposite the origin of the ant is indeed receding from teh ant
    it will recede slower and slower till it begins to move toward the ant?
    Last edited: Nov 16, 2011
  19. Nov 16, 2011 #18
    how can that be?

    the rate at which a point opposite the origin of the ant is receding depends only on the amount of space between it and the ant.
    The amount of space is increasing.
    Therefore the rate at which it is receding should increase not decrease.
  20. Nov 16, 2011 #19


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    The 'c' is a red herring. This isn't a relativity problem. Even it it were there are probably regions of our universe that are receding from us faster than light. We just haven't seen them yet. And may not ever, if the universe is really in accelerated expansion.
  21. Nov 16, 2011 #20


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    Slow down. You worked it out correctly. You already pointed out the flaw in reasoning like that in post 13. You have to consider the expansion of the space behind the ant as contributing to his progress.
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