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Math Help - Prove folk saying false

  1. #1
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    Prove folk saying false

    There is a common folk saying that "As long as you keep moving you will eventually get there." Show that this is false by giving an example of a motion for whic velocity is always positive, but for which total distance traveled, even in all eternity, is less than any arbitrarily prescribed positive number.

    All help appreciated!
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  2. #2
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    Quote Originally Posted by Alice96 View Post
    There is a common folk saying that "As long as you keep moving you will eventually get there." Show that this is false by giving an example of a motion for whic velocity is always positive, but for which total distance traveled, even in all eternity, is less than any arbitrarily prescribed positive number.

    All help appreciated!
    I am presuming you have learnt improper integrals.

    Look at, for example, a function like v(t) = e^{-t} which plots the velocity of an object against time.

    Now, you will be aware that the displacement is given by the area under the graph (which can be found by integrating the function). Then, as time approaches infinity, we can evaluate the distance travelled:

    \int_0^{\infty} e^{-t} dt = \lim_{t \to \infty} \int_0^t e^{-t} = \lim_{t \to \infty} [-e^{-t}]_0^t = 0 + 1 = 1

    So even as time becomes infinite, the velocity remains positive (though infinitely small) and the distance travelled remains finite.

    We can confine the finite distance to any arbitrary value \alpha by using the velocity function:

    v(t) = \alpha e^{-t}

    It is also worth noting that there are other functions that also have finite areas (and v is positive for all t) for example:

    v(t) = \frac{1}{(x+1)^2}
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  3. #3
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    Quote Originally Posted by Alice96 View Post
    There is a common folk saying that "As long as you keep moving you will eventually get there." Show that this is false by giving an example of a motion for whic velocity is always positive, but for which total distance traveled, even in all eternity, is less than any arbitrarily prescribed positive number.

    All help appreciated!

    Using infinite series, suppose an object travels at second n the distance \frac{1}{n^2} meters, so travelling "for all the eternity" (i.e., when n increses to infinite) we get

    \sum\limits_{n=1}^\infty\frac{1}{n^2}=\frac{\pi^2}  {6} meters

    If you want to make the above less than any prescribed number \epsilon>0 then just make the object advance \frac{\epsilon}{2n^2} meters every second...

    Tonio
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