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Instantaneous velocity doesn't actually exist?

  1. Sep 12, 2010 #1
    can anyone verify my thought process here?

    so, instantaneous velocity is like average velocity in that it is a slope between two points on a graph of position as a function of time

    but the two points in a problem with instantaneous velocity are made to be extremely close, almost the same point in fact (a derivative)

    so in the end, our result for instantaneous velocity is really just an approximation (as are all derivatives)

    is this right? it seems to have some philosophical implications that we don't get an exact identity...

    also, since i am already making a post--is the difference between average speed and average velocity just that average speed is the absolute value of average velocity?

    thanks in advance
  2. jcsd
  3. Sep 12, 2010 #2
    No instantaneous velocity does exist in the way in which you are referencing. The whole idea of calculus is the idea of convergence. As you shrink your interval to a smaller and smaller value it does converge, exactly, to a value -- that value, in this example, is the instantaneous velocity.
  4. Sep 12, 2010 #3


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    absolutely not. Average velocity is displacement/time; instantaneous velocity is rate of change of dispalcement with respect to time.
    No! Average speed is distance travelled/time; average velocity is displacement/time. If you travel from A to B, a distance of 10 m, in 2 seconds, and then back again from B to A, in 2 seconds, your average speed is 20/4 = 5m/s. Your average velocity is 0 , since you haven't displaced from your original position.
  5. Sep 12, 2010 #4
    thanks for the reply--but doesn't a limit never actually get to the value (for example, in a limit where x-->a, x never actually equals a?) you can have a graph where the limit is x approaching 5, but for the value of 5 the function f(x) can be anything, like 113 (i think)
  6. Sep 12, 2010 #5
    ok, but isn't the way to calculate the rate of change of displacement with respect to time to use the derivative of the slope (dx/dt), conceptually minimizing the distance between the two points to almost zero?
  7. Sep 12, 2010 #6
    Yes, correct. I don't think you were totally wrong. You can liken average velocity and instantaneous velocity because they have the same definition, that is, the rate of change of displacement with respect to time. Only in the latter, the slope must be taken as an infinitesimal change in displacement divided by an infinitesimal change in time.
    Last edited: Sep 12, 2010
  8. Sep 12, 2010 #7


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    That is incorrect. They do not have the same definition. Average velocity is final displacement/time. Instantaneous velocity is the rate of change of displacement with respect to time (ds/dt). Suppose a particle moves in a straight line easterly 10 meters at constant a speed of 5m/s. So it takes 2 seconds to travel the 10 m. Its instantaneous velocity at t=2 seconds is 5m/s east. It's average velocity over that time is also 5 m/s east. In this case, they are the same.

    But suppose that particle is moving in a straight line easterly and accelerating at a constant rate of 1m/s^2 as it travels, starting from rest. Its instantaneous velocity after say 2 seconds is v=at = (1)(2) = 2m/s (exactly) east, and it will have travelled s =1/2at^2 = 2 m east in that time. (Note that ds/dt = at, it's instantaneous velocity). But its average velocity over that period is final displacement/time = 2/2 = 1 m/s east. They are not the same.
  9. Sep 12, 2010 #8


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    You misunderstand: a derivative is an exact, single-point slope of a curve - in this case, an instantaneous velocity. That's one of the main points of a derivative. You can see that once you've calculated a derivative, you have a new equation in which you enter one "x" value.
    Yes, but a derivative doesn't have any limits in it!
  10. Sep 12, 2010 #9
    But if you were to find the slope of the displacement vs time curve at the point t=2 seconds and you took dx to be infinitely small, and also dt to be infinitely small, you would also get 2m/s. You can arrive at the correct answer for instantaneous velocity using the concept of a slope... so in that sense you can liken them. But they certainly don't have the same definition, I don't know why I said that.
  11. Sep 13, 2010 #10
    No. The result is exact. The simplest example I can think of is that .9+.09.+.009... is exactly equal to 1. This can be proven using mathematical induction.

    Regarding the OP, I think you are confusing an integral with a derivative.
  12. Sep 15, 2010 #11
    ok, i think i have a new but related question about this topic--when you take the derivative (dx/dt) and you let t-->0, doesn't the change in distance also end up going to 0? i know i'm misunderstanding something but it just seems like shrinking a value to almost nothing--how does it help?
  13. Sep 15, 2010 #12
    oh, and rebound i'm not sure i understand why it's an exact value--isn't the idea to work with two points that are extremely close together so a curve looks like a line? but we're still working with slopes and spaces between two points, right? unless t actually = 0, which is not only undefined but also what i'm confused about in my above post...
  14. Sep 15, 2010 #13
    Yes, the change in distance is negligible, but once you divide it by the negligible change in time, the result can be very big. For example 0.000000001 / 0.0000000008 is 1.25.
  15. Sep 15, 2010 #14
    Interesting thought but a little late. Zeno of Elea, born about 490 BC was the first to discuss this idea.

    From Wikipedia:
    In the arrow paradox (also known as the fletcher's paradox), Zeno states that for motion to occur, an object must change the position which it occupies. He gives an example of an arrow in flight. He states that in any one instant of time, for the arrow to be moving it must either move to where it is, or it must move to where it is not. However, it cannot move to where it is not, because this is a single instant, and it cannot move to where it is because it is already there. In other words, in any instant of time there is no motion occurring, because an instant is a snapshot. Therefore, if it cannot move in a single instant it cannot move in any instant, making any motion impossible.
  16. Sep 15, 2010 #15
    ok, yes i see! but what is the resolution of Zeno's paradox? in one of Feynman's lectures (using the other example in which Achilles can run 10 times as fast as the tortoise but he can never catch the tortoise) he says "although there are an infinite number of steps (in the argument) to the point at which Achilles reaches the tortoise, it doesn't mean that there is an infinite amount of time."

    I sense that the answer is in there, i'm just not sure what it means...
  17. Sep 15, 2010 #16
    There is no paradox.
    A "snapsot" removes a fundamental component of reality... time, from flowing.
    As such, the total system is not perceived correctly. Thus invalid, and can portend rather bizarre conceptual results which do not fit reality at all.
  18. Sep 15, 2010 #17

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    Zeno talked about several paradoxes. The arrow paradox in post #14 is distinct from the Achilles and tortoise paradox in post #15. The key thing to remember about any paradox (philosophical paradoxes in particular) is that if the paradox deduces that the impossibility of some event that obviously does occur, the problem lies in the thinking that led to the deduction rather than the obvious event.

    The resolution to the arrow paradox, "In other words, in any instant of time there is no motion occurring, because an instant is a snapshot," is non sequitur, or in less polite terms, utter BS.

    The resolution to the Achilles versus tortoise paradox is that Zeno did not comprehend that an infinite sum can converge to a finite value. That shouldn't be too surprising given that the necessary mathematics to understand that concept didn't exist in Zeno's time.
  19. Sep 15, 2010 #18
    I really can't answer your question any better than this without teaching you calculus from scratch.

    [Edit] Sorry. Try this page though, http://en.wikipedia.org/wiki/Fundamental_theorem_of_calculus
    Last edited: Sep 15, 2010
  20. Sep 15, 2010 #19
    As I read somewhere, Zeno just missed discovering calculus. Instead of taking the lim ds/dt → 0, he went directly to dt = 0 and asked what is the difference between a moving arrow and a stationary arrow at one instant in time. That question wasn't answered until Einstein revealed that the moving arrow is slightly foreshortened.
  21. Sep 15, 2010 #20
    ok i think i'm with it, finally. infinite sum, finite value. thank you all for the insight
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