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Why does the Average angular velocity and Instantaneous angular velocity give different results?

  1. Nov 3, 2014 #1
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    Can somebody explain to me the reason why? Thanks!
     
  2. jcsd
  3. Nov 3, 2014 #2
    Chances are your velocity was changing. The instantaneous = the average only when it is absolutely constant.
     
  4. Nov 3, 2014 #3
    Say there was an object that was experiencing a varying angular acceleration decreasing over time to zero. Would that mean its velocity continues to increase/change over time?
     
  5. Nov 3, 2014 #4

    haruspex

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    The instantaneous is the average over an arbitrarily short time interval. In general, an average velocity can be over a substantial interval. In notation, d is used for an infinitesimal change, ##\Delta## for a general change. So ##\frac{\Delta x}{\Delta t}## is an average velocity over time ##\Delta t##, while ##\frac{dx}{dt}## is the instantaneous velocity.
     
  6. Nov 3, 2014 #5
    If the acceleration is nonzero, then velocity is changing, regardless of whether the acceleration varies. It is only when acceleration = 0 that velocity is constant.
     
  7. Nov 3, 2014 #6
    Ah I see. So because the velocity is changing over time, we get a curved position vs time graph (representing velocity) to show that its increasing. and so if we take the instant angular velocity at a single time it wouldnt equal with the average angular velocity (between two time intervals) because like you said, that velocity changes for every time, correct? Just wanted to make sure I follow what you're telling me.

    Thanks!
     
  8. Nov 3, 2014 #7
    It sounds like you've got it right.
    I'll provide an example for you.
    suppose you have velocities v=(1,2,3,4) at times t=(1,2,3,4)
    the average velocity over the whole time is 2.5.
    The average velocity over the first 3 seconds is 2.
    The instantaneous velocities are only defined at a point, so for instance at t=4, the instantaneous velocity is 4.

    make sense?
     
  9. Nov 3, 2014 #8
    Thanks you're the best!
     
  10. Nov 3, 2014 #9

    haruspex

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    Right, except there is certain to be some instant in the interval at which the instantaneous velocity equals the average over the interval (mean value theorem).
     
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