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Methods for calculating average velocity

  1. Feb 15, 2015 #1
    1. The problem statement, all variables and given/known data
    A child is pushing a merry-go-round. The angle through which the merry-go-round has turned varies with time according to θ(t)=γt+βt3, where γ= 0.383 rad/s and β= 1.00×10−2rad/s3.

    Calculate the average angular velocity ωav−z for the time interval t=0 to t= 5.50s.

    2. Relevant equations
    Given

    3. The attempt at a solution
    Okay so I solved all the questions above this one which gave me the final and initial angular velocities and I got ωi = 0.383 rad/s and ωf = 1.29 rad/s.

    When I calculate average angular velocity, why can't I use (ωi + ωf)/2
    Instead, I have to use (Θf - Θi)/5.50 to get the correct answer. Shouldn't the first equation yield the same result because an average is the sum of the velocities divided by 2?
     
  2. jcsd
  3. Feb 15, 2015 #2

    Nathanael

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    Homework Helper

    Nooo this is not true at all!

    This only applies to one very special case: constant acceleration.

    The average speed (over an interval of time) is the constant speed which produces the same displacement (over the same interval). So if you imagine a graph like y(x) or v(t) then the average is the constant (a horizontal line) which produces the same area (over a certain interval of x or t) as the area under the actual curve.

    In the special case where y(x) or v(t) has a constant slope (a.k.a. constant acceleration) the average happens to be the midpoint of the line and thus is (y(a)+y(b))/2
    (You might like to convince yourself that the midpoint between to points on a graph with a constant slope is the only constant value which gives the same area between the two points.)

    I hope I'm not making it confusing, but words tend to do that with visual ideas. It is important to remember that the (y(a)+y(b))/2 or (vi+vf)/2 is just a special case which only applies for constant slope (constant acceleration).
     
  4. Feb 15, 2015 #3
    If the angular acceleration is constant, can (vi+vf)/2 be used?
     
  5. Feb 15, 2015 #4

    Nathanael

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    Homework Helper

    Yes. If and ONLY if the angular acceleration is constant.
     
  6. Feb 15, 2015 #5
    What do you think is the more general case of avg velocity?

    Think: if I went 100 miles in 2 hours, my avg was?
     
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