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Using average

  1. May 13, 2008 #1
    Now we've all been taught how to use the average. Let me give 2 examples to those who don't know.

    Example 1: Say an object moves with velocity 3t in the time t=0 till t=2. Find distance covered.
    Initial velocity = 0.
    Final velocity = 6 disp. unit/ time unit.
    Avg. Velocity = 3 disp. unit/ time unit.
    Distance covered = Avg. velocity x time = 6 disp. units.
    Using s = ut +1/2 a[tex]t^{2}[/tex] we get 6 again. Amazing!

    Example 2: Force acting on a box of mass 1 unit is 3t in the time t=0 till t=2. Find work done by the Force. Box is initially at rest to your frame.
    No other forces act on it.

    Initial force = 0
    Final force = 6 units.
    Avg. force = 3 units.
    Now avg. accn. = 3 units [mass = 1]
    As in previous sum, displacement = 6 units.
    Work done = 3 x 6 = 18 units. This comes out fine if you work it out the normal way also.

    Now onto my questions.

    If you noticed both were linear variations. How do I find the average of any polynomial function? I would find that VERY useful. For instance I found out for a cos/sin function average is 1/[tex]\sqrt{2}[/tex] of the co-efficient of the cos function. Isn't that fantastic?

    Also one more. I was given a problem that the charge density of a sphere varies as [tex]\beta[/tex]t. But when I tried average, it doesn't work although it seems to be a linear variation.

    Why doesn't it work?
     
  2. jcsd
  3. May 14, 2008 #2
    Can you expand a bit on the problem at hand?
     
  4. May 14, 2008 #3

    berkeman

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    Staff: Mentor

    You use integral calculus in the general case. Just like a simple average is the sum of the elements divided by the number of elements, a generalized average is the quotient of two integrals.
     
  5. May 14, 2008 #4

    marcusl

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    Science Advisor
    Gold Member

    To calculate the average (also called mean), integrate the function over the range of interest and divide by that range.
     
  6. May 14, 2008 #5

    berkeman

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    Staff: Mentor

    For example, see the end of this:

    http://math.cofc.edu/lauzong/Math105/Section 5.4 Applying Definite Integral.pdf


    .
     
  7. May 16, 2008 #6
    I see. Thats awesome! Average is such a nice way of going about the problem. What about the sphere of charge? Why can't I average that?
     
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