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Relationship between velocity, acceleration, and a circle?

  1. Jan 18, 2016 #1
    The perimeter of a circle is 2πR (R=radius). [ref]
    Acceleration = Δv/Δt (v=velocity, t=time). [ref]
    Motion mathematics can always be reduced to multiple independent one-dimensional motions. [ref]
    The distance an object travels while accelerating = vit + at2/2 (a=acceleration, vi=initial velocity). [ref]
    1. If a circle centered at (0,0) has a radius of 2m, then it has a diameter of 4m and a perimeter of 4π m.
    2. If an object is moving clockwise around this circle once every 4 seconds, then that object has a speed of 1π m/s.
    3. At the top of the circle, the object has an (x,y) velocity of (π,0); let this be t0.
    4. At the right of the circle, the object has an (x,y) velocity of (0,-π); let this be t1.
    5. Δt = t1-t0 = 1 second.
    6. Δv = v1-v0 = (π,0) - (0,-π) = (-π,-π).
    7. a = (-π,-π) / second (for that particular time interval).
    8. So displacement on the X-axis = π m/s * 1 s + ( (-π m/s2) * (1s)2 / 2 ) = π m + -π/2 m = π/2 ...
    But the object travelling the perimeter of the 2m-radius circle every 4 seconds should be at (2,0) 1 second after (0,2). What am I doing wrong - displacement, acceleration, or something else/more?


    (other references)
    - velocity calculator
    - kinematic equations
  2. jcsd
  3. Jan 18, 2016 #2
    Δv = v1-v0 = (π,0) - (0,-π) = (-π,-π).
    This is wrong.
    Δv= (π,π). By basic arithmetic, your pies are positive. 0 - - π = π, and π - 0 = π
  4. Jan 18, 2016 #3
    You can only use the equation ## s = ut + \frac 12 a t^2 ## when acceleration is constant.
  5. Jan 18, 2016 #4
    Thanks for catching that. I think that it should then be:
    Δv = v1-v0 = (0,-π) - (π,0) = (-π,-π)
    and that I had the v0 and v1 incorrectly switched; the outcome is correctly still (-π,-π).
  6. Jan 18, 2016 #5
    Ah, I was trying to take the average acceleration and treat it as constant.
  7. Jan 19, 2016 #6
    Plots of the actual component numbers show just how off an assumption of constant acceleration for each component is for this problem.

    Last edited: Jan 19, 2016
  8. Jan 19, 2016 #7
    Thank you very much! That relationship between position, velocity, and acceleration exposes quite a bit of my problem.

    [ref] For anyone reading this post later, acceleration is the derivative of velocity and velocity is the derivative of position. The graphs posted by @spamanon show all 3 of those for my circle (red position, blue velocity, black acceleration).
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