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[DYNAMICS] Acceleration as a function of position.

  1. May 30, 2013 #1
    1. The problem statement, all variables and given/known data

    Initial velocity = 17 m/s
    Radius of curvature = 13 m
    Acceleration as a function of position = -0.25s m/s^2
    Time elapsed = 2 s

    Solve for the magnitude of the acceleration.

    2. Relevant equations

    a = v (dv/ds)

    a^2 = a(t)^2 + a(n)^2, wherein a(t) is tangential acceleration and a(n) is normalized acceleration.

    3. The attempt at a solution

    a*ds = v*dv

    int(-0.25s*ds) = int(v*dv)

    v^2 = -0.25s^2

    From this point I have hit significant difficulty, as you cannot take the square root of a negative number. Presumably there is another method to solve this equation that I have neglected, but it hasn't occurred to me, and my professor is notoriously poor at returning emails. Any thoughts?
     
  2. jcsd
  3. May 30, 2013 #2

    rcgldr

    User Avatar
    Homework Helper

    Shouldn't that be

    1/2 v^2 = c - 0.25s^2

    where c is a constant of integration. Assuming c is greater than .25s^2, then 1/2 v^2 is positive.

    Also, you might consider using another name for position or distance, to avoid confusion since s is often use to represent time in seconds.

    1/2 v^2 = c - 0.25 d^2
     
  4. May 30, 2013 #3
    Understood. It would then be 1/2 v^2 = C - 0.125 d^2, would it not? Let me try it out.
     
  5. May 30, 2013 #4
    Unless I have misinterpreted something horribly, this was my process:

    s denotes position.
    t denotes time.

    -0.25s ds = v dv
    C -0.125s^2 = 1/2 v^2
    When s = 0, v = 17.
    C = 1/2 (17)^2
    C = 144.5
    144.5 - 0.125s^2 = 1/2v^2
    289 - 0.25s^2 = v^2
    v = sqrt(289 - 0.25s^2)
    ds / dt = sqrt(289 - 0.25s^2)
    (289 - 0.25s^2)^(-1/2) ds = dt
    2*sin^-1(0.029412s) = t
    t = 2
    s = 34sin(1)
    s = 28.61

    v = sqrt(289 - 0.25s^2)
    v = sqrt(84.367)
    v = 9.185

    a(n) = v^2 / ρ
    v = 9.185
    ρ = 13
    a(n) = 6.48956

    a(t) = -0.25s
    s = 28.61
    a(t) = -7.1525

    a = sqrt(a(n)^2+a(t)^2)
    a = sqrt(42.114+51.1583)
    a = 9.66

    The answer was correct!

    Many thanks to you.
     
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