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A(t), v(t), r(t) converting, integration and derivatives

  1. Feb 22, 2012 #1
    1. The problem statement, all variables and given/known data

    Given that the acceleration vector is
    a(t) = (-16cos(-4t))i + (-16sin(-4t))j + -2tk,
    the initial velocity is v(0) = i + k
    and the initial position vector is r(0) = i + j + k, compute:
    The velocity vector v(t) = ___i + ____j + ____k
    The position vector r(t) = ___i + ____j + ____k
    2. Relevant equations

    v(t) = r'(t) = ∫a(t)dt

    3. The attempt at a solution

    v(t) = ∫a(t)dt = ∫a(t) = (-16cos(-4t)) i + (-16sin(-4t)) j + -2t k dt
    v(t) = 4sin(-4t) i + (-4cos(-4t) j + (-t^2) k

    r(t) = ∫v(t) = ∫4sin(-4t) i + (-4cos(-4t)) j + (-t^2) k dt
    r(t) = -cos(-4t) i + sin(-4t) j + (-1/3)t^3 k

    I've double checked my integration but can't figure out what I did wrong. None of the parts of my answers are correct.
  2. jcsd
  3. Feb 22, 2012 #2


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    Science Advisor
    Homework Helper

    No, you left out the arbitrary constants of integration,

    v(t) = ∫a(t)dt = ∫a(t) = (-16cos(-4t) +C1) i + (-16sin(-4t) +C2) j + (-2t +C3) k dt

    Use the value of v(0) they gave you to figure out what the constants are.
  4. Feb 22, 2012 #3
    Ok, thanks so much! I got it now.
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