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Simple dynamics problem, I can't seem to get the answer to

  1. Feb 21, 2009 #1

    rock.freak667

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    Homework Helper

    1. The problem statement, all variables and given/known data

    A car travels around a circular track having a radius of 300m such that when it is at point A, it has a velocity f 5m/s, which is increasing at the rate of [itex]\dot{v}=0.06t m/s^2[/itex]. Determine the magnitudes of the velocity and acceleration when it has traveled one-third the way around the track

    2. Relevant equations

    n,t-coordinate system

    [tex]a=\sqrt{a_t^2+a_n^2}[/tex]

    [tex]a_n= \frac{v^2}{r}[/tex]

    3. The attempt at a solution
    Since the radius r=300m, the total distance the car will travel is [itex]2 \pi r= 600\pi m[/itex]

    So I want to find v and a when the distance = 200pi

    Now at A, [itex]\dot{v}=a_t=0.06t[/itex]
    Initially at A,t=0 and v=5

    so

    [tex]\int^{v} _{5} = \int^{t} _{0} 0.06t dt[/tex]
    [tex]v=5+0.03t^2[/tex]

    Thus
    [tex]\int ^{s} _{0}= \int ^{t} _{0} (5+0.03t^2) dt[/tex]

    [tex]s=5t+0.01t^3[/tex]

    When [itex]s=200\pi[/itex]

    [tex]200\pi=5t+0.01t^3[/tex]

    Which I do not know how to solve since there is no rational root.

    Was I going on the correct track?
     
  2. jcsd
  3. Feb 21, 2009 #2

    Doc Al

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    Staff: Mentor

    There is a real solution. (Yes, you're on the right track.)
     
  4. Feb 21, 2009 #3

    rock.freak667

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    Homework Helper

    Well I had to use an online calculator to get t=35.579993691668676.

    But my normal calculator doesn't have a function for cubic equations. How would I normally solve it without a computer? I know the rational root theorem but if I were to use the newton-raphson method, I'd spend a lot of time finding a starting point and doing the iteration.
     
  5. Feb 21, 2009 #4

    Doc Al

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    Staff: Mentor

    Solving a general cubic equation by hand is a bear. There's an analytic solution, but I wouldn't dare attempt it from memory. (I too have been spoiled by fancy calculators.) Here's one version: http://mathworld.wolfram.com/CubicFormula.html" [Broken]
     
    Last edited by a moderator: May 4, 2017
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