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Homework Help: Changing the Variable of Integration

  1. Feb 2, 2010 #1
    1. The problem statement, all variables and given/known data

    Suppose the acceleration of a particle is a function of x, where ax(x) = (2.0 s-2)x.

    (a) If the velocity is zero when x = 1.0 m, what is the speed when x = 2.7 m?


    (b) How long does it take the particle to travel from x = 1.0 m to x = 2.7 m?


    2. Relevant equations

    a = integral v dt = integral (integral (x)) dt


    3. The attempt at a solution

    This CAN be solved as a differential equation, but we haven't done those in my Calc course yet, so I have no idea how to solve it that way.

    On the other hand, I know the problem is that acceleration is a function of x, hence a(x), and that it needs to be a function of time in order to change it over to velocity and then displacement (if needed). So, I tried figuring that out and go to this point:

    a = [tex]\frac{dv}{dt}[/tex] = ([tex]\frac{dv}{dx}[/tex] * [tex]\frac{dx}{dt}[/tex])

    [tex]\int\frac{dv}{dx}[/tex] = [tex]\int2x[/tex][tex]\frac{dx}{dt}[/tex]

    v = x2 + c

    Do I have this set up correctly? And, if so, wouldn't the integral (and thus the velocity function) end up being x2 + some constant? And would that constant be related to part b, or inherited from the given info?
     
  2. jcsd
  3. Feb 2, 2010 #2

    rock.freak667

    User Avatar
    Homework Helper

    Well you have that

    a= (dv/dx)(dx/dt) and dx/dt is v.

    So really you have that

    v(dv/dx)=2x

    or v dv = 2x dx

    so integrate it now.

    Then yes, you must use the conditions given in part a to get the constant of integration.
     
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