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Homework Help: Motion in a force field

  1. Aug 18, 2006 #1
    I wasn't sure whether to put this in the math section or the physics section because it's a bit of an overlap problem.

    I want to know how to find the position as a function of time of a particle given its acceleration as a function of position. I know this is some sort of differential equation but I'm confusing myself with it.

    Any advice?
     
  2. jcsd
  3. Aug 20, 2006 #2

    benorin

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    If a(t) and x(t) are the acceleration and position of the particle as a function of time, then x''(t)=a(t)

    Edit: What the [tex]\frac{d^2x}{dt^2}=a(t)[/tex] is up with the LaTeX?
     
    Last edited: Aug 20, 2006
  4. Aug 20, 2006 #3
    You're not understanding what I'm asking.

    You are not given a(t). You are given a field assigning a force or acceleration to every point in space.

    Also, you're given the particle's initial position and velocity.

    The goal is to find the position as a function of time.

    To make things simple, let's just look at this situation for motion along a line.
     
    Last edited: Aug 20, 2006
  5. Aug 20, 2006 #4

    quasar987

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    What Newton's second law says is that given the force field [itex]\vec{F}(\vec{r})[/itex] asigning a force to every point in space, the path [itex]\vec{r}(t)[/itex] of a particle of mass m in this force field is a solution of the differential equation: (or rather of the 3 following coupled ode:)

    [tex]\vec{F}(\vec{r(t)}) = m\frac{d^2\vec{r}}{dt^2}(t)[/tex]
     
  6. Aug 20, 2006 #5
    As quasar said, you just solve the system
    of differential equations. In general you're going
    to have to do it numerically.

    For simple 1-D potentials, though, you use
    a standard trick

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

    So for the differential equation
    ma = f(x)
    we have
    m v (dv/dx) = f(x).
    mv^2/2 - mv0^2 /2 = integral(f(s), s=x0..x) .

    Now put g(x) = sqrt( v0^2 + 2/m int(f(s), s=x0..x) )
    so that we gave
    v = g(x)

    then you have v = dx/dt = g(x)
    which is seperable so that

    int( 1/g(s), s = x0.. x) = t - t0

    you invert this to get x = x(t).
     
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