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Formulas for constant power acceleration

  1. Sep 29, 2012 #1

    rcgldr

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    This isn't homework. A friend asked about this so I decided to work out the formulas, but wanted to know if this was already done by someone here (otherwise I'll do the math).

    p = power (constant)
    a = acceleration
    v = velocity
    x = position
    t = time
    f = force

    Assume an object is initially at rest, at position zero and time zero:

    v0 = 0
    x0 = 0
    t0 = 0

    f = m a
    p = f v
    f = p / v

    first step

    a = f / m = dv/dt = p / (m v)
    v dv = (p/m) dt
    1/2 v2 = (p/m) t

    [tex] v = \frac{dx}{dt} = \sqrt {\frac{2\ p\ t}{m}} [/tex]

    This is continued to find x as a function of t, then t as a function of x

    Then determine f(x) = p / v(x)

    and finally show that work done is

    [tex]p\ t_1 = \int_0^{x_1} f(x) dx [/tex]
     
  2. jcsd
  3. Sep 30, 2012 #2

    mfb

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    You can simply integrate your equation for v(t) and get x(t). That can be used to get v(x) and the other expressions.
     
  4. Sep 30, 2012 #3

    rcgldr

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    I know that, was just wondering if someone here had already done this in a previous thread. The previous threads I did find never actually completed the formulas. I'll go ahead and do this later.
     
  5. Sep 30, 2012 #4

    mfb

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    I am sure this has been done before, it is a nice and easy problem in mechanics and can be solved with very basic concepts.
     
  6. Sep 30, 2012 #5

    rcgldr

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    [tex] p = f \ v [/tex]
    [tex] a = \frac{dv}{dt} = \frac{f}{m} = \frac {p} {m\ v} [/tex]
    [tex] v\ dv = \frac{p}{m}\ dt [/tex]
    [tex] \frac{1}{2}\ v^2 = \frac{p}{m}\ t [/tex]
    [tex] v = \frac{dx}{dt} = \sqrt {\frac{2\ p\ t}{m}} [/tex]
    [tex] dx = \sqrt {\frac{2\ p\ t}{m}}\ dt [/tex]
    [tex] x = \sqrt {\frac{8\ p\ t^3}{9\ m}} [/tex]
    [tex] t = \sqrt[3] {\frac{9\ m\ x^2}{8\ p}} [/tex]
    a as a function of t:
    [tex] a = \frac {p} {m\ v} = \frac {p} {m\ {\sqrt {\frac{2\ p\ t}{m}}}}
    = \sqrt {\frac{p}{2\ m\ t}}[/tex]
    v as function of x:
    [tex] v = \sqrt {\frac{2\ p\ \sqrt[3] {\frac{9\ m\ x^2}{8\ p}}}{m}}
    = \sqrt[3] {\frac{3\ p\ x}{m}} [/tex]
    a as a function of x:
    [tex] a = \frac{p}{m \ v} = \frac{p}{m \ \sqrt[3] {\frac{3\ p\ x}{m}}}
    = \sqrt[3] {\frac{p^2}{3\ m^2\ x}} [/tex]
    f as a function of x:
    [tex] f = m\ a = \sqrt[3] {\frac{m\ p^2}{3\ x}} [/tex]
    work done versus x:
    [tex] w = \int_0^x \sqrt[3] {\frac{m\ p^2}{3\ x}} \ dx
    = \sqrt[3]{\frac{9\ m\ p^2\ x^2}{8}} [/tex]
    work done versus time:
    [tex] w = \sqrt[3]{\frac{9\ m\ p^2\ \left( \sqrt {\frac{8\ p\ t^3}{9\ m}} \right )^2}{8}}
    = p \ t [/tex]
     
    Last edited: Oct 1, 2012
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