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Step Force

  1. Jul 19, 2008 #1
    [solved] Step Force

    Never mind, i think i figured it out
    1. The problem statement, all variables and given/known data
    Suppose a body of mass m is subject to a constant force [tex]F_1[/tex] that acts for a length time [tex]t_1[/tex] and then the force suddenly changes to a constant [tex]F_2[/tex]

    Show [tex]v(t) = v_0 + \frac{F_1 t_1}{m} + \frac{F_2(t-t_1 )}{m}[/tex]
    [tex]x(t) = x_0 + v_0 t_1 + \frac{F_1 {t_1}^2}{2m} + (v_0 + \frac{F_1 t_1}{m}(t-t_1 ) ) + \frac{F_2}{2m}(t-t_1 )^2 [/tex]

    2. Relevant equations

    3. The attempt at a solution

    I'm having a hard time with the step force. Here's how I get v(t)

    [tex]\int_{v_0}^v dv = \int_0^{t_1} F_1/m dt + \int_{t_1}^{t} F_2/m dt[/tex]
    [tex]v(t) = F_1/m t_1 + F_2/m (t - t_1) + v_0[/tex]
    Which is fine, but then when it gets to x(t) is where I am having problems:

    [tex]\int_{x_0}^{x} dx = \int_{0}^{t_1} F_1/m t_1 dt + \int_{t_1}^{t} F_2/m (t-t_1) dt + \int_0^{t_1} v_0 dt + \int_{t_1}^t v_0 dt[/tex]

    Which yields:
    [tex]x(t) = x_0 + v_0 t_1 + (v_0)(t-t_1) + F_1/m {t_1}^2 + \frac{F_2 (t-t_1)^2 }{2m}[/tex]

    I know that in order to have a step function you need to get rid of [tex]F_1[/tex] after time t_1 because it no longer acts on the object. So I could just add [tex]F_1 t_1 / m (t-t_1)[/tex] but I'm not really sure where the [tex]F_1 t_1/m (t)[/tex] force would come into play.

    So my question is: Am I doing the analysis correctly? It seems like taking care of the step function is proving to be a difficultly.
    Last edited: Jul 19, 2008
  2. jcsd
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