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Pressure in an air cannon- ODE

  1. Oct 3, 2008 #1
    Pressure in an air cannon-- ODE

    1. The problem statement, all variables and given/known data
    Okay, I'm designing an air cannon, and need to calculate the muzzle velocity. The only propelling force is the one caused by the pressure of the compressed air.

    2. Relevant equations
    I've been working with Boyle's law:
    [tex]P_0 V_0 = P_1 V_1[/tex]

    3. The attempt at a solution
    From here, I can express pressure as a function of distance down the barrel
    [tex]P_1 = \frac{P_0 V_0}{V_1}[/tex]
    [tex]P_1 = \frac{P_0 (\pi r^2 x_0)}{\pi r^2 x} [/tex]
    here x represents the initial length of of the chamber inside the barrel where the pressure builds up plus the distance down the barrel.
    From this, I get:
    [tex] P = \frac{P_0 x_0}{x}[/tex]
    And working with the definition of pressure and Newton's second law:
    [tex] \frac{m a}{\pi r^2}=\frac{P_0 x_0}{x}[/tex]
    [tex]a = \frac{P_0 x_0 \pi r^2}{m x}[/tex]
    This is the part I can't get passed:
    [tex]x'' = = \frac{P_0 x_0 \pi r^2}{m x}[/tex]
    I haven't yet taken a course on differential equations, so I don't know how solve this: all of the values, except x, are constant, so there has to be a general solution to
    [tex]x'' = \frac{1}{x} [/tex]
    Can anyone tell me what it is?
  2. jcsd
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