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I'm trying to model how an object of mass M will accelerate for a given pressure. I'm assuming that no gas escapes and that the temperature doesn't change, making both R and T constant. Here's my work so far:

$W=nRT * ln(\frac{V2}{V1})$

$W = F * ΔD$

$F * ΔD = nRT * ln(\frac{V2}{V1})$

$F = \frac{nRT * ln[\frac{D2}{D1}]}{ΔD}$

$F = M * A$

$A = \frac{nRT * ln[\frac{D2}{D1}]}{ΔD * M}$

This should be the acceleration as the mass leaves the barrel, and as D2 is moved closer to D1, the acceleration at each point should be given. This is supported by the fact that as D2 approaches D1, the acceleration approaches 0. Does this equation check out?

Thanks!

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You are modeling the expansion process as being both isothermal and reversible. For the process to be reversible the expansion needs to occur very slowly . That means the acceleration would have to approach zero.

Hope this helps

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  • $\begingroup$ That seems to make sense. Do you have any idea how to model a system that isn't at equilibrium then? If not, what simplifying assumptions would be acceptable to make the system model-able? $\endgroup$
    – Eric Jestel
    Nov 21, 2021 at 22:19
  • $\begingroup$ @EricJestel, how "realistic" do you want your model to be? If you assume constant acceleration, a kinematic equation will work. However, if you want to model the burning of gun powder, pressure inside the gun barrel, etc., you are going to need a lot of information and will probably have to use a numerical model. $\endgroup$
    – David White
    Nov 21, 2021 at 22:34
  • $\begingroup$ @EricJestel I can only address your model from a thermodynamic standpoint. But if you want rapid expansion, I would model it as an irreversible adiabatic expansion. $\endgroup$
    – Bob D
    Nov 21, 2021 at 22:48
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As the air cannon works quickly, you can reasonably assume that no heat is exchanged with the surroundings --- in which case the expansion adiabatic so $PV^\gamma$ is constant.

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