Author: **SpudBlaster15** » Fri Nov 11, 2011 8:53 pm

Pressure is force divided by area (In this case, cross sectional area), or P = F/A. Rearranging the equation and combining it with F = m*a, you get a = P*A/m for the acceleration of the projectile at constant pressure.

Unfortunately, this expression is not immediately useful for modelling launcher performance, as the pressure on the base of the projectile is not constant (Barring an exceptional case such as a tanker truck sized chamber connected to a pen tube barrel, where the pressure drop is negligible). The simplest way to work around this is to use average barrel pressure to find an average acceleration. In an adiabatic system, the final pressure of a decompressed gas is given by P<sub>final</sub> = P<sub>initial</sub>/(V<sub>final</sub>/V<sub>initial</sub>)<sup>1.4</sup>. From this, the pressure at the muzzle can be calculated, which allows an average value to be determined from the initial chamber pressure.

This type of approach isn't very accurate, but it will get you in the ballpark.

A much more accurate method is to use an iteration based model. For this, the calculations are performed in a series of time steps, where the projectile acceleration, displacement and change in velocity are computed for a fixed pressure over a very short time interval (0.1ms or so), then a new pressure is calculated based on the decompression of the gas over the displacement that occurred during the previous step, and the cycle is repeated until d<sub>total</sub> = l<sub>barrel</sub>. At the end of the cycle, the total velocity change is equal to the muzzle velocity of the projectile.

The most effective approach (And the one used in GGDT) is to design a system of ordinary or partial differential equations to model the conditions inside the launcher, and solve the system using numerical methods. You probably won't be using this technique, as it requires a fairly significant background in CFD.

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