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I'm building a pneumatic airgun and I'm trying to find out what pressure/barrel lenght I have to use for a given muzzle velocity. I've done a bit of number crunching but I keep coming up with insane barrel lenghts, and that's not even considering any friction. Here's what I did:

I want to accelerate a 0.2 gram 6mm bb to a velocity of 300m/s using compressed air.

I take a pressure difference of 30 bar (3,0 * 10^6 Pa) so a total pressure of 31 bar and a surface area of 2.83 * 10^ -4 m^2. P = F/A, therefore F = PA

3 * 10^6 * 2.83 * 10^ -4 = 84,9N, so when I initially pull the trigger the air pressure is pushing my bb forward with 84,9N of force. If the pressure behind the bb at the moment I pull the trigger is 31 bar and at the moment it exits the barrel is 1 bar, this leaves me with an average force of 84.9/2 = 42,45N.

a = F/M (where a = acceleration), F is 42.45 and M = 2 * 10^ -4, 42.45/2 * 10^ -4 = 212250 m/s^2

At that rate of acceleration, my bb reaches 300 m/s after accelerating for 0.0376 seconds. The thing is, by then it has already traveled 300/2 * 0.0376 = 5.64 meters, which is a bit unpractical.

Modern air rifles use a lot less air to accelerate much heavier projectiles to similar velocities with 50/60cm barrels.

What am I doing wrong here?