Forces in a paintball gun

You guys are going around in circles a bit and the approaches both work (though the second, not quite as described): if you know the force and the distance (we were given the distance, we'd need the CO2 pressure and how/if it varies as the ball moves through the barrel), you easily calculate the work and since work is equal to kinetic energy, you can use the kinetic energy equation to calculate the final speed. In my opinion, that is the most direct method.

Alternately, you can use newton's mechanics equations, a=f/m, s=at, and d=st, with some integration, to find the same thing. You don't know the time: you have to calculate it.

 

In any case, you need the mass of the paintball.

I would use the KE approach and just assume that it is accelerated by a constant force down the entire length of the barrel. That will surely underestimate the true maximum force, but should be a decent first-order approximation.

If you want a more advanced approximation then you can consider the CO2 to be an adiabatically expanding ideal gas and assume that no CO2 leaks around the ball in the barrel. That should give you a force that varies down the barrel length in a reasonable manner.

 

Here is the Hyperphysics page with the equation for an http://hyperphysics.phy-astr.gsu.edu/hbase/thermo/adiab.html#c1".

You know the total work done on the ball so start with an initial guess for the starting volume, add the volume of the barrel to get the final volume, and use that to solve for K. Then you can use K to solve for the pressure as a function of the volume, which will give you the force on the paintball as a function of distance down the barrel.