Physics
What makes these things work?
Pressure differential. Plain and simple. If you have an object in the middle of a pressure differential it will move at a rate determined by many factors. In the tank you have a high pressure, outside you have low pressure, in the middle you have your projectile, residing in the barrel. When the valve opens the pressure wants to equalize with the least resistance possible, and that way is through the barrel, and guess what's in the middle? Thats right, the projectile. What the barrel does, is keep the high pressure behind the projectile for a given amount of time thereby increasing acceleration. The more air that can get to the projectile the faster it goes, the bigger the barrel i.d. the more force is exerted on it (remember pounds per SQUARE INCH) the more force the more acceleration (remember force=mass x acceleration or better put for this case acceleration = force/mass). There are limits to how far you can go before velocity starts to drop off. What I mean is, you can only have so long a barrel for a given chamber, or for that matter once you reach a certain speed all the extra barrel length will do nothing but slow the projectile down. The barrel drag is a constant, but the force the projectile sees keeps getting smaller with time, so at a certain point the barrel drag will overcome the force pushing it and start slowing the projectile down. Another thing to note is air drag, the faster you push the projectile the more air drag becomes, think of it like this, stick your arm out the window of a car at 20 mph, not much. Now get the car moving 90 mph, get the picture.
Forumulae
area of a circle = 3.14 x radius^2
volume of a cylinder = area of the circle x length of cylinder
chamber to barrel ratio = volume of chamber / volume of barrel
pressure at the muzzle of barrel = pressure in chamber x volume in chamber
volume of barrel +volume in chamber
average pressure of gun = chamber pressure + pressure at the muzzle
2
average force acting on the projectile = average pressure of the gun x area of projectile
kinetic energy of the projectile = weight in pounds x velocity in fps^2
64.4