[AdamC]

Everybody else before me is correct. we're operating now with 2 1.5" cylinders, but there isn't much luck right now. Pneumatics operating with FIRST regulation parts and pressures are not strong enough to launch a trackball BY THEMSELVES! You can create a mechanical linkage powered by a cylinder, but just punching the cylinder will not work.
The main issue is the hoses we are required to use. The flow-rate through the hoses is not fast enough to fill the cylinder fast enough to "punch" it. You will apply a very powerful force to the trackball with a beefy cylinder, but it will not impart a powerful enough impulse against the ball.
If you want to crunch the numbers, the formula is: Area*Pressure = Force Output. To calculate area, you simply find the area of a the cylinder head (pi*.5bore^2).
You can crunch the numbers further with some kinematics equations to find the resulting velocity. If you choose to pursue the idea (which I highly recommend you don't), invest in a cylinder that will provide you with about 25% to 50% more force than you technically require to launch the ball the desired height.
I will GREATLY envy you if you manage to pull the pneumatic launcher off, and I'll want to see some pictures.