[StevenB]

The topic of which motors to use for a shooter has been beaten to death in several threads, most often discussing CIMs vs Fisher-Price motors. A dozen teams have posted videos of shooters, and there are tons of pictures from 2006 which also involved shooting foam balls. Take a look at those and you'll at least get a good reference point. The things I've seen most commonly are Fisher-Price motors with gearing around 1:5 and CIMs around 1:1.

The best thing you can do is to build something quickly and test it. There are tons of variables involved: the mass of your wheels, the squeeze you put on the balls, the contact time, the shape and material of your wheels, etc. You really won't know how it will perform until you build it and shoot some balls, and you can't easily use physics to figure out what ratio you need.

Also, I would caution you against deciding on a wheel size and designing the rest of the shooter around them. It's quite possible that you'll get a particular gear ratio and have to work the wheel size around that. You won't necessarily be able to get the ideal ratio, but you may be able to vary the size of your wheel.