[Tom Line]

Part of the reason you haven't gotten a reply is lack of identification. Team number, etc.

The other reason is that you're essentially asking people to engineer a system for you. I'll give you some help and give you some of the tools you need, but determining the answers are up to you.

How high is the basket?
How far away is the hoop?
How high will you be launching your ball from?
What speed is required to get your ball that far? (you need the 3 above questions answered to figure out this answer - ask your physics teacher).

Once you know how fast your ball needs to be fired, you can figure out how fast your wheels need to be spinning. Again - this is a basic physics question. Once you know how fast to spin your wheels, figure out what motor you'd like to use (the more powerful, the faster it will spin up to speed) then figure out the gear ratio required based on the 50% of the free speed.

If you need help with this, I'm sure any of your teachers will be able to provide assistance. It's all fairly basic newtonian physics.