[mathking]

For what it's worth, my first impressions as a mathematician and former basketball coach... (and as a mentor who was around for Aim High)

Do some math first. You want to work out trajectories that bring the ball down in the center of the rim. The amount of arc is a function of the height above or below the rim and the distance to the rim. If you just start building prototypes you will likely make several designs which have fatal flaws. A little math ahead of time can avoid some of the mistakes.

Second, have someone with some skill at free throws shoot a few with this ball. Shoot some video (from a variety of angles) of the shots to give you some idea about trajectories. You might also want to watch a good free throw shooter shoot some shots with a regular basketball.

There balls deform a lot more than a basketball or baseball. That deformation will vary greatly depending on the mechanisms you use for shooting. This will have a lot of effect on the air resistance and flight characteristics of the ball. So there will be no getting around doing a lot of testing with prototypes.

The (horizontal) angle at which you are shooting at the rim will matter. In a theoretical world where your shooter is always the same, this isn't the case. But when you are shooting at an actual hoop, you are going to want to shoot as close to straight on as you can. There will be shots that are a little too long. If you are shooting close to straight on a decent percentage of the shots are going to bounce off the backboard and come back to the rim. This means you want to err a little bit on the side of too far rather than too short in your shooting.

Figuring out how quickly you can shoot without having the ball interfere with one another is another important thing to figure out. If you shoot too quickly you are likely to have one or two balls bounce out off of a preceding ball.