Robotics

If you wanna go simple with it, just put a spring mechanism inside a tube that you can pull back and attach it to a platform where the ball will be placed. You can use projectile motion equations and spring equations to figure out how far back you have to pull the platform and release and can make a bunch of markings on the tube for it. Most of the parts can be printed using an FDM printer, except the spring. Get a nice industrial grade spring to give the ball a good launch and the mechanism won't cost you that much to make, depending upon the vendors and won't even require any power source. However, if you want something automated that can shoot multiple balls at a time and can be digitally controlled, I'd suggest making something similar to a tennis ball launcher that people use to practice, although making that might be significantly more complex and expensive.

Take a look.

LIMS2-AMBIDEX Whole Body Motion Test

You need to collect some stats from top players so you can emulate the launch angle they use to hit the hoop and work back from there. Once you have the optimum launch angle and you know the weight of the ball you can plot the trajectory and force you need to get the ball to a certain distance and height. The only variables are the ball speed, launch angle and gravity to plot the trajectory. The ball speed at launch and the ball weight will allow you to calculate acceleration to get the sping force needed.

Based on the attached image, I think the ball thrower should be placed under the basket such that a player will not have to go after the ball after making a shot. The thrower will simply throw the ball towards the player, thus saving time.

If this is the case, then the thrower should be made automatic and be able to through practically any ball that falls on its net.

But Zak would need to clarify this

Exactly, that’s the idea! The ball thrower should be placed under the basket to automatically return the ball to the player after each shot, making training efficient. 

Basketball thrower:Your shots are so good!

Me:I'd love to shoot you sometime!

when we can start this project sir