Calculating trajectory of ball rolling down slope and off launch ramp
The variables that I can think of for the ramp problem are as follows:
Initial height - given
Exit ramp height - given
Exit ramp angle - given
Slipping - are the balls released from the top of a shallow ramp or are they poured onto a steep ramp? It's important to know if they go down the ramp with or without slipping. If they do slip, you need to figure this energy bleed into your energy equations.
Friction - dynamic friction factors and rolling friction. When slipping occurs, use the dynamic mu to get the ball rolling and bleed energy until it stops slipping. When it stops slipping, use the rolling friction to bleed energy from the ball.
Air Friction - after the ball leaves the ramp
My plan of action to attack this one would be as follows:
1. I wouldn't want to do the calcs to get the ball rolling using friction. To get around that, I'd make a smooth, relatively long radius curve at the top of the ramp to make sure the balls have no slipping at any time on the ramp. You can calculate what radius the entry curve needs to be, how steep the ramps can be, and how tight the valley curve can be using rotational inertia and friction equations.
2. Set up the equation: Potential Energy (PE) at the top of the ramp equals the Kinetic Energy of the ball's translation (KEt) plus the kinetic energy of the ball's rotation (KEr) minus the energy lost due to rolling resistance. KEt and KEr are related by the radius of the ball and can be set at the exit ramp - you don't have to worry about going into the valley. In other words, your PE is the mgh from the top of the start ramp to the lip of the exit ramp.
3. That equation should give you the velocity exiting the ramp. Use your ballistics equations, adding in air friction, to see how far it'll go.
Two problems with this analysis: rolling friction and air friction. You'll have to do some serious research to get any decent numbers on these. The key may be to find out how sensitive the final trajectory is to them. It may be that these frictions are so small that they don't affect the outcome that much.