[sjspry]

Get a camera (preferrably high speed) to watch the act of shooting.

Our team is using a flat belt of material to accelerate the disk in a linear shooting design; as such, we must simply make the shooter long enough to accelerate the disk completely. This is very quick process.

To model this with more accuracy, we think of the forces acting upon the disk. The disk experiences forward acceleration due to rolling, which is based on the angular acceleration from the belt, which is related to the torque of the motor.

The motor accelerates the belt based on the moment of inertia of the belt system, which is deterined experimentally; consider the motor and belt as a single system, with the motor just being the active torque on the belt.
The belt accelerates the disk (and the disk accelerates the belt) based on the moment of inertia of both the disk and the belt - use conservation of momentum.
The angular speed of the disk is related to its rate of travel along the shooter, assuming no slipping.

(1) T[belt] = I[belt] * a[belt] ; relation of torque of belt to angular acceleration.
(2) L[belt] = I[belt] * w[belt] ; relation of angular momentum of belt to angular velocity.
(3) L[disk] = I[disk] * w[disk] ; the above for the disk, assumed to be zero as w[disk] will be zero before the collision.
(4) L[belt] + L[disk] = L[belt + disk] ; conservation of momentum (might be unnecessary?)
(5) T[belt+disk] = I[belt+disk] * a[belt] ; now the motor is accelerating the belt and disk.
(6) T[belt] = T[disk] = T[belt+disk] ; they're equivalent (redundant, but note this).
(7) a[belt+disk] = T[belt+disk] / I[belt+disk] ; acceleration of the system.

Okay I'm going to stop because I'm running out of time and I think I messed up, but now you need to find out how long the (possibly variable) force must be applied to reach equilibrium (running speed, when the disk is fully accelerated) and the distance of this act. Then, you take the angular acceleration of the belt, integrate to turn it into angular velocity, then you need to turn that into tagentital velocity, integrate for position, and see if what you get is feasible (time to equilibrium <= acceleration possible in your shooter's distance.

Feeding the disk into the mechanism with the belt already spinning will cause slippage. This probably is not an issue; if the time it takes to accelerate the disk is 1/2 (probably much less) of the total time it takes to shoot, approximately 1/2 (probably much less) of that would be spent slipping in my estimate. If you wish, you could model the force during the time spent slipping as related to the difference in tangenital (as opposed to angular) velocity of the disk and belt.

F = k * (v[belt] - v[disk]) ; where k is a constant.

If your mechanism is not a flat belt, but wheels, the approximation should be the same. At first I thought the varying distance to the wheels from the center of mass of the disk would cause varying force/acceleration, but this shouldn't be the case (I think). Bumping as the disk transition wheels would probably have more of an effect on the acceleration of the disk.