Shooting Physics

[Sparks333]

Hello!

So, did a bit of back-of-the-napkin physics, and came up with a highly simplified model that can be described in terms of a simple equation. Keep in mid this makes a LOT of assumptions - stuff like the edge of the frisbee does not slide, the amount of energy the motor imparts on the system during launch is negligible, etc, but this ought to get you in the ballpark.

The energy of a shooter prior to shooting is:

1/2 * I * omega_start^2

Where I is the inertia of the motor/wheel (with a multiplier effect for gearboxes) and omega_start is the angular velocity of the wheel prior to shooting. The energy of the shooter and the frisbee together after shooting is

(1/2 * I * omega_end^2) + (1/2 * M_frisbee *V_frisbee^2)

where I is the same as before, omega_end is the angular velocity after the frisbee has left the shooter, M_frisbee is the frisbee mass, and V_frisbee is how fast the frisbee has left the shooter.

Still too many unknowns. So, you know that when the frisbee leaves the shooter, one side is going the same speed as the wheel, and the other side is rotating against the side of the shooter, and not moving. Therefore, we know that the velocity of the frisbee is half of the velocity of the edge touching the shooting wheel, or

V_frisbee = omega_end * r_wheel / 2

where r_wheel is the radius of the shooter wheel (all of these are in metric, by the way).

Combining and simplifying the two equations, we get:

I * omega_start^2 = I * omega_end^2 + ((M_frisbee/4) * omega_end^2 * r_wheel^2)

As an experiment, if you take a standard 8" AndyMark pneumatic wheel (r = 0.1016 meters, mass of 0.5126 kg --> moment of inertia calculated to be about 0.00265 kg m^2) and hook it straight onto a CIM and let it spin up to maximum power (277.89 radians/sec) and throw in a 0.2 kg frisbee, this equation predicts that you'll end up with a screamer of a frisbee heading out at 13 meters per second, and have a shooter wheel spinning at about 254 radians/sec. With this equation, the only difference between a curved shooter and a linear shooter is the moments of inertia - in a linear shooter, with multiple wheels, the inertias add linearly, whereas since a curved shooter has one wheel, it gets used as is.

Don't expect to actually get the numbers I've posted here in real life - there are way, waaaay too many simplifications for it to work that well - but this isn't a bad upper limit on performance.

Good luck, have fun!

Sparks