What length of linear track (the geared kind) would be sufficient for a full-field Linear Puncher?

Well think about it, the gear pulls the track when it does not have the slip, so it depends on how many teeth that you cut off and what gear you are using (60 or 36 tooth). Find the circumference of the gear and measure out how much geared track you will need (pieces).

There is the simpler way of someone just telling you how much track but this way you get to use math too!

Perhaps I should rephrase my question: How far back should I cock back the puncher in terms of gear teeth?

In my experience, that is more of a test it yourself question. Try shaving 9 teeth off and putting elastics on, and see how it works for you(shave off more if you want to). Then adjust to where you are getting a consistent pull that isn’t burning out the motors.

With the simple physics, where k is spring constant x is distance pulled back, v is final velocity, f is friction, m is mass, and = is “is proportional to”, .5kxx-fx=.5mvv so v=Sqrt((.5kxx-fx)/(.5m)) so you want a long pullback, strong springs or lots of springs to increase spring constant, less friction and less mass to increase v. Reload time, t is much more complicated and I’ll try the proportions tomorrow maybe but more x and k increases t while more max speed, less friction, less weight, and more torque decreases t. This is simple physics that is inaccurate with real data but they give the right ideas unless I did the math wrong

We use the length of 2 of the linear gears on a slide

27 - 30 teeth of pullback is what most teams use.