Ether, what would it take to get you to come over here and join my team?
You mathematical contributions to the Chief Delphi forums this season have been fantastic.
Regarding your model, and the (neglected) friction, I've learned it does play a BIG factor. Tonight I did some testing and visually saw much longer acceleration times than your model predicts. Anyhow, your previous calculations inspired me to try out a few things of my own, so last night and this morning I built up a minibot with the following specs:
~2.6-2.7 lbs
2 motors, direct drive
0.42 inch rollers
Approximately 6.5lbs normal force to the pole (magnets).
Mostly charged battery (had been charging for an hour or two) measuring 14.5 Volts when pulled off the charger.
I have friction in bushings supporting the end of the rollers, and friction of a limit switch against the pole. My best time was 1.6 seconds.
One thing I found interesting was that on a battery which measured 11.98 Volts (no load), the minibot didn't have enough force to accelerate itself at all. On this rather dead battery, with a thumb on one roller (for as long and hard as I could without burning myself), battery voltage was dropping to 9.5ish under load. So, the lesson here is that if your minibot battery measures 12 Volts, it's dead.
I predict the teams on Einstein will have their deployments perfected, and the winner will be the team who can get their minibot the lightest, and tuned to operate at its peak performance point.