"Gearing for speed" criticism

[Manoel]

Top speed is one of the most common metrics used in the FIRST Robotics Competition; yet, the problem is that the great majority of teams design their drive trains considering only the steady-state speed of the system, easily calculated by determining the operating point of the torque-speed curve of the motor (sometimes using a "desired" operating point such as the rule-of-thumb 75% of free speed, seldom using the wheel's coefficient of friction). Given the power-to-weight ratio we're allowed by the rules, the time constant of our robots is not negligible and, thus, it is important to consider the dynamics of the system.
From time to time some people do point out to unexperienced designers that "yeah, your final speed is great, but how long does it take to get there?". This remark only gives a qualitative analysis, though. I propose a different way to look at the problem: How far will your robot travel in x seconds?

Consider a 2012 FRC robot with maximum weight (120 lbs + 15 lbs battery and 15 lbs bumpers) and this configuration:
4 CIM motors driving two CIMple gearboxes, N (N doesn't really matter as long as they are all powered) HiGrip FIRST Wheels with a 26 tooth sprocket attached. The following graph demonstrates the distance traveled in three seconds by a robot with varying coefficients of friction (from 0.8 to 1.5) and gearbox sprocket (8 - OK, no such thing - to 26 teeth):

(distance equal to zero means the gearing is such that the robot cannot overcome friction)


It can be seen that, for each coefficient of friction, there's an optimal sprocket selection that will make the robot move as far as possible in a given time frame. AndyMark states that the coefficient of friction of the HiGrip wheels is around 0.95-1.0*. Let's take a look at the distance a robot travels in three seconds using those wheels:

The gearing selection by AndyMark, with a 12-teeth sprocket, is pretty close to the optimal one, 10-teeth (in a three-second run, that's about 9 inches you lose in the worst-case). I would love to hear from Andy or Mark if that's just a fortunate coincidence or if they did in fact take that into account. If so, why choose the 12-tooth over the 10-tooth sprocket?
Another thing that can be readily seen is that if a good-willing but uninformed designer decides to "increase" the speed of his robot by buying a larger sprocket he actually ends up with a robot that covers LESS ground in the same amount of time.

* Notes on coefficient of friction: http://www.chiefdelphi.com/forums/sh...d.php?t=107759