[Frank Neuperger]

By the time the tape arrives, it is close to free speed. It is ~ 2 to 2.5 time constants into an approximately exponential velocity profile. i.e. it reaches 85% to 90% of free speed. I say ~ only because it is close but not exactly the classic
vel=Vf*[1-e^(-t/Tau)] model where Tau is constant. Tau, the time constant in this case is variable because Tau is a function of the (moving) mass of the tape (and the stall torque of the motor).. In this case, the tape (moving)mass increases by a factor of ~2 from the mass at standstill and the mass at full extension.

For a given tape /hook mass, free speed, stall torque and other fixed variables, there is an optimal gear ratio for shortest transit time. We did not have a chance to optimize this as all of this was crammed into the 3 weeks before nationals. Varying wheel diameter has the same effect. It would have been nice to have the time to try several wheel diameters.

Yes there are gains to be made by changing gear ratio. If you step up the wheel speed, you decrease the starting torque (decrease time constant) but increase final free speed if you can reach it...... If you step down the wheel speed, you increase starting torque but decrease free speed. It is an optimization problem. Either model the system to find the peak performance or do some trials.