Do the rules require the potential energy from the mass to be transformed into rotational kinetic energy via spinning an axle? If not, I would have just designed a small car with hard wheels and a steep ramp in the back. I would then simply drop the mass onto the ramp, propelling the car forward. The rear axle would have to be behind the last point on the ramp to prevent the car from pulling a wheelie. I know this is also getting into momentum/impulse, but it is utilizing gravitational potential energy.
Was this optimizing for speed or for distance? For distance, there are a lot of principles that would help that didn't appear to be targeted for optimization. (As much as one can optimize with cardboard and balsa-wood for a school project). 1) The weight moving fast means that some of the weight's potential energy is spent accelerating the weight and not the car. While you may want a low gear ratio to get moving, you should quickly convert to a high gear ratio to minimize energy wasted on the weight's velocity. A spiral or a very very thin winch drum where the connecting rope wraps like a VHS tape may be best. 2) The weight of the car matters to an extent, but overall a heavier car will then drift further, slown down less by a given amount of drag, so the stability of the car and its ability to move in a straight line consistency probably matters more than weight considerations. Same story for the wheels - CD's are great for low rotational inertia, but it seems like some cars had problem with friction on the wheels, causing them to steer off-course. Slipping is a huge waste of energy. Momentum put into the wheels will pay itself back later while coasting. 3) The primary loss of forward motion is drag from the air, and friction on the axles. Ball bearings or plastic with some graphite or other lubrication would probably extend the motion of these cars considerably. The thinner towers are better, as are more aerodynamic bases. It's not much, but you don't have much energy to play with, and it has to be going somewhere to make the car stop, so these are the only two avenues that keep it from plodding along forever.
