I recall having to build something similar whilst at university many years ago, however that was elastic band powered and required a brake. With my team we ended up building a lightweight aluminium monocoque with a single rear wheel on a ball-bearing mounted axle (drive on one side, brake on the other), the front axle had individual spigots with ball bearings mounted in the wheels.
Rear wheel was very groovy, and light. Large diameter, thin tri-spoke made from Perspex. We HAD to fit a ‘tyre’ (rubber band), as it would spin up... with the tyre on it generated enough grip to pick up the front wheels....
We ended up with a variable ratio gear on the drive side.
Vehicle was required to cover a 12m track as quickly as possible, with points for accuracy (hence the brake). Great fun!