Car from Household Materials
You can make an elastic motor for a car out of any elastic band, and the car itself out of things you find in the house. A foam meat tray, or a well glued piece of paper will make a great chassis. You should reinforce a more flimsy base with something like old pens or popsicle sticks. A pair of straws attached to the bottom of the chassis are perfect for holding the axles--a pair of skewers--in place. The axles should have an exposed section on them, to which you can attach the elastic band--so cut a notch into the center of the rear axle. The next two challenges are attaching the wheels and the elastic to the axles. This depends on the type of wheel and the materials the axle is made of. Some kind of putty will hold the wheels in place, and tape or staples will connect the elastic to the axle. The best way to connect the elastic to the car is by unclipping one arm of a binder clip, slipping the elastic into it, and rebuilding the binder clip.
Car from LEGO Pieces
Making a car out of LEGO pieces will give you a head start on building the car. The wheels are already built and the chassis will be strong enough to handle the elastic band. You will need to make the chassis first, and the best materials to use are Technic pieces--as you will be able to make a really solid chassis with these pieces, and won't have to worry about attaching the axles anywhere special. Make sure that the rear axle has a 1-inch gear on it, to which you will attach the elastic band. To use the vehicle, hook the elastic to one of the teeth of the gear, and roll the axle backward to wind the band. When you release the car, the elastic will unwind and eventually detach.
Races and Competitions
Experiment with the vehicle, making several different styles. Try making one out of household materials, as in Section 1, and one out of LEGO as in Section 2. Then, see which works better. Set up a racecourse by laying a couple of yardsticks end to end, and then make a starting line with masking tape. Make a second line of about 18 inches to 2 feet before the starting line to set up the winding point--this is the spot where you will start winding your cars so that they are starting with the same amount of stored energy. Next, decide if you're aiming for speed or for distance, and let the races begin.