I love the first answer - it would work, but it is actually gravity-powered! But your project doesn't make sense the way you put it. Why use batteries but no electric components? Do you mean you can't use resistors, capacitors or inductors, but you can use an electric motor? If you can use a motor, stop it by having your power wire loosely connected, or connected to a switch. Tie a 10m string to the wire or switch such that when it gets tight, it turns the car off.
If you can't use a motor, maybe they want you to build something like a magnetic railgun. Google that. You could do it with a line of permanent magnets and a switching (alternating) electromagnet. Or use something like a solenoid that accelerates a weight out the back - a reaction-powered car.