The Cub Scouts has an annual competition called the Pinewood Derby. Each scout builds a wooden car that will race down an inclined track (inclined at some angle θ) powered only by gravity. The goal is to build a car that will make it down the slope as fast as possible. The total mass of the car (including the wheels) must be ≤ M, and the radius of the wheels of the car must be R, though the wheel weight can be any fraction of the total you like. In this problem you will investigate whether it is better to make the wheels heavy or light (with ballast added to the car body).
Derive an expression for the time it takes the car to reach a certain position x down the incline in terms of R, θ, g, M, and the mass mw of one of the four wheels. From this expression, make a prediction as to which type of car (one with heavier or lighter wheels) will get to the bottom of the track fastest. Ignore air resistance and assume that each of the four wheels rolls without slipping while going down the track.