Hi Guys,
I just had to write to let you know that you really dropped the ball on the Pinewood Derby question today. Although you were correct in saying that the kinetic energy of the cars is all calculated in the same manner as 1/2(mv^2), you neglected the fundamental objective of the endeavor: to take a vehicle with a POTENTIAL energy (when it is resting at the start gate), convert it to a maximum KINETIC energy (by the time it reaches the flat part of the track), and then maintain as much of the kinetic energy as possible until the car reaches the finish line (which means that you must maintain as much velocity as possible, as at these low speeds quantum effects are of no concern). This problem may be most simply explained by saying that when an object falls in AIR (not a vacuum), its acceleration and final velocity are both dependent on its air resistance, other resistance terms, mass and the initial height from which it is dropped. The potential energy of the car (treating it as a simple lumped mass) is given by (mgh), where m=mass, g=gravity, and h is the height differential from beginning state to final state. This formula determines the maximum amount of energy that the car can possess as it reaches the flat portion of the track, since no other source of energy is permitted. SO, in order to maximize the change in potential energy, one must MAXIMIZE the weight AND MAXIMIZE the height of the weight from the final resting height, while MINIMIZING resistance terms. I believe (if memory serves correctly) that the cars start facing down an incline, and end basically flat. Therefore, placing the maximum weight at the very rear of the car would provide for the maximum potential energy change, and thus a maximum of kinetic energy when the car reaches the flat.
Also important, of course, is the aerodynamic shape of the car. The
car should have as little drag as possible so that (a) there is less
drag to hinder the car's fall from start to finish; and (b) the car may
continue to conserve as much of its kinetic energy from
the beginning of the flat to the finish. (And definitely lube those
"nail" axles to lower the friction!)
Finally, as a vehicle dynamicist, I must mention that though putting all of the the weight in the rear of the car would provide the maximum change in potential energy (if my memory of the track is correct), it is NOT a good idea from the standpoint of vehicle stability or, possibly, from a rolling resistance/friction point of view. If you use the oversimplified analogy of a car as a dart on wheels, then you can see that the car is less likely to continue in a perfectly straight path in this manner, and would more easily stray from its course due to disturbances. (Rubbing the guide rail is a good way to eat up that precious energy--maybe a little graphite on the insides of the wheels is warranted?) Also, as the resistance of the plastic wheels turning on the "nails" and that of the wheels on the wood surface is likely to be somewhat more than directly proportional to load, concentrating all of the mass over one axle would likely increase rolling resistance.
As you can see, there are many variables to be optimized in this situation; finding the optimal solution may require the fabrication of many expensive prototypes and lots of (wo)man-hours to "tweak-in" the perfect design. If the den mother involved were able to scrape together, say, $130,000 for a detailed analysis of this fascinating dynamics problem, my employer would be happy to work out a simulation environment for solving these issues.
Please let me know if she is interested.
;*) ;*) ;*) ;*) ;*) ;*) ;*) ;*) ;*) ;*) ;*) ;*) ;*) ;*)
Thanks,
Eric Bowman
