A little background: a Pinewood Derby car is a small wooden car that races down an inclined track, powered only by gravity. You are allowed to add weight to the car up to a certain limit.
Here is a recommendation to add the weight to the upper back of the car, to maximize the potential energy. My own gut feel is to add it so that the weight is evenly distributed between all 4 wheels. I'd like some advice that is well grounded in science.
UPDATE: Having totally misunderstood what a pinewood derby car was, my previous answer probably wasn't very helpful (I thought it was like a go-cart). Here is a proper answer:
First off, let me say the recommendation you read is still wrong, for reasons that will be seen in a minute.
The car starts off inclined at an angle $\theta$, with the additional mass a distance $x$ behind the front axle, and a distance $h$ above the line passing through both axles. The change in height of the additional mass relative to the front axle is then given by $d_x + d_h$, where $d_x$ is the decrease in relative elevation of the point a distance $x$ behind the front axle on the line between the two axles, and $d_h$ is the decrease in relative elevation between the mass relative and this point. Clearly, $d_x = x \sin(\theta)$ and $d_h = h (\cos(\theta) - 1)$. Notice that $d_h$ is negative, since the change in relative elevation actually increases for positive $h$. Thus the total change in height (which is proportional to the additional potential energy) is given by $\Delta H = x\sin(\theta) - h(1-\cos(\theta))$. Thus, the best position for the additional mass is very low down (even below the line of the axles if possible), at the back.
Having been in the derby and lost as a kid, and as parent, it is true that the angle tapers back, so you could gain a small amount of potential energy by placing it back as far as possible. I think the biggest issue is keeping the wheel(s) from vibrating in the track and generating extra friction. The wheel is solid plastic with a hole drilled through it, and a nail goes through the axial hole, into a slit on the bottom of the wood block. I've seen people polish and/or oil the part of the nail that contacts the wheel, with little apparent results. I suspect it probably has more to do with how well aligned the axles are to the direction of motion and how well balanced the wheels are.That may be largely luck of the draw. Perhaps one should time all possible combinations of wheels and nails (4 factorial squared), and select the fastest one. You might also try both combinations of wheel orientations, but that gives you 96*96 possible combinations, which is too many to try them all.
Aside from position of weight, I would add that what you really want to reduce is the the rolling resistance of the car. One of the first areas I would target is the contact surface area of the wheels with the track. I would also reduce the amount of weight at the edges of the wheel. For instance, instead of using wide, fat wheels, I would use wheels that taper down into a v-shape. This will minimize the energy being used to spin up the wheels, as well as reduce the energy being transferred through friction to the surface. I would also do whatever it takes to reduce friction in whatever bearing mechanism there is between the wheels/shaft/body of the racer. In addition, I would work to reduce as much weight in the wheels as possible. Again, the energy absorbed by the wheels needs to be minimized as much as possible.
So think along those lines...minimize rolling resistance, mimimize wheel weight, minimize wheel weight at the edge of the wheel. Maximize velocity at the bottom of the ramp.
addendum I will add one little caveat, after looking at the track, there is an optimization problem here between energy lost due to friction, and the need to store energy to keep the vehicle moving. A spin up of the wheels will store energy, which may be needed if your friction is high enough that you need that energy in the flatter portion of the track. In any case, think in terms of transfer of energy from potenial to kinetic, what is stored on in the wheels, and what is lost due to friction, air resistance and rolling resistance.
Place the weight up front. Pull a hole shot and hope they don't catch you at the bottom. Worked for me! I also dimpled the car, all over, like a golf ball.
