There is really only one thing that makes the car move-gravity.
I drew a simplified diagram of the situation. I simplified the car to just a point mass. There are 3 forces on the car: mg- the force from the gravity of the earth, N- the normal force from contact with the track, and f- the frictional forces acting opposite the direction of travel.
The car only accelerates in the downhill direction of the track, I have designated this direction as the +x direction, which makes the normal force in the +y direction. The only force with a component in the direction of acceleration is mg, we can break it into x and y components to get a better look at what's going on.Since the car doesn't move in the y direction we know that the magnitude of mg in the -y direction is equal to the magnitude of the normal force in the +y direction and those forces cancel.
We are now left with only x forces:
The rolling frictional force from the wheels on the axle and wheels on the track can be represented with a constant k multiplied by the magnitude of the normal force which is the same as the y component of mg. If we assume that the race will take place in a vacuum then this is the only friction that we need to consider. This gives us an interesting result:
Well, it might be that I have neglected important factors in my simplifications. I should put the race back in an air filled environment so the spectators don't have to bring their own oxygen. This will introduce another frictional force from the air but from what I've seen so far in my classes, this force has little effect in many cases and often times is neglected. Either we have in vain been putting weights on our cars for years or conventional wisdom is correct and air drag makes enough difference for us to bother with attaching weights.
It all depends on what the coefficient of friction is, which I believe involves the density of air and the shape and density of the car. This can be found experimentally. With that term added our expression now looks like this:
The force of the drag is proportional to velocity. This makes me very curious now and I want to get that track and do a bunch of tests with varying weights and other things to find out how much effect this really has.
The placement of the weight may have something to do with the stability of the car also, to figure all these things out I'd have to do a more complicated analysis and look at the car as something more than just a point mass. I'm not going to do this, and now that I think of it, the urge to do a bunch of tests is leaving me too. I'm feeling content enough to just get those wheels turning smoothly, put weights on the cars till they are 5 ounces and enjoy the race.
I would like your input on the matter though. If you happen to have any insight on these things please let me know, I'm sure I've missed something.