Puzzler Solution
Putting one car behind another has a flaw. The rear car has greater potential energy and would reach the flat portion of the track with a higher speed than the front car if it were unimpeded. This would be true for either car. Lacking a flat portion for the track, friction would not be the significant factor that it is on a “standard” track. If differences wre noted with this technique then one of the cars needs alignment or more graphite.
Harry Elliott
1961VI
Not sure if the effects from “drafting” behind the first car would be equal in each trial. A slower car due to drag could appear much faster and a car with low drag doesn’t the same benefit. Of course when the test is performed in the vacuum of space (and it very likely was) the effect is negated.
… When you can’t solve the puzzler one needs to resort to throwing rocks at those that get it pretty much right (great puzzle by the way)
Seriously, he wasn’t talking about drafting (I thought of that.) He meant that the rear car will necessarily start “higher up the hill” than the lead car, and thus travel faster on the flat.
A better solution: the car that traverses the course the fastest will (very likely) be going fastest at the end. So, have both cars smack a wooden block on a string at the end. The one that knocks the block into the highest arc had the most kinetic energy, the most speed (assuming equal mass of cars) and almost certainly traversed the course the quickest.
Seriously, he wasn’t talking about drafting (I thought of that.) He meant that the rear car will necessarily start “higher up the hill” than the lead car, and thus travel faster on the flat.
A better solution: the car that traverses the course the fastest will (very likely) be going fastest at the end. So, have both cars smack a wooden block on a string at the end. The one that knocks the block into the highest arc had the most kinetic energy, the most speed (assuming equal mass of cars) and almost certainly traversed the course the quickest.
If both cars are typical pinewood derby cars, both runs will be just about the same. Drag will cause the car in front to go slower.
Although gravity accelerates both cars at the same rate (the slight difference in starting positions won’t have any significant effect), pinewood derby cars are almost always weighted to the maximum weight allowed so that the car has more kinetic energy to overcome the effects of drag and friction.
For a typical pinewood derby car, the acceleration due to drag will be about twice that due to friction. Drag is proportional to the frontal area of the car. The frontal area of the car in back is close to zero because it is behind the other car. Therefore, the car in front will be slowed by both drag and friction. The car in back will be mostly slowed by friction alone. Therefore, the car in back will generally go fast enough to keep up with the car in front.
Unless one car is very ell designed and the other very badly designed, we have to call this one BOOOOOOOGUS. 
– Roger
What also gums things up is that stuff “bounces” off of wood (think Louisville slugger.) That means, if there is indeed a gap between the two cars at the finish, then either the trailing car was slower, OR it was faster (and knocked the slower car forward by bouncing off of it.)
I still think the best solution…if it’s a given we can’t just time the two cars on independent runs…is to measure the kinetic energy at the finish line.
“I still think the best solution…if it’s a given we can’t just time the two cars on independent runs…is to measure the kinetic energy at the finish line.”
I think that’s it! Let the cars roll off the end of the track and see which one goes the farthest.
– Roger
“(the slight difference in starting positions won’t have any significant effect)”
It has a huge affect. When my son was in Cub Scouts, he won the Pinewood derby every year, even the year he built the car all by himself. The trick was to put ALL the weight at the back of the car to give it the most “fall” distance.
All the other cars had their weight either at the front or the middle. In the final heats, the other cars would be neck and neck with him to the bottom of the track, but at the transition to the flat, his car would jump out 1’ to 2’ ahead of everyone else, and hold the lead to the end.
“The trick was to put ALL the weight at the back of the car to give it the most “fall” distance.”
Putting the weight in the back will move the center of mass of the car a little towards the back, but it doesn’t make it fall any farther. Moving the Center of Gravity makes the car more stable (which might reduce the drag) and it might lesson the friction on the wheels and axles (by taking weight off the front axle).
– Roger
Think about dropping two identical balls from two different heights. Before you drop them, the higher one will have more potential energy and “fall distance,” but the ball closer to the ground will always hit the ground first.
– Roger
The essence of the puzzler solution is that you can estimate which car has more friction loss by comparing their relative rates of acceleration.
Start by assuming no friction or drag. Both cars will accelerate at the same rate due to gravity. As long as they are falling the distance between them will remain the same. If you measure the velocity of each based on how far each one traveled it will also be the same. Pick an arbitrary point along the fall and measure the first car’s velocity. When the second car comes along it will be traveling faster because it has fallen further but it won’t catch up to the first car because the first car is accelerating at the same rate. Include friction and drag. If the friction and drag are equal for both cars the same applies. Both cars will accelerate at the same, only slower rate. As long as both cars start from 0 and accelerate at the same rate the distance between them will remain the same.
When the cars transition from falling to moving horizontally their acceleration due to gravity is gone. Assuming equal friction losses and given enough distance the car falling from the higher position will pass the lower car due to its higher velocity.
If you can believe the above then you will see why moving the center of mass toward the rear of the car improves its performance. The cars may travel the same distance over all but the one with the higher center of mass in effect starts from a higher position.
This is slightly off-topic, but I think it interesting nonetheless.
Aristotle taught that heavier objects fall faster than lighter ones. This view was held to be true up until the time of Galileo. Galileo showed, through a simple thought experiment, that the Aristotelian hypothesis leads to an irresolvable contradiction.
Not only did Galileo prove experimentally that Aristotle had been wrong (see leaning tower of Pisa), but he also pointed out a logical contradiction in Aristotle’s own reasoning. Simplicio, the stupid character, mouths the accepted Aristotelian wisdom:
Simplicio:
There can be no doubt but that a particular body … has a fixed velocity which is determined by nature…
[and heavier bodies fall faster than lighter bodies]
Salviati:
If then we take two bodies whose natural speeds are different, it is clear that, [according to Aristotle], on uniting the two, the more rapid one will be partly held back by the slower, and the slower will be somewhat hastened by the swifter. Do you not agree with me in this opinion?
Simplicio:
You are unquestionably right.
Salviati:
But if this is true, and if a large stone moves with a speed of, say, eight [unspecified units] while a smaller moves with a speed of four, then when they are united, the system will move with a speed less than eight; but the two stones when tied together make a stone larger than that which before moved with a speed of eight. Hence the heavier body moves with less speed than the lighter; an effect which is contrary to your supposition. Thus you see how, from your assumption that the heavier body moves more rapidly than the lighter one, I infer that the heavier body moves more slowly.
I have always liked logical arguments like the above in physics and mathematics.
Sounds like an old episode of Cat Talk.
Yes, shifting the CG back could cause the car to be going a little faster at the transition to horizontal. You can calculate that, in a vacuum, the faster car would quickly catch up and pass the slower one even if the difference in starting height is small. For example, dropped in a vacuum from about one meter up, an object falling two inches farther would catch up and pass the other object within about 1/2 of an inch (excuse me for mixing units) if the acceleration suddenly changed to zero.
In air, however, the drag increases at twice the rate of increase of the speed of the object and the cars aren’t falling free. So, in practice, I suspect that lowering the drag (and to a lesser extent, the friction) on the car would have more of an effect on performance than moving the CG back.
And, contrary to what I may have said before, moving the CG back could make the car less stable and increase the drag and/or friction.
Nevertheless, the Puzzler answer remains bogus. Unless one of the cars is terribly designed, the car placed in front on the track is always going to appear to be slower because of the drag on the car in front.
All true comments in the purest sense…however, I think running the race several times switching the positions would give the observer enough info to account for the above and still determine which was faster. If the back car is bumping the first, it would be obvious. Plus, I don’t think the drag is enough to affect the outcome. The faster car would always ride on the slower’s rear end when behind and pull away when it’s in front.
I think this test would not ALWAYS determine the fastest, but it would often enough that I wouldn’t call this bogus. But the methods of testing the kinetic energy might be more accurate.
I won one Derby and placed 3rd in another as a kid…pure luck in each case, although the year I won, my car was weighted to the limit, weights were in the middle and definitely hurt the aerodynamics. Based on that, I’d say weight is more important than drag in determining outcome.
jadebox, you are missing one critical point. The car with the weight in the back is NOT starting out behind the car with the weight in the middle. The cars start at the same point. Its the weight that falls farther, not the car. The further the weight falls, the more speed it develops.
The car with the weight in the back will continue to accelerate during the transition to the horizontal part of the track where the car with the weight in the middle will start coasting sooner. Putting the weight in the back only helps when you have reduced the drag and friction to the least amount possible. Drag and friction have a far greater affect on the outcome of the race.
“… the car placed in front on the track is always going to appear to be slower because of the drag on the car in front.”
Actually, running two cars back to back decreases the air drag on both cars, because the vacuum drag on the rear of the lead car is broken up by the following car. When such a situation occurs in stock car racing, it is observed that both cars can travel faster than either car alone. Often (typically?) the low pressure drag on the rear of a car exerts a greater retarding force than the high pressure drag on the front nose of a car.
And, we haven’t talked about the rotational inertia of the wheels. That’s why they have races, to see who has the best configuration.
A professor of mine told the following story:
In ancient Greece a group of philosophers spent the day arguing whether or not a local farmer could pick up a particularly large rock. A stranger wandered into town and hearing their discussion suggested they ask the farmer to try. They killed the stranger.