– Feature Article – What is Maximum Velocity?
– Pinewood Derby Car Showcase
– Memory – Modeling the Nissan Skyline
What is Maximum Velocity?
By Michael Lastufka
Pinewood derby racers inflict body, wheels and axles with all manner of inhumane treatments to exorcise friction, the pinewood demon. Somewhere, tribologists (friction-ologists) tell us there is a limit beyond which no amount of surface modification can reduce it further. Is this limit knowable? Can it be realized? Economically? Practically? My personal quest to discover this limit has not yet lead me to the definitive materials reference; surely some industrial juggernaut studied aluminum and plastic ‘journal bearings’, plastic rolling on wood and air resistance at low Reynold’s numbers?!
Each source of friction has a number associated with it between 0 and 1, closer to 0 than 1. I don’t believe technology will lower these numbers to 0, but it is easier to examine what happens if they are 0 than guess at the slightly greater than zero limits. Perhaps someday pinewood cars will take advantage of maglev technology or an air cushioned track. Would it be a ‘car’ without wheels?
In the following, I assume that there are wheels, and that they spin up frictionlessly (a physical contradiction). The supposition of no axle, tread, or air friction leads to a simple time equation. Remember the distance equation, x = vt? Dividing by v gives, t = x/v. The initial speed of the car is zero and by the time it gets to the end of the ramp x distance away, it’s V. The average speed is then V/2. When there is no friction, acceleration is constant so the time to the bottom of the ramp is the distance divided by the average speed, or t = 2x/V. Since there is no friction on the flat runout, the car continues along at speed V. So, t = 2x/V + f/V where f is the distance from the bottom of the ramp to the end.
Things aren’t quite as happy as it seems though. The velocity, V, depends on three car factors, mass – m, outside wheel radius R and the wheel’s moment of inertia I (assuming the wheels still spin up). Also, the distance on the ramp, x, and flat, f, are not the length of the ramp or of the flat. Instead, they are the trajectories of the car’s center of mass, and depend on the location of the car’s center of mass – the point (Cx, Cy) measured from the tip of the nose and above the axles. A race time can be summoned up by supplying values for the 5 parameters. But which ones?
The time equation is a bit involved at this point, but by picking small but reasonable ranges for the parameters a much simpler equation can be produced by something called a Yates analysis. For the math geeks out there, it linearizes and normalizes the complicated equation near a point, which in our case has 5 components. The central point I used in this study was m = 0.01295 ozs2/in (5 oz weight), R = 0.594 in (typical wheel radius), I = 0.000046892 ozins2 (Awana wheel), Cx = 3.5 in (middle of car), Cy = 0.39 in (debatably typical). The maximum velocity this typical car would achieve with no friction is about 1.9 seconds on a 32 foot track with a 9.2 foot-long ramp tilted at -33 degrees to horizontal.
The parameter ranges I wanted the simpler equation to cover were:
m between 0.012432 to 0.013468 ozs2/in, that is 4.8 to 5.2 oz in terms of weight.
R between 0.5095 to 0.6785 inches.
I between 0.000029451 to 0.000064333 ozins2, an Awana wheel with a 1/4 inch of tread cut off and a normal BSA wheel.
Cx between 2.047 to 4.953 inches, about 2 inches from the front and rear.
Cy between 0.08575 to 0.69525, almost even with the axles and a little higher than the center of an uncut block.
Using these ranges, I got a much simpler equation that proved valid to better than 2.5 thousandths of a second for twice the range it was created for – that’s most legal pinewood cars. The average of all race times within the range of parameters was a race time of 1.9065 seconds. The new equation also showed that the only significant interaction between factors was between I and R. The analysis also answered the big questions, “Which is most important?” and “What frictionless car configuration is the fastest?”.
The run time was reduced by …
0.014824 seconds per moment of inertia decrease of 0.000017441 ounce-inch-seconds squared (lighter wheels)
0.011827 seconds per wheel radius increase of 0.0845 inches (larger diameter wheels).
0.006470 seconds for every 1.453 inches the center of mass is moved backward (rear weighted car).
0.002626 seconds for every 0.3 inches the center of mass is raised.1
0.001589 seconds per weight increase of 0.2 ounces (heavier car).
So a fast frictionless car would have plastic cut off the tread and somehow added back on to increase wheel diameter. It’s center of mass would be as close to the rear axle as possible and raised up without destabilizing the car. It would weigh as much as legally possible without causing the plastic wheels to deform. Sounds like a good friction-challenged car!
If you measure m, R, I, Cx and Cy from your car, you can use this simple multilinear equation to find its maximum velocity (on the track mentioned above) when you have finely figured out how to eliminate all the friction!
t = 1.931758025 – 3.068244274 m – 0.010812495 R + 2486.009049 I – 0.004452577 Cx – 0.008617302 Cy – 2754.332148 R I
1I always thought that raising the center of mass didn’t matter much, or it reduced the energy the car attains at the bottom of the ramp, hence it hits the flat at a slower speed. This is true, but raising the mass does shorten the car’s trajectory by a couple of inches. A car with friction slows down on the flat and the advantage of the shortened trajectory is wiped out. But when there is no friction, and when the trajectory is short enough, even though the car with the higher CM has a slower velocity, it still wins over a lower CM car.
Michael Lastufka is the source of all things mathematical as it relates to pinewood derby cars. You can learn more from Michael at: www.lastufka.net/lab
Pinewood Derby Car Showcase
The Finger: Frank & Keith Tonra
My son Keith (Wolf) and his older brother Frankie (Webelo 2) started to think about their car months before the derby (like the day after the last one). Keith wanted to do a black hearse with skulls and look scary cool, and Frankie wanted a 3D Frankenstein car and to win it all. But when we talked about the car’s looks versus speed, things changed. Keith’s car would be big and bulky like the hot dog last year so he wanted a nose, so he could ‘win by a nose’. The nose as a wedge car seemed like a good idea but proved to be hard to shape, and still look like a nose. So, the Finger car was born. Frankie settled on speed over looks, and made a lightning bolt wedge car that took second in his den and strangest shape in the pack. Keith’s Finger Car came in second for Fan Favorite, won Most Humorous Car and finished forth by ten thousands of a seconds. The boys had a great time, were proud of the cars and are already at work on their next cars.
Cool Flag: Scott & Taylor Morris
Here is my daughter Taylor’s Awana Grand Prix car from this year. Taylor designed her car and did most of the work herself – other than the power saw work. We used many of the ideas and techniques found on Maximum Velocity in building the wedge car for speed. Taylor wanted the car to be stylish too, so she opted for the Freedom Flag body skin. Taylor’s car didn’t lose one heat and took 1st Place for speed, and got lots of attention for the “cool flag”. She has already started designing next year’s car.
Solar Car: Anton & Andrew Petrou
The car you see here was inspired by the University of Michigan’s solar race car, which we saw on display during a visit to the Chicago Museum of Science and Industry. Using this as an inspiration, we went to Michigan’s web site and learned about how they minimized wheel friction, rolling resistance and aerodynamic drag. Sound familiar? This ended up being the perfect way to teach my son about the physics of a well designed pinewood derby car.
There was, however, a real challenge, which was how do you get enough weight in such a thin tail and still achieve an optimum balance point. We ended up sawing the shape of the car, and used the body to form a sand casting mold. All the materials needed were a bucket and some moist playground sand. To keep things safe, I melted the lead and poured it into the mold, and all of this was done outdoors. Once cooled, the lead was virtually the same shape as the back of the car. All we had to do was cut off the of the car equivalent to the length of the molded lead and then epoxy it to the back. All it needed was some wood putty, paint, and voila, a Solar Car is born. The side view says it all, since the car is in no way attached to the white post, instead it is balanced on top of it!
We worked hard on building the Solar Car, and at the Pack Derby it won all 15 races and ended up finishing 1st in the Pack. To this day, I think this particular experience continues to motivate my son towards a career in engineering – just like Mom and Dad.
Pinewood Derby Memory
Modeling the Nissan Skyline
For the 2006 RA Derby Race at our church my 10 year old son, Jonathan, decided to make his car look like the Nissan Skyline in the movie “2 Fast 2 Furious”. I told him that it would not be easy to make a car of that detail, but he was set on making that car. We received the kit in early November so we had plenty of time to work on the car. He drew the design he wanted on the block of wood, and I cut it out on the band saw (so he would have all his fingers for the sanding!). I also drilled a rough opening under the wing. He then went to work on the final shaping and sanding. His work progressed nicely, and he was becoming happy with his design. After he completed the sanding I poured lead into the routered bottom to weight the car. The car was now ready for paint. The base coat of paint was applied and let dry. He then started to work on the side graphic paint, hood stripes and wing paint. He took his time and worked very hard at making everything symmetrical on both sides. I was pleased that he was paying attention to details. It was then onto painting the windows and lights. All was looking good and he was on the way to being finished a week before the race. Then disaster struck. One week before the race he was applying the final clear gloss coat of paint. As the clear went on all the blue paint on the graphics and wing started to lift and develop an alligator texture. He was very disappointed. With six days until the race he had to re-sand and refinish the car, Since this was the second attempt at painting the car it went a lot faster. He had the car finished on Friday night before the Saturday race. Although the painting was rushed the outcome was almost as good as the first paint job. We went to the race Saturday. The car did not perform very well on the race track, but when he won 1st Place in his age class for design and craftsmanship he was very happy. Now it is on to the Association Race in two weeks; we’ll see what happens there.
Jonathan and Joe Coyne
P.S. The poor performance on the track is another story, all the usual work was done on the axles and wheels, and the weight was at the max but it still ran slow. I am convinced that big body cars just do not perform like small sleek designs. I helped my son’s friend with polishing his axles and wheels identical to my son’s wheels. His friend’s car was a very thin sleek design that was only 3/4 inch high near the rear of the car. His car won every heat, and he took home the 1st Place Race trophy.
Perhaps you can help me here with an unbiased response. Yesterday our Pack held its annual PD. One boy’s car was checked in and went on to win his division’s 1st Place award. During the lunch break, one of our scout leaders noticed that the car had non-BSA wheels (not detected during the check-in process). We voiced our concerns to the other leaders about the validity of his winning the division.
Our Cubmaster stepped in and said that because the car passed the check-in, the win would stand. While we did not disagree with the Cubmaster’s opinion, we did feel strongly that the boy should not be allowed to continue the race for the Pack Championship with those same wheels. The mother became terribly upset and it was frustrating for everyone. The Cubmaster said he would race with the same wheels that he was checked in with, and no changes would be made. The boy went on to win the entire derby for the Pack. Given this information, do you have an opinion as to the outcome?
This is a tough situation, but I believe the pack leadership handled it correctly. Since the non-approved wheels were obvious, there was every reason to expect that they would have been caught at the official inspection. Since the inspectors passed the car, they essentially ‘approved’ the configuration. Whether or not the car did well in the race is not really an issue. The inspectors passed the car, so it should be allowed to run and compete normally.
In the future, the inspectors need to be better trained as to what is and is not legal. A checklist and an example of a valid car is really helpful for this.
There is one case where I would consider disqualifying the car: if one of the child’s parents was the inspector who passed the car. Hopefully this was not the case.
I shaved all my wheels with the Pro-Wheel Shaver XT and thought the weight differential would be minimal when I put them on the wheel balancer. However, it looks like the heavy side of the wheel is always the side with the most writing on it. Should I sand the lettering of the outside of the wheel before balancing?
I would certainly check with your race officials before removing the lettering. Some groups want the lettering intact. But if you are allowed to remove it, then certainly do so.
However, I just checked a set of speed wheels with the balancer, and the heavy side is not in the same place on each wheel. So removing the lettering may or may not eliminate the need to balance the wheels.
My race is this Friday and I’m taking my axles to a jeweler to polish them with rouge. The jeweler’s question to me was whether or not after polishing the axles with the rouge, if he should put the axles into a solution and then into a steam bath to remove all of the rouge and any other particles that may be on the axles. I wasn’t sure if the rouge was just for polishing and could be removed without issue, or if the rouge actually adds some slickness and is needed to mix with the graphite for speed. Any thoughts?
Red Jeweler’s rouge is basically small particles of iron oxide that are used for polishing metal. You do want to remove all of the residue, so I would let the jeweler clean them after polishing.