Mass is the property of all matter which deals with the question "How much of it is there? It is measured in grams. We sometimes talk about mass as "weight" (pounds), although this is not scientifically correct. (Weight is the force which a mass produces when it is attracted by gravity. This is why 1000 grams of lead weighs about 2.2 pounds on Earth but only about 0.4 pounds on the Moon.)

Mass applies equally to matter, regardless of its state: solid, liquid, gaseous or plasma. We are concerned here only with matter in its solid state.

The amount of mass determines how that matter will react when a fixed force is applied to it. In particular, how rapidly will that amount of matter change its speed?

Energy of motion (kinetic energy) is the product of the square of velocity (speed) and mass.

Potential energy for motion is the product of the height (of the available fall), mass and the acceleration of gravity.

Note that the law of conservation of energy tells us that the sum of our Car's kinetic energy and potential energy is a constant, provided that there are no losses such as from friction. This means that Potential energy at the starting line (when the car is at rest) is equal to its Kinetic energy at the finish line (when is has completed its "fall").

This lets us write some equations relating height of the starting line above the finish line and the velocity at the finish line. Note that mass is not a part of the final equation!

Broadly speaking, "inertia" is a mass's resistance to change in velocity. Our Car is subject to two kinds of inertia: "linear" and "angular". Our Car expends its initial potential energy overcoming these two types of inertia.

Linear inertia affects our Car by resisting the pull of gravity. It also keeps our Car moving once gravity has ceased pulling it down the track. It is a characteristic of our Car's mass, and, I'm afraid, there is nothing we can do to reduce it.

Angular inertia mainly affects our Car's wheels. It depends on the mass of the wheels and on how the mass is distributed relative to the center of rotation. Mass at the rim of the wheel has more effect than mass at the hub.

Since the wheels must turn if our Car is to progress down the track, some part of our energy will be used up making the wheels turn faster. That part of the energy is not available to make our Car go faster! How much energy is used depends on the inertia of each wheel and on how many wheels turn.

What happens if only three of the wheels turn as our Car rolls down the track? What happens if there is less mass on the rims of the wheels? (Note that the rules usually limit wheel mass removal.)

There is another bit of angular inertia involved, but (I think) it has a very small effect. The entire Car rotates about 30 degrees as it moves down the first (approximately) 12 feet of track. The rotation is about an axis that is horizontal and perpendicular to the direction of motion. The angular inertia is rather large (compared to the wheels), and this all happens in less than a second, so there could be meaningful use (and, so, loss) of energy. What distribution of weight in the Car would minimize this part of angular inertia? What would happen if the track had additional slope changes, such as a 12" high "hump" two thirds of the way down the track?

Air resistance is one of the frictions which our Car must expend its energy to overcome.

This is important! Mass is NOT a factor in our Car's air resistance (aerodynamics). The amount of energy which our Car loses due to air resistance is not affected by our Car's mass. Air resistance will extract the same amount of energy whether our Car weighs 1 ounce or 5 ounces!

This is important, too! In the face of the same air resistance (aerodynamic) energy loss, a lighter Car loses more speed than a heavier Car. This means that your car should weigh as much as the rules permit, unless aerodynamic losses can be almost totally eliminated. (There may be some trade-off involving wheel-axle friction when aerodynamic losses get low enough.)

Try to apply this analogy: If I have $1 and lose 50 cents, I have lost half of my "buying energy". But, if I have $5 and lose 50 cents, I have lost only one tenth of my "buying energy".

Your Car's shape will have a dramatic effect on air resistance. Anything that "sticks out" or "grabs air" will increase air resistance. Pretty things, such as wind shields and "spoilers", may look good, but they will slow your Car down.

Your Car should be shaped to move the air out of the way "gently" rather than abruptly. "Thin" (flat) designs seem to be much more effective than "fat" designs. (Note that there is some interplay between optimum aerodynamics and optimum weight distribution.)

**Relationship to Cross-section.**

A good rule of thumb is to shape your Car so that it moves as little air out of the way for as short a time as possible. This can be accomplished more easily if the car occupies a smaller volume, i.e. is more dense. Since metal (e.g. lead) is much more dense than wood, as much wood as possible should be replaced by lead. This will allow you to pack your allowed 5 ounces into a volume much smaller than if you relied totally on wood.

**Relationship to Surface Texture.**

As your Car zips through the air, every "nook and cranny" into which air may flow will increase the friction with air. My rule of thumb is to have as little surface as possible and to have that surface as smooth as possible. Smooth means that the surfaces are sanded to remove large irregularities and painted to fill in the small irregularities.

Oscillation is rhythmic motion. One source of oscillation in your Car is from wheels that are "out of round", i.e. that have run-out. The energy used up in creating oscillation depends on the magnitude and the frequency of the oscillation.

Wheels spin at a rate proportional to the Car's velocity. The frequency of the oscillation matches the rate at which the wheels spin. So, the faster that your Car tries to go, the more your Car is inhibited by oscillation.

The usual way to minimize oscillation is to gently sand (or cut) the tread of the wheel tread as the wheel is spun on a mandrel. (A mandrel is a device for holding the wheel while the whole assembly is spun by an electric drill. I prefer to use a drill press because that way I can move the surface of the wheel tread past the abrasive (cutter). This keeps the tread exactly parallel to the axle hole.)

**Oscillation "Disaster":**

Here is an example of how oscillation can dramatically impact a car. A few years ago, I constructed a car for our "open class" adult competition. The 2 inch diameter wheels were made from (copper-clad fiber glass) printed circuit board, and the hubs were made from short telescoped sections of brass tubing, stepping down from 1/8" O.D. The 1/8" tubing was cemented in place using epoxy. Super glue held the telescoped sections in place. The inside diameter of the hub was an excellent fit for an axle made from a straight pin, i.e. sewing pins of about 0.026" diameter.

I did a fairly good job of assembling the wheels so that the hubs were perpendicular to the wheels. But there was some error.

The axles / straight pins were pushed into the wood frame. Small holes were drilled so that the pins knew where to go!

The car was exceptionally fast, and readily won first place in the district's adult competition. However, the pins were easily bent by the stopping mechanism at the end of the track, and I found that before each race I had to bend the axles back where they belonged, aligning them "by eye."

The obvious "cure" was to provide support for both ends of the axle. Right? Well, in this case, wrong!

With this "improvement" the axle was no longer able to flex to compensate for wheel construction errors. The result was a significant side-to-side oscillation whose natural frequency resisted the car's attempts to exceed a particular wheel rpm. The result was a very poor performance.

Should you be interested in doing a computer simulation of the performance of a Car on a track, the following information may be of use to you. (Hint: divide the track up into short segments, e.g. 6 inches each, and compute the approximate time for your Car to pass through each segment and its velocity at the end of each segment. Writing this program in Basic is an excellent way to experiment with Car structure variations.)

**Relationships:**

- Force = Mass x Acceleration
- Weight (force) = Mass x Gravity constant
- Distance = Velocity x Time
- Energy (potential) = Mass x Height x Gravity constant
- Energy (kinetic) = Mass x Velocity ^ 2
- Momentum = Mass x Velocity
- Friction = Force normal x Coefficient of Friction

- Time: seconds
- Mass: grams
- Acceleration: centimeters / second ^2
- Velocity: centimeters / second
- Distance, height: centimeters
- Energy: ergs
- Force: dynes, gram x centimeters / second ^2
- Torque: dyne centimeters

**Constants and conversions:**

- Gravity: 975 centimeters / second 2
- Mass: 1 ounce = 28.3495 grams (on Earth)
- Distance: 1 inch = 2.54 centimeters
- Energy: 1 erg = 1 dyne x 1 centimeter
- Force: 1 dyne = 1 grams x centimeters / second ^2

**Fundamental Concept of Pinewood Derby Car Racing:**

It isn't how **fast** your Car runs... It's how **quickly** your Car reaches the finish line. "Fast" helps, but it is important to gain speed quickly and hold that speed through the "flat" to the finish line.

**Fundamental Equation of Pinewood Derby Car Racing:**

Kinetic Energy (speed) = Potential Energy - Lost Energy

Design your Car so that it runs in the steepest, smoothest, straightest and slickest lane, no matter which lane you use.

- Car as long as rules allow.
- Rear wheels as far back as rules allow.
- Front wheels forward as far as rules allow, without affecting rear wheel location.
- Car's center of mass as far back as car stability allows.
- Wheel alignment "dead-on".
- Weight as close to maximum allowed as possible.
- Car's cross section as small as possible.
- Wheels "in round", balanced, and all contact surfaces polished.
- Axle contact surfaces polished.
- Hub contact area close to wheel's axis.

**Examine rules for changes every year, especially limits or "boundary conditions", for performance implications.**

Latest update: 7/17/2002

Copyright 1995, 1997, 2002 © by Stan Pope. All rights reserved.