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Re: [Phys-l] Pinewood Derby Car Weight....



Some of you can probably predict what I'm going to say.

SCALING LAWS !

Scaling laws have been an important part of physics since
Day One of science as we know it, i.e. since 1638.

They are grievously under-emphasized in most high-school
science texts ... even though they are simple, powerful,
elegant, and more age-appropriate than much of the stuff
that is emphasized...............



In the case of pine cars:
-- The PE scales like mass and like height.
-- The KE scales like mass and like velocity squared.

Therefore IF (big IF!) these are the only two contributions,
then velocity scales like the square root of height, and like
mass to the zeroth power. That is, velocity is independent
of mass.

The remaining zeroth-order contribution is height, and as
others have pointed out, this is *not* the same for all cars.
The cars start out on a sloping ramp, but some (indeed most)
of their time on a nearly-flat straightaway, so moving the
center-of-mass to the rear is a winning strategy. Without
this, you really don't stand a chance of being competitive.

You want to move as much mass as possible to the back, but
there are some irreducible masses (such as the front wheels)
that you can't move back ... so to move the CM you want to
have as much lead as possible waaay in the back. (I'd be
tempted to use osmium, but that's disallowed under most
versions of the rules.) Just thinking about lead mass as
a percentage of total mass is a sufficient reason -- all
by itself -- for running the total mass right up to the
legal limit.

This is a sufficient answer to the original question. The
winning car will be right at max mass, for reasons having
to do primarily with mass distribution and only secondarily
with the magnitude of the mass.

A good teacher would probably shut up now, so as not to detract
from the main point which has already been covered. However,
as the saying goes, I can resist anything except temptation.....

Let's assume the remaining contenders have already figured out
the main point, and pushed the mass distribution as far as
it can go.

The aforementioned terms (KE and PE) are still the largest
terms, but if they are the same for all the contenders,
then they contribute nothing to the margin of victory.
That is, the zeroth-order terms drop out of the equation.
The races will be won by small margins, based on first-
order and higher-order terms.

This is a classic small-difference-between-large-numbers
scenario. Therefore the discussion of the zeroth-order
terms -- while absolutely a necessary starting place --
cannot be the end of the discussion.

Let's be clear: You have to start by checking the zeroth order
contributions to see whether they drop out ... but when you
find that they do drop out, you have to proceed to the first-
order and higher-order terms.

In the case of pine cars, there are many, many additional terms.
Some of these terms drop out, too.

*) For example, if you take a simple model of sliding friction,
the force scales like weight (and therefore like mass) times
the coefficient of sliding friction ... so once again adding
mass contributes nothing to the velocity-scaling law.

However, as others have mentioned, anything you can do to lower
the /coefficient/ is advantageous. Hence the truing and
polishing of axles, and -- if allowed -- fancy lubricants.

*) As others have mentioned, aerodynamic drag is in a completely
different category from the contributions mentioned above. It
scales like area, like velocity squared, and like some dimensionless
coefficient of drag that depends on the detailed shape. (Forward-
facing concave "parachute" shapes are not recommended!) It
is conspicuously independent of mass, so this is a good (albeit
secondary) answer to the original question: Added mass helps
swamp aerodynamic drag.

*) Accurate steering is essential, as others have pointed out.

*) One thing I didn't see mentioned in this thread so far: Race
car suspension is very important, both for full scale and for
models. The pine car track is not glassy-smooth. If the car
hits a "speed bump" it may fly up into the air. Whatever
energy was imparted to this vertical mode is likely to be
dissipated and never seen again, thereby reducing the car's
overall energy (and speed). The typical pine car suspension
is grossly too stiff ... which means that adding mass brings
it somewhatecloser to where it should be. This is another
sound (albeit secondary) answer to the original question.

*) Et cetera. There are lots more contributions to the energy
budget.