ABSTRACT (means the message is long):
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The air resistance data are no longer "quick and dirty". Propagation of
random errors of d --> "differences of differences" is very instructive
and worth emphasizing. The bottom line is that the R=F(v) curves are
quadratic.
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The method, as before, is based on the ultrasonic detector (from Vernier)
connected to a Mac computer. Let us see the difficulties. To find the air
resistance force, R, we use the self-evident formula: R=m*(9.8-a), where
a is the "instantaneous" acceleration and m is the mass of the falling
object. If the terminal v is reached then a=0 and R=m*g, as it may be for
a folling coffee filter or a parachute. Here are the data for a ball whose
mass was 0.550 kg and whose diameter was 9 inches. The data sampling rate
was 40 per second; the averaging was set to 15.
Why is velocity decreasing at t>1.75? This is due to averaging. Suppose you
collected 50 data points. Single measurements are never accurate so the
averaging is imposed. At any given position d is the average (of what is
measured and of 14 data point before it). This means that the first 15
distances (when v>0) must be rejected. The same is true for the last 15
distances. (you will see v>0 when the object is already at rest) In our
case distances between t=1.45 and 1.70 are "real" while those outside of
this region are "phony". To get a broader range of v one must reduced the
averaging span (the software allows for averaging over 3, 5, 7, 9 and 15).
But less averaging leads to broader fluctuations in the values of a. As in
the case of a camcorder, this has to do with the fact that aceelerations
are calculated as "difference of differneces". Slight improvement can be
obtained by additional averaging in the 4th column; for example, by using
the mean of 9.48, 9.44 and 9.38 instead of 9.44, etc.
Do not forget that high accuracy of a is essential because R is calculated
as m*(9.8-a). If a=9.44 (+/- 3%) then R is 0.2 (+/- 100 %). We think this
is a good topic for a student research project. The air resistance forces
acting on the falling ball can be approximated by a smooth R=0.0128*v^2
curve. An experiment with a spherical balloon (of the same size as the ball)
loaded with a small mass would help to collect data at smaller v. It is
easier to get good data when R is the main player, not a small contributer.
The rest of our data were collected with coffee filters loaded with aluminum
disks of known masses. The mass of each filter (about 1 gram), and of each
disk, was known to better than 1%. We have 15 data points between v=0.95 m/s
and v=4 m/s. The smooth curve over that region is R=0.100*v^2. The method is
the same as in the case of the ball but working with small a (zero when m is
less than 3 grams) is much easier for obvious reason.
Ludwik Kowalski and Richard Hodson
P.S.
Time zero corresponds to the moment at which the instrument is activated;
the object is usually released about one second later.