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Re:air resistance



ABSTRACT (means the message is long):
********
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.

t(s) d(m) v(m/s) a(m/s^2) R(N)

1.400 0.608 1.773 9.63
1.425 0.655 2.013 9.65
1.450 0.708 2.254 9.65
1.475 0.668 2.494 9.64 0.09
1.500 0.835 2.734 9.60
1.525 0.904 2.975 9.61 0.10
1.550 0.983 3.215 9.59
1.575 1.066 3.454 9.54 0.14
1.600 1.155 3.691 9.48
1.625 1.251 3.927 9.44 0.20
1.650 1.352 4.162 9.38
1.675 1.458 4.395 9.33 0.26
1.700 1.573 4.632 9.31 0.27
1.725 1.692 4.865 9.33
1.750 1.816 5.035 7.83
1.775 1.945 4.971 1.27

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.