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Re: air resistance



On Wed, 26 Nov 1997, LUDWIK KOWALSKI wrote:

Several people asked for the row data backing the claim that for v<4 m/s
the dependance of R on v is roughly linear. Here are the data from another
try. This time maximum speed is less than 3 m/s; we do not control the
initial balloon locations very carefully. Data collection rate was 50 per
second (not 20) aveaging was 15 (not 7). High averaging reduces fluctuations
of accelerations (due to errors) but prevents us from measuring small v
(first 15 data points were ignored). Numbers were copied from the table
generated by Mac Motion, do not criticize us for digits which are clearly
not significant.

Low v data will be collected with a smaller mass, perhaps later today.

mass=97 grams ---> R=0.097*(9.8-a).
This run indicates that R=0.1*v, (not 0.08*v).

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

0.56 0.520 1.324 8.379 0.137
0.58 0.549 1.492 8.171 0.158
0.60 0.581 1.655 8.058 0.169
0.62 0.617 1.815 8.009 0.174
0.64 0.655 1.974 7.859 0.188
0.66 0.695 2.122 7.392 0.233
0.68 0.739 2.274 7.356 0.237
0.70 0.786 2.420 7.231 0.249
0.72 0.837 2.570 7.322 0.240
0.74 0.891 2.716 7.272 0.245
0.76 0.948 2.859 7.202 0.252

Ludwik,

I don't think you can conclude that air resistance is a linear
function of speed from these data. I fit your d vs. t data with a
simple type of "predictor-corrector" method (which predicts a new
v, uses it to predict a new a, and then uses the average of the
old a and the projected new a to calculate the "actual" new v and
also uses the average of the old v and the new v to calculate the
new d). I found a best fit with a drag per unit mass (in SI
units) of 0.283*v^(2.500). The detailed results were as follows:

t(s) d(m) v(m/s) a(m/s^2)
0.56 .521 1.294 9.261
0.58 .549 1.477 9.049
0.60 .580 1.656 8.802
0.62 .615 1.829 8.520
0.64 .654 1.996 8.207
0.66 .695 2.157 7.867
0.68 .740 2.310 7.504
0.70 .787 2.457 7.123
0.72 .838 2.595 6.730
0.74 .891 2.726 6.330
0.76 .947 2.848 5.927

The d data is reproduced with an rms error of about 0.001 m--
presumably well within tolerances. Notice, however, that the
acceleration values differ markedly from those you report. I
suspect that the averaging method used (by MacMotion?) to
calculate the velocity and acceleration values is simply not to be
trusted.

Just to see what I would get from a less robust numerical method,
I repeated the analysis using both the standard and modified Euler
methods (modified = new d calculated from *new* v instead of old
v). Using the standard Euler method, I found a best fit with a
drag per unit mass of 0.925*v^(1.116) and with the modified Euler
method I got 1.184*v^(.8019). In both cases I also reproduced the
d data with rms errors of about .001 m. Both of these results
would seem to indicate something like linear drag, but the
numerical methods are clearly inferior.

Conclusion: The indicated drag exponent is a strong function of
the analysis method, so I wouldn't place much faith in *any* of
these results. It is likely the case that data over such a small
range of time and velocity simply cannot yield conclusive results
unless it is taken much more frequently and/or accurately.

John
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A. John Mallinckrodt http://www.intranet.csupomona.edu/~ajm
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