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Re: Air resistance



On Wed, 10 Dec 1997, Vern Lindberg wrote:

John,
Precisely what are you fitting? d versus t? With what function? Did you
integrate to get d-t formula for different resistance functions or what?

I am performing a numerical integration (via spreadsheet) of the
equations of motion

d/dt d(t) = v(t) and d/dt v(t) = g - r*[v(t)]^n

based on the following:

1) An assumed value for g of 9.8 m/s^2

2) Five fit parameters: do = d(1.4s), vo = v(1.4s), g, r, and n

3) A constant time step of .025 s

4) A numerical method that

a) Calculates the current acceleration, a_curr
b) Predicts the next velocity as v_pred = v_curr + a_curr * dt
c) Predicts the next acceleration, a_pred, based on v_pred
d) Determines the next velocity as
v_next = v_curr + (1/2)(a_curr + a_pred) * dt
e) Determines the next d as
d_next = d_curr + v_curr * dt + (1/2)[(2/3)a_curr + (1/3)a_pred] * dt^2

(The 1/2 factor in the equation for v_next and the 1/3 and 2/3
factors in the equation for d_next are the result of linearly
interpolating the acceleration from a_curr to a_pred over the
interval.)

N.B. I have checked the results of this numerical method against
the results provided by the far more robust numerical methods
built into Maple and find that the two agree precisely to less
than the resolution of the data provided by Ludwik.

I then adjust the values of the parameters (generally keeping g
fixed at 9.8 m/s^2) to minimize the sum of the squares of the
residuals between the d values of the model and those of the data.

Of course, by dividing my sum of squares of the residuals by the
square of an *assumed* constant uncertainty in the d values, I can
obtain a hypothetical value for chi-squared (as suggested by Jack
Uretsky earlier). Since we expect chi-squared to be approximately
equal to the number of degrees of freedom (12 in this case), I can
get an idea of the uncertainty in d (assuming the fitting function
is valid). It turns out to be about 0.8 mm, pleasingly in line
with the implicit resolution of the data. Anything less than
about 0.7 mm generates a chi-squared that would be too large to
support the validity of the fitting function.

Based on the above, I have now done a detailed chi-squared
analysis on the value of n and find that it is minimized at n =
2.9 and is extremely well fit by a cubic polynomial in n over the
region from n = 0.5 to 7. Based on the behavior of chi-squared
with an assumed uncertainty in d of 0.8 mm, I conclude that the
value of n is somewhere between 0.7 and 6 at the 95% confidence
level.

John
-----------------------------------------------------------------
A. John Mallinckrodt http://www.intranet.csupomona.edu/~ajm
Professor of Physics mailto:ajmallinckro@csupomona.edu
Physics Department voice:909-869-4054
Cal Poly Pomona fax:909-869-5090
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