Back to
the N=2 (Quadratic Drag) case

5. Sample Results

To illustrate the properties of these solutions, let us make some graphs of
the N=1 and N=2 solutions, with unit time constant and the same terminal
velocity, as a guide to identifying them and possibly fitting data. The two
results are:

Linear drag: The velocity for the N=1 case was given in Eq. (7) [in
Section 3]:

where T1 [defined in Section 3] is the time constant
for the "decay" of the velocity as it approaches the terminal velocity. This
function is shown with the blue curve in the figure below for the
Vmax = 1 and T1 = 1 case, which are
effectively "natural" units that are easily scaled to any problem.

Quadratic drag: The velocity for the N=2 case was given in Eq. (10) [in
Section 4]:

where T2 [defined in Section 4] is the time constant
function is shown below with the red curve and values of Vmax
= 1 and T2 = 1, again effectively in "natural" units that are
easily scaled to any problem.

Figure 5.1:

Plot of v(t) vs. t
for the N=1 and N=2 solutions
with unit time constants and same terminal
velocity.

Notice that the quadratic drag case (red curve) rises much more quickly and
reaches the terminal velocity sooner. However, part of the difference you see is
due to the fact that the time constants have different meaning for these two
functions. They are not equal when t = 1 as we would wish if we were
using the speed relative to the terminal velocity at t = 1 as a data
point.

We can make the quadratic drag speed at t = 1 equal to that for linear
drag (whose value is 0.63212 at t = 1) by choosing T2 =
1.34239. This gives the result shown below.

Figure 5.2:

Plot of v(t) vs. t
for the N=1 and N=2
solutions that have the same terminal
velocity and
the same velocity at t = 1.

You can see that the velocity function for quadratic drag (red curve) still
rises faster than that for linear drag, but the differences are not so dramatic.
This shows that if you were to measure Vmax and the time when
v(t) reaches some intermediate value, you do not have enough information
to choose between these two functions. Further, you can see that you need pretty
good data at some other times to make that choice between models.

I say more about this in the next section.

Next,
Fitting data to get CD.

Jim Carr
2000-02-17