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Topic: Re: applications of a simple differential equation
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mark snyder

Posts: 1,300
Registered: 12/3/04
Re: applications of a simple differential equation
Posted: Sep 22, 1997 10:58 PM
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I would like to apologize to the members of the list for saying something
which is not completely correct, and is misleading.

The question was about the differential equation y' = - ky. I stated that
the motion of an object through the air was governed by the related
equation v' = g - kv. This is not false, but it is misleading. The
equation in question (Stokes' equation, if the object is a sphere) is
regnant when the object is "very small," like the water droplets in fog, or
the spores of plants. In that case, the constant k is proportional to a
length L characteristic of the object, divided by the mass, i.e., to 1/L^2,
so that the terminal velocity (v for v' = 0) is proportional to L^2. But
when the object is large, like a horse, a mouse, or a person plummeting
through the air, this is a very unrealistic approximation to the true

The actual situation is very complicated, and depends on the "Reynolds
number" (I will not define that: see any book on fluid mechanics, e. g.,
that of Batchelor), among other things. In physics books (e.g., Becker,
Theoretical Mechanics, 128ff), one generally treats the case given above,
and the slightly more realistic case in which Newton's Resistance Law is

v' = g - (Ak/m)v^2

where k is some constant, m is the mass of the object, and A is the
cross-sectional area of the object perpendicular to the direction of
translation. This diffeq can be solved as follows:

Multiply both sides by dt, then separate the variables to obtain:

dv/[(mg/kA) - v^2] = kAdt

Then integrate to obtain:

v = sqrt(mg/Ak). tanh(t*sqrt(gkA/m))

The limiting velocity is the limit as t --> infty, which is also obtained
by setting v' = 0:

v_(term) = sqrt(mg/kA)

For two objects of the same density and same shape, oriented the same way,
m is proportional to L^3, and A to L^2, so the terminal velocity (i.e.,
speed) will be proportional to sqrt(L). How this applies to realistic
situations (a horse, a mouse, or the elephant in "Dumbo Drop," for
instance) depends on the extent to which this equation governs the motion.
My impression is that in general, one runs wind-tunnel tests and does
things numerically (the hardest part of which is, of course, convincing the
horse to get into the wind-tunnel).

Perhaps a good calculator exploration for AP Calc students would be to
write the generalized equation v' = g - Kv^n and compare how the approach
to the terminal velocity depends on n. Or to calculate the trajectory of a
projectile fired from the surface of the earth, assuming that there is a
resistive force Kv^n directed opposite to the projectile's motion,
comparing this with the parabolic shape of the trajectory in the absence of
air resistance.

The subject of how objects move in fluids is a very complex one, and I am
far from an expert in these matters; perhaps there is someone on the list
with some practical experience who can say more, or correct me?

mark snyder

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