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Re: applications of a simple differential equation
Posted:
Sep 22, 1997 10:58 PM


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 motion.
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 valid:
v' = g  (Ak/m)v^2
where k is some constant, m is the mass of the object, and A is the crosssectional 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 windtunnel tests and does things numerically (the hardest part of which is, of course, convincing the horse to get into the windtunnel).
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



