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Aristotle is sometimes quoted as claiming that a body falls at a speed proportional to its mass. Let us suppose that Aristotle’s experience was based on objects falling in air (with turbulent flow drag) essentially at their terminal speeds.
a. If Aristotle were considering spherical objects, say, all of the same size but of different materials (hence different densities), what would the actual relation between terminal fall speed and mass be? That is, if we were to write vt= K1f1(m), where f1(m) is a function only of mass and K1 contains all the factors independent of mass, what would the function f1(m) be?
b. Now suppose Aristotle were considering spheres all of the same material and density, but different radii, so that mass varies with radius. If we once again write vT= K2f2(m), where f2(m) depends only on mass and K2 contains all the mass independent factors, what would the new function of f2(m) be?

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