Solved: Free fall An object in free fall may be modeled by

Chapter 7, Problem 62

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Free fall An object in free fall may be modeled by assuming that the only forces at work are the gravitational force and resistance (friction due to the medium in which the object falls). By Newtons second law (mass * acceleration = the sum of the external forces), the velocity of the object satisfies the differential equation m # v_1t2 = mg + f 1v2, where f is a function that models the resistance and the positive direction is downward. One common assumption (often used for motion in air) is that f 1v2 = -kv2, where k 7 0 is a drag coefficient. a. Show that the equation can be written in the form v_1t2 = g - av2, where a = k>m. b. For what (positive) value of v is v_1t2 = 0? (This equilibrium solution is called the terminal velocity.) c. Find the solution of this separable equation assuming v102 = 0 and 0 6 v1t22 6 g>a, for t 0. d. Graph the solution found in part (c) with g = 9.8 m>s2, m = 1 kg, and k = 0.1 kg>m, and verify that the terminal velocity agrees with the value found in part (b).