The velocity v of a freefalling skydiver is well modeled

Chapter , Problem 12

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The velocity v of a freefalling skydiver is well modeled by the differential equation m dv dt = mg kv2, where m is the mass of the skydiver, g is the gravitational constant, and k is the drag coefficient determined by the position of the diver during the dive. (Note that the constants m, g, and k are positive.) (a) Perform a qualitative analysis of this model. (b) Calculate the terminal velocity of the skydiver. Express your answer in terms of m, g, and k. Exercises 1315 consider an elementary model of the learning process: Although human learning is an extremely complicated process, it is possible to build models of certain simple types of memorization. For example, consider a person presented with a list to be studied. The subject is given periodic quizzes to determine exactly how much of the list has been memorized. (The lists are usually things like nonsense syllables, randomly generated three-digit numbers, or entries from tables of integrals.) If we let L(t) be the fraction of the list learned at time t, where L = 0 corresponds to knowing nothing and L = 1 corresponds to knowing the entire list, then we can form a simple model of this type of learning based on the assumption: The rate d L/dt is proportional to the fraction of the list left to be learned.Since L = 1 corresponds to knowing the entire list, the model isd Ldt = k(1 L),where k is the constant of proportionality.