The Study of Free Fall Provides One Context to Consider $g$

question 46

Essay

The study of free fall provides one context to consider differential equations. In the simplest case, in the absence of air or other resistance, physicists assume that the rate of change of velocity of a body is constant. But it is more realistic to consider the presence of air resistance. Assume that $g$ is the constant of acceleration due to earth's gravity. Suppose that air resistance is proportional to the velocity of the falling body.(a) Explain why the differential equation $\frac { d v } { d t } = g - k v$ , where $k$ is a positive constant, would be a reasonable model for velocity under these conditions.
(b) When does $\mathcal { V }$ increase most rapidly? Justify your answer.
(c) Consider the equation in part (a). What would happen to the rate of change of velocity, $\frac { d v } { d t }$ , as $t$ increases? Justify your conclusion.
(d) Make a sketch of a possible solution for this differential equation.

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