# Suppose the model in Problem 40 is modied so that air resistance is proportional to v 2

## Question:

Suppose the model in Problem 40 is modied so that air resistance is proportional to v^{2}, that is,

M dv/dt = mg - kv^{2}.

See Problem 17 in Exercises 1.3. Use a phase portrait to find the terminal velocity of the body. Explain your reasoning.

**Data from problem 17**

For a first-order DE dy/dx = f (x, y) a curve in the plane defined by f (x, y) = 0 is called a nullcline of the equation, since a lineal element at a point on the curve has zero slope. Use computer software to obtain a direction field over a rectangular grid of points for dy/dx = x^{2} - 2y, and then superimpose the graph of the nullcline y = ½ x^{2} over the direction field. Discuss the behavior of solution curves in regions of the plane dened by y < ½ x^{2} and by y > ½ x^{2}. Sketch some approximate solution curves. Try to generalize your observations.

**Data from problem 40**

Terminal Velocity In Section 1.3 we saw that the autonomous differential equation

M dv/dt = mg - kv,

where k is a positive constant and g is the acceleration due to gravity, is a model for the velocity v of a body of mass m that is falling under the influence of gravity. Because the term 2kv represents air resistance, the velocity of a body falling from a great height does not increase without bound as time t increases. Use a phase portrait of the differential equation to find the limiting, or terminal, velocity of the body. Explain your reasoning.

## Step by Step Answer:

**Related Book For**

## A First Course in Differential Equations with Modeling Applications

**ISBN:** 978-1305965720

11th edition

**Authors:** Dennis G. Zill