PROBLEM 16E

How High?—Nonlinear Air Resistance Consider the 16-pound cannonball shot vertically upward in Problems 36 and 37 in Exercises 3.1 with an initial velocity v0 = 300 ft/s. Determine the maximum height attained by the cannonball if air resistance is assumed to be proportional to the square of the instantaneous velocity. Assume that the positive direction is upward and take k = 0.0003. [Hint: Slightly modify the DE in Problem 15.]

REFERENCE : Problems 36 and 37 in Exercises 3.1

How High?—Linear Air Resistance Repeat Problem 36, but this time assume that air resistance is proportional to instantaneous velocity. It stands to reason that the maximum height attained by the cannonball must be less than that in part (b) of Problem 36. Show this by supposing that the constant of proportionality is k = 0.0025. [Hint: Slightly modify the DE in Problem 35.]

How High?—No Air Resistance Suppose a small cannonball weighing 16 pounds is shot vertically upward, as shown in Figure 3.1.10, with an initial velocity v0 = 300 ft/s. The answer to the question “How high does the cannonball go?” depends on whether we take air resistance into account.

(a) Suppose air resistance is ignored. If the positive direction is upward, then a model for the state of the cannonball is given by (equation (12) of Section 1.3). Since the last differential equation is the same as where we take g = 32 ft /s2. Find the velocity v(t) of the cannonball at time t.

(b) Use the result obtained in part (a) to determine the height s(t) of the cannonball measured from ground level. Find the maximum height attained by the cannonball.

Solution

Step 1 of 5

In this problem we have to determine the maximum height attained by the 16 pound cannon ball

Which is shot vertically upwards considering the air resistance is proportional to square of the instantaneous velocity.

We have