Raindrops Keep Falling . . . When a bottle of liquid refreshment was opened recently, th...

Raindrops Keep Falling . . . When a bottle of liquid refreshment was opened recently, the following factoid was found inside the bottle cap: The average velocity of a falling raindrop is 7 miles/hour. A quick search of the Internet found that meteorologist Jeff Haby offers the additional information that an “average” spherical raindrop has a radius of 0.04 in. and an approximate volume of 0.000000155 ft³. Use this data and, if need be, dig up other data and make other reasonable assumptions to determine whether “average velocity of . . . 7 mi/h” is consistent with the models in Problems 35 and 36 in Exercises 3.1 and Problem 15 in this exercise set. Also see Problem 36 in Exercises 1.3.

(reference problem 15 in exercise 3.1)

A small metal bar, whose initial temperature was 20° C, is dropped into a large container of boiling water. How long will it take the bar to reach 90° C if it is known that its temperature increases 2° in 1 second? How long will it take the bar to reach 98° C?

(reference problem 36 in exercise 3.1)

How High?—No Air Resistance Suppose a small cannonball weighing 16 pounds is shot vertically upward, as shown in Figure 3.1.12, 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.

Suppose air resistance is ignored. If the positive direction is upward, then a model for the state of the cannonball is given by d²s/dt²= - g (equation (12) of Section 1.3). Since ds/dt = v(t) the last

differential equation is the same as dv/dt= - g, 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.