# Towing Icebergs, Falling Dominoes, and Other Adventures in Applied Mathematics

ROBERT B. BANKS
Pages: 344
https://www.jstor.org/stable/j.ctt1r2dk1

1. Front Matter
(pp. i-vi)
(pp. vii-viii)
3. Preface
(pp. ix-xii)
4. Acknowledgments
(pp. xiii-2)
5. 1 Units and Dimensions and Mach Numbers
(pp. 3-14)

About twenty to twenty-five thousand years ago, an enormous meteor hit the earth in northern Arizona, approximately sixty kilometers southeast of the present-day city of Flagstaff. This meteor, composed mostly of iron, had a diameter of about 40 meters and a mass of around 263,000 metric tons. Its impact velocity was approximately 72,000 kilometers per hour or 20,000 meters per second. With this information, it is easy to determine that the kinetic energy of the meteor at the instant of collision was$e = (1/2)m{U^2} = 5.26 \times {10^{16}}$joules. This is about 625 times more than the energy released by an ordinary atomic bomb.

This...

6. 2 Alligator Eggs and the Federal Debt
(pp. 15-23)

About two hundred years ago, an English clergyman-economist named Thomas Malthus published a series of essays (1798, reprinted 1970) in which he contended that populations grow according to the law of geometric progression. That is, if a population of some locale has a certain magnitude at a particular moment, then that population will double itself at the end of a specified time period, and this periodic doubling of population will continue indefinitely. For example, if the population is$N = 1$at time$t=0$, then after each specified time period,${t_2}$, the population will be 1, 2, 4, 8, 16, and so...

7. 3 Controlling Growth and Perceiving Spread
(pp. 24-30)

We saw in the previous chapter that things which multiply according to geometric progression, or equivalently, exponential growth, can increase to very large amounts in relatively short times. Good examples are the number of alligator eggs, the amount of the federal debt, and your savings account earning compound interest.

Something always spoils the fun and here it is: Things cannot grow exponentially forever. A mathematically inclined proud father keeps a record of the height of his son during the period the son is from one to five years old. He then comes up with a great formula:$H = 28.2{e^{0.09t}}$, where$H$...

8. 4 Little Things Falling from the Sky
(pp. 31-41)

There are always things falling from up there somewhere: smoke particles, dust, and volcanic ash; rain, snow, sleet, hail, and other objects that do not deter the postal service; leaves, bombs, Wall Street bankers, and skydivers. Some of us have been targets of seagulls as we walk along the beach. Others have caught baseballs sailing out of Wrigley Field as the Chicago Cubs hit yet another home run. Still others, enjoying coffee and newspaper in the kitchen, are distracted when a small meteor crashes through the ceiling. Isaac Newton, impacted by apples, devised the universal law of gravitation. Even the...

9. 5 Big Things Falling from the Sky
(pp. 42-53)

Within the category of big things falling from the sky, we have those phenomenal objects we call asteroids, comets, and meteors. And we all know that things descending from the heavens do not get much bigger than these.

In the introduction of chapter 1, we looked briefly at the incredible impact of a very high-velocity meteor in Arizona about 25,000 years ago. We shall examine this awesome event in more detail later in this chapter. In the meantime, let’s carry out some computations involving objects somewhat less spectacular than meteors.

Right now, our main task is to determine the kinetic...

10. 6 Towing and Melting Enormous Icebergs: Part I
(pp. 54-67)

The next topic we are going to examine involves the towing of icebergs. Now even though this rather intriguing method of obtaining fresh water has been talked about for quite a long time, nothing has ever been carried out with respect to actual engineering projects. However, it may well be that as the world gets much more crowded and people get much more thirsty, this source of fresh water will begin to look considerably more attractive. Then we may see some progress.

In any event, at the start of our iceberg towing studies we need to determine where in the...

11. 7 Towing and Melting Enormous Icebergs: Part II
(pp. 68-78)

Continuing our problem of towing an iceberg from Antarctica, we recall that the mightyIowahad securely attached a long cable to an enormous iceberg and is ready to leave the Ross ice shelf (longitude 150°W, latitude 75°S) for its 12,340-kilometer journey to Long Beach (longitude 118°W, latitude 34°N).

The 50,000-tonIowawith its 200,000 shaft horsepower, has already accelerated to the$U = 2.0$knots constant towing speed, pulling the 1,700 times more massive iceberg behind it. The captain of the ship now intends to follow the great-circle route since this is the shortest distance to Long Beach. Questions: (1) what...

12. 8 A Better Way to Score the Olympics
(pp. 79-92)

The subjects of probability and statistics are extremely important areas of the broad field of mathematics. In this chapter, and those that follow, we shall look at several topics which show how statistics and probability are used to analyze many kinds of phenomena and events. A complete list of the practical applications of statistics and probability would be endless: everything from the probability that it will rain tomorrow to your likelihood of winning at Las Vegas and from the annual cost of your life insurance to your chances of being kicked in the head by a horse or struck by...

13. 9 How to Calculate the Economic Energy of a Nation
(pp. 93-108)

In this chapter, as we continue our study of statistics and probability, we shall examine a number of topics involving economics, physics, and geography.

Let’s pick up where we left off in our previous chapter. Recall that we had generated an equation that allegedly predicts the score a nation should be expected to receive at the Olympic Games. By “score” we mean the summation of the weighted values of the gold, silver, and bronze medals a nation receives at the Games. The equation we got is

$$S = 0.125{P^{1/3}}{G^{2/3}}$$, (9.1)

in which$P$is the population of a nation in millions of...

14. 10 How to Start Football Games, and Other Probably Good Ideas
(pp. 109-120)

We are now going to look at a few problems that are based on some simple concepts of probability. The first problem shows how you can determine the numerical value of π = 3.1416 by simply dropping a needle or a grain of sand onto a sheet of paper laid flat on a table. The second problem gives you the equation for designing a coin that, when tossed, is just as likely to land on its edge as it is on one side or another. The third problem produces a secret formula that will enable you to make millions, at...

15. 11 Gigantic Numbers and Extreme Exponents
(pp. 121-132)

“The bigger the better, the more the merrier.” These words seem to characterize the fascination we have about gigantic things and enormous numbers.

The Empire State Building is 1,250 feet high. The Queen Mary weighs 75,000 tons. The largest recorded iceberg covered an area of 10,000 square miles. The Panama Canal required the excavation of 250,000,000 cubic yards of rock. Over the past 1,000,000 years about 80,000,000,000 people have lived. The distance to the nearest star is 25,000,000,000,000 miles. And so on.

In this chapter we are going to look at a few situations involving awesomely large numbers. As a...

16. 12 Ups and Downs of Professional Football
(pp. 133-149)

“The rich get richer and the poor get poorer.” That may be true for a lot of things but it’s certainly not true as far as team performance in the National Football League is concerned. We shall come back to this point later on.

In chapter 2, we introduced the following simple relationship, which describes so-called Malthusian or exponential growth:

$\frac{{dN(t)}}{{dt}} = aN(t)$, (12.1)

where$N(t)$is the magnitude of something at time$t$, and$a$is a growth coefficient. This equation says that therateat which the magnitude of the “something” is increasing is directly proportional to its present magnitude....

17. 13 A Tower, a Bridge, and a Beautiful Arch
(pp. 150-167)

Since the dawn of civilization, mathematics has been utilized to provide information needed to design and build massive structures. The earliest example is the Great Pyramids of Egypt.There was substantial knowledge of geometry and trigonometry at the time these spectacular feats of engineering were carried out (2650 to 2500 B.C.). The Pyramids are still there, of course, and are in remarkably good shape considering their forty-five centuries of exposure to the ravages of nature and mankind.

The civil and military engineers of Rome constructed massive buildings, bridges, aqueducts, and forts throughout the empire. Architects and engineers of the Middle Ages...

18. 14 Jumping Ropes and Wind Turbines
(pp. 168-178)

In the previous chapter, in our study of the famous Gateway Arch in Saint Louis, we analyzed the mathematical curve called the catenary. You will remember that this is the curve that describes the shape of a sagging flexible cable. We obtained the following equation for the catenary:

$y = H\left[ {1 - \frac{{\cosh (x/a) - 1}}{{\cosh (L/2a) - 1}}} \right]$, (14.1)

in which cosh is the hyperbolic cosine and$a = {T_0}/{\rm{p}}g$, where$p$is the mass of the cable per unit length,$g$is the force per unit mass due to gravity$\left( {g = 32 \cdot 2ft/{s^2} = 9 \cdot 82m/{s^2}} \right)$, and${T_0}$is the tension force in the cable at the lowest point. This equation, shown in figure 14.1,...

19. 15 The Crisis of the Deficit: Gompertz to the Rescue
(pp. 179-188)

In preceding chapters we have seen how mathematics can be utilized to provide answers to many kinds of practical problems of the real world. Without question, among the most important categories of such problems are those associated with the economics and finance of our nation and our federal government.

In this regard, many people think we presently face the following very serious problem. The problem is that for quite a long time our government has been giving out money much faster than it has been taking in money. Nearly everyone believes that this is not a good idea.

Some say...

20. 16 How to Reduce the Population with Differential Equations
(pp. 189-200)

It seems as if everything grows: The trees in the park, your monthly fuel bill, people, your taxes, the federal debt, the amount of pollutants in the atmosphere, the number of cars on the freeway, the populations of cities, nations, and the world, and so on. Sometimes things grow at a steady rate and sometimes they speed up or slow down. Fuel bills rise fast with the onset of winter and decline in the spring; the reverse is true of plants and trees. Everyone knows that teenagers grow at incredibly high speeds, thirty-somethingers have leveled off, at least in height,...

21. 17 Shot Puts, Basketballs, and Water Fountains
(pp. 201-218)

One of the most ancient problems involving mathematical analysis is the problem of determining the trajectories—the flight paths—of projectiles. Historically, this branch of science has been known asballistics. It is impossible to determine when mankind first became interested in the subject. It is safe to say that prehistoric cavemen, shooting rocks from slingshots and throwing spears at saber-toothed tigers, did not worry too much about the physics and mathematics involved. However, without doubt, the military engineers of Rome had extensive empirical knowledge concerning how high and how far their catapults could throw big stones. Likewise, the artillery...

22. 18 Balls and Strikes and Home Runs
(pp. 219-233)

In our study of trajectories in the preceding chapter, it was established that when an object, for example, a tennis ball, is moving through a vacuum (i.e., no air), it is not difficult to determine its path. The problem is easy because there is only one force acting on the object: gravity force. In this case, the trajectory is simply a parabolic curve. However, when the object is moving through the air, it is usually necessary to consider also the effect of another force: drag force due to the air. Generally, this makes computation of the trajectory much more difficult....

23. 19 Hooks and Slices and Holes in One
(pp. 234-242)

We have completed our analysis of baseball trajectories and move on to an examination of the trajectories of golf balls. In many ways the two studies are similar. As an introduction to our analysis, we list some of the characteristics of golf balls:

circumference,$C$: 5.10 to 5.28 inches

diameter,$D$: 1.65 inches, 0.138 feet, 0.042 meters

weight,$W$: 1.62 ounces, 0.101 pounds, 0.046 kilograms

density,$\rho$: 2.296 slugs/ft³, 1186 kg/m³

roughness elements: 330 to 400 small dimples spaced uniformly over the ball’s surface

In our analysis of the flight path of a baseball, recall that we derived the equations...

24. 20 Happy Landings in the Snow
(pp. 243-253)

In the preceding chapters we examined a number of topics dealing with trajectories of objects moving through a vacuum or through air. We conclude our coverage of the subject with the analysis of another kind of object moving through air: the ski jumper.

As in the case of baseballs and golf balls, important aerodynamic forces are encountered in ski jumping. As the ski jumper accelerates down the in-run ramp and then springs into the air at the end of the take-off table, to begin the free-flight portion of the jump, drag forces and lift forces due to the air are...

25. 21 Water Waves and Falling Dominoes
(pp. 254-269)

In this chapter we are going to explore a few topics involvingwavesandwave motion. We will not be able to do much more than scratch the surface; the scope of the subject is extremely broad.

Think of how many different kinds of waves there seem to be and how commonplace and crucial many of them are in our everyday lives. Here is a list of various kinds of waves—to which you can probably add some:

light waves

sound waves

electromagnetic waves

ocean waves

earthquake waves

flood waves

traffic waves

atmospheric waves

epidemic waves

population waves

Over a...

26. 22 Something Shocking about Highway Traffic
(pp. 270-282)

In this chapter we are going to look at some topics concerning phenomena of wave action on roads and highways. Not many people realize that such things exist. When we hear about wave motion, compression waves, expansion waves, shock waves, and the like, we usually think about ocean waves, sound waves, or even earthquake waves. Yet nearly every time we drive along the highway or expressway we experience “traffic waves” in one form or another. So let’s start our chapter with the following little episode of life involving you and your car.

You are about two hundred yards from a...

27. 23 How Tall Will I Grow?
(pp. 283-299)

The basic question is: How tall will I grow? We begin our answer to this question with a presentation of the results of measurements of the heights$H$of a large number of American boys and girls, as cited by Tanner (1978). These measurements are given in table 23.1. In addition to the height data, the table also shows height velocities,$H$; this quantity is the annual increase of height of the children. As the table indicates, both$H$and$H$depend on time (i.e., age)$t$.

The title of the table states that the tabulated quantities correspond to “fiftieth...

28. 24 How Fast Can Runners Run?
(pp. 300-320)

Humans have been running and racing for as long as there have been humans. An early example: Prehistoric cavemen, without doubt, displayed remarkable running abilities as they endeavored to widen the gap between themselves and hotly pursuing saber-toothed tigers.

Nearly everyone has heard of the race called the marathon, the length of which is 26 miles and 385 yards. This is the distance a messenger allegedly ran, in the year 490 B.C., from the plains of Marathon to the city of Athens to convey the news of the Greek victory over the Persians. And so on throughout history.

Now even...

29. References
(pp. 321-326)
30. Index
(pp. 327-328)
31. Back Matter
(pp. 329-330)