In Praise of Simple Physics

The Science and Mathematics behind Everyday Questions

Paul Nahin, Paul J. Nahin

• Publisher: Princeton University Press
• Year: 2016
• Print ISBN: 9780691178523, 9780691166933
• eBook ISBN: 9781400880515
• Topic: Physical Sciences
• Subtopic: Mathematics General

Fun puzzles that use physics to explore the wonders of everyday life

Physics can explain many of the things that we commonly encounter. It can tell us why the night is dark, what causes the tides, and even how best to catch a baseball. With In Praise of Simple Physics, popular math and science writer Paul Nahin presents a plethora of situations that explore the science and math behind the wonders of everyday life. Roaming through a diverse range of puzzles, he illustrates how physics shows us ways to wring more energy from renewable sources, to measure the gravity in our car garages, to figure out which of three light switches in the basement controls the light bulb in the attic, and much, much more.

How fast can you travel from London to Paris? How do scientists calculate the energy of an atomic bomb explosion? How do you kick a football so it stays in the air and goes a long way downfield? Nahin begins with simpler problems and progresses to more challenging questions, and his entertaining, accessible, and scientifically and mathematically informed explanations are all punctuated by his trademark humor. Readers are presumed to have some background in beginning differential and integral calculus. Whether you simply have a personal interest in physics' influence in the world or you're an engineering and science student who wants to gain more physics know-how, this book has an intriguing scenario for you.

In Praise of Simple Physics proves that if we look carefully at the world around us, physics has answers for the most astonishing day-to-day occurrences.

1. How’s Your Math?

What would life be without arithmetic, but a scene of horrors?

—Sydney Smith¹ (in a letter dated July 22, 1835)

In this opening chapter I’ll discuss several examples of the kind of mathematics we’ll encounter in “simple physics” questions that may (or anyway could) occur in “ordinary life.” These are questions whose intent, I think, anybody can understand but that require at least some analytical thinking to answer. The math examples are very different from one another, with their only “unifying” (if I may use that word) feature being a progressively increasing sophistication. The central question to ask yourself as you read each example is, do I follow the arguments? If you can say yes, even if you can’t initially work through the detailed analysis yourself, then your understanding is sufficient for the book.

Example 1

Our first example of analytical thinking requires no formal math but, rather, logic and a bit of everyday knowledge (lit lightbulbs get hot). Think about it as you work through the rest of the examples, and, as with the wind-and-airplane problem at the end of the preface, I’ll give you the answer at the end of the chapter.

Imagine that you are in a multistory house with three electrical switches in the basement and a 100-watt lightbulb in the attic. All three switches have two positions, labeled ON and OFF, but only one of the switches controls the lightbulb. You don’t know which one. All three switches are initially OFF. One way to determine the controlling switch is with the following obvious procedure: Flip any one of the switches to ON, and then go up to the attic to see if the bulb is lit. If it is, you are done. If it isn’t, go back to the basement, turn one of the other OFF switches to ON, and then go back up to the attic to see if the bulb is lit. If it is, then the switch you just turned ON controls the bulb. If the bulb is not lit, then the switch that has never been ON controls the bulb. So, you can obviously figure out which switch controls the lightbulb with at most two trips to the attic.

There is, however, another procedure that guarantees your being able to make that determination with just one trip up to the attic. What is it?

Example 2

This question also requires no real math but, again, logical reasoning (although it does require an elementary understanding of kinetic and potential energy). Suppose you fire a gun, sending a bullet straight up into the air. Taking air resistance into account, is the time interval during which the bullet is traveling upward greater than or less than the time interval during which the bullet falls back to Earth? You might think you need to know the details of the air-resistance drag-force law, but that is not so. All you need to know is that air resistance exists.² You may assume that the Earth’s gravitational field is constant over the entire up-down path of the bullet (it remains the same, independent of the bullet’s altitude). As in Example 1, think about this question as you work through the rest of the examples, and I’ll give you the answer at the end of the chapter. Hint: Potential energy is the energy of position (taking the Earth’s surface as the zero reference level, a mass m at height h above the surface has potential energy mgh, where g is the acceleration of gravity, about 32 feet/seconds-squared), and kinetic energy is the energy of motion (a mass m moving at speed v has kinetic energy

).

Example 3

This question does require some math, but it’s really only arithmetic involving a lot of multiplying and dividing of really big numbers. In the 1956 science fiction story “Expedition” by Fredric Brown (1906–1972), the following situation is the premise. There are 30 seats available on the first rocket ship trip to colonize Mars, with the seats to be filled by selecting at random 30 people from a pool of 500 men and 100 women. What is the probability that the result (as in the story) is one man and 29 women?

We start by imagining the 30 seats lined up, side by side, from left to right, and then we compute the total number of distinguishable (we assume each person is uniquely identifiable) ways to fill the seats without regard to gender from the pool of 600 people. That number, N₁, is³

Next, if N₂ is the total number of distinguishable ways to fill the 30 seats with exactly one man and 29 women, then the probability we seek is

. We calculate N₂ as follows:

there are 30 different ways to pick the seat for the one man

and

there are 500 different ways to pick the one man for that seat.

So,

The formal answer to our question is then

I use the word formal because we still don’t have a single number for P.

The factorials in this expression are all huge numbers, numbers that are far too large for direct calculation on a hand calculator (my calculator first fails at 70!). So, to make things more manageable, I’ll use Stirling’s asymptotic⁴ approximation for

. Then,

Each of the factors in the curly brackets is easily computed on a hand calculator, and the result is

The premise in Brown’s story is therefore highly unlikely. No matter, though, because while it is so unlikely as to be verging on the “just can’t happen,” it is not impossible, and besides, it’s a very funny story and well worth a willing suspension of disbelief.⁵

Example 4

Quadratic equations are routinely encountered in mathematical physics (you’ll see an example of this in Chapter 9), and here’s an example of a quadratic in the form of a type of problem that many readers will recall from a high school algebra class. Readers may take some comfort in ...

Table of contents

• Cover Page
• Title Page
• Copyright Page
• Dedication Page
• Contents
• Foreword
• Preface with Challenge Problems
• 1: How’s Your Math?
• 2: The Traffic-Light Dilemma
• 3: Energy from Moving Air
• 4: Dragsters and Space Station Physics
• 5: Merry-Go-Round Physics and the Tides
• 6: Energy from Moving Water
• 7: Vectors and Bad Hair Days
• 8: An Illuminating Problem
• 9: How to Measure Depth with a Stopwatch
• 10: Doing the Preface Problems
• 11: The Physics of Stacking Books
• 12: Communication Satellite Physics
• 13: Walking a Ladder Upright
• 14: Why Is the Sky Dark at Night?
• 15: How Some Things Float (or Don’t)
• 16: A Reciprocating Problem
• 17: How to Catch a Baseball (or Not)
• 18: Tossing Balls and Shooting Bullets Uphill
• 19: Rapid Travel in a Great Circle Transit Tube
• 20: Hurtling Your Body through Space
• 21: The Path of a Punt
• 22: Easy Ways to Measure Gravity in Your Garage
• 23: Epilogue: Newton’s Gravity Calculation Mistake
• Postscript
• Acknowledgments
• Index