How to Fall Slower Than Gravity
Paul Nahin is known for his many books written to popularize mathematics, but readers familiar with his work, know that there is always quite some mathematics involved, and it is not always the simplest of problems or computations that he describes. This book is a sequel to In praise of simple physics in which he introduces, like in this one, problems from physics with solutions. The MacGuffin for this book, as Nahin writes in the preface, is a letter published in the Boston Globe in which it is contested that in an exam for college placement it is required to know about quadratic equations. The title was Who needs to know this stuff?. Instead of writing an angry answer, Nahin wrote this book to illustrate that mathematics and physics in combination with basic laws of physics can solve real life problems. As a motto for the book, it opens with a problem of Lord Rayleigh from the 1876 mathematical tripos in Cambridge. Those who excelled in these exams had a bright future, whatever they chose as their profession. I doubt that this book will convince the authors of the letter in the Globe that quadratic equations are useful to know for everybody, but if the reader has taken calculus and physics courses at an introductory level, and is intrigued by the power of mathematical physics, then this book will give nice examples of what is possible and it has some challenges for the reader too. This is to illustrate that anyone who had these elementary courses (which is about anybody whatever his or her later profession turns out to be) should in principle be able to solve such problems. Several examples are already proposed in the long preface, which sets the tone for the rest of the book.
The bulk of the book consists of 26 problems (with variations). Some of these are classic and have appeared elsewhere. If they are simple exercises as in introductory courses, then the reader is mainly on his own to solve the problem. For the more demanding problems, Nahin goes through a discussion leading towards a solution and at the end leaves some challenges for the reader. Such a challenge can be a variation of what has been discussed or a step that has been skipped or a similar problem, or it is asked to write a computer program to check some of the numbers numerically. Usually the challenge is simpler than what he has gone through already. An extensive discussion of the solutions to these challenges are found in the second part. Not all solutions can be found analytically, so sometimes a short matlab code is included if necessary. These programs are not always written in the most elegant programming style, but most importantly they are quite readable and they just do what they need to do.
To give an idea about what kind of problems are discussed, here are some examples.
- The first one is a classic problem of launching a projectile over a wall. This typically involves a parabolic trajectory and thus the quadratic equation pops up.
- The second is a fun problem, seemingly impossible to solve: before noon, snow starts falling at a constant rate and a snowplow starts clearing a long road at noon at a constant volume per hour. The second hour it travels half as far as in the first hour. When did it start to snow? Hard to believe, but Nahin gives an analytic solution that results in the exact moment (up to the second) that it started to snow.
- There are some problems involving Monte-Carlo simulation because an analytic solution is infeasible, and hence programming a simulation is required.
- On the other hand, there are problems related to combinatorics where straightforward programming is excluded because numbers become too large (unless extended precision is used), and so these require the analytic derivation of asymptotic formulas.
- A classic more involved example is to find the tangential speed and time when someone falls off a slippery log (assumed to be a perfect cylinder).
- A 1967 paper of Nahin is recycled in a discussion of NASTYGLASS. That is theoretical glass that acts as a filter cutting off all electrical power below a certain threshold but leaving intact what is larger. Looking at a nice picture through this glass is supposed to make it ugly (hence the name).
- The problem described by the book's title is about the physics of a raindrop that is accumulating mass as it falls though a humid environment. In its simplest form it will accelerate at only one quarter of the gravitational constant.
- As we progress in the book, the problems become more involved with longer elaborations by Nahin. Some earlier problems return like rocket launch but now launching underwater, and it is explained how Enola Gay could launch the atomic bomb and escape the blast.
- It is shown how to compute ζ(6) using only undergraduate mathematics assuming Fourier series and the Dirac impulse are known (which he supposes to be available at the end of the undergraduate level).
- He also connects ζ(s) to prime numbers and cryptography. This connection can be verified experimentally by computing with a simulation the probability that two (or more) randomly selected numbers are coprime.
- After an excursion via cubic and quartic equations, in the last problem, the quadratic equation turns up again in a model to detect a fault in an undersea cable and in a Wheatstone test bridge.
In some appendices, extra material is provided about continued fractions, and the problem by Lord Rayleigh mentioned in the beginning of the book.
The different problems can be considered independently in most cases, although there are cross references for some. What Nahin offers is a bit of an unusual mixture of explanations and analysis of physical phenomena and some related problems for the reader. Much is in the style of his previous book In simple praise of simple physics. Whether we should classify all the discussed problems as "practical" in the sense that people are confronted with these in everyday life is highly disputable, but they all do involve basic laws from physics (although somewhat simplified) and it is illustrated (at this elementary level) how mathematics helps a physicist to solve such problems. As Nahin writes in his preface: "Millions of students are enrolled worldwide in calculus and physics courses, and the majority of them will not become mathematical physicists, but this does not mean that they cannot enjoy the power of mathematics making sense of a physical world".