Practice may or may not make perfect, but physics without practice is sterile. The laws of physics tell us how the universe works. To truly understand these laws we must see them at work. We must see, for example, how the laws of motion and the laws of universal gravitation combine to generate the elliptical paths of the planets. Without this confirmation, the laws of physics are just the laws of physics and not the laws of nature.
The basic laws of physics are not particularly difficult to understand. However, their application to anything but the simplest situation is difficult. To determine the planetary orbits, for example, we must solve a pair of coupled, nonlinear, second-order differential equations. Although a mathematical solution to these equations is difficult, a numerical solution is straightforward. Furthermore, the ease of obtaining a numerical solution is more or less independent of the nature of the forces, or, for that matter, the nature of the physical situation. We may apply the same numerical methods to projectile motion, planetary motion, space travel, harmonic oscillators, radioactive decay, foxes eating rabbits eating grass, and many others.
While numerical solutions are straightforward, they present two problems: They are tedious and difficult to interpret. Now the computer can surely remove the tedium. Computers thrive on tedium. When the computer is presented with a long repetitious task in arithmetic you can almost feel it quiver with excitement as it nervously awaits your command to begin.
But once the computer has determined the numerical solution, you find it has created a monster. You are deluged with a mass of data and must cope with the second problem generated by numerical solutions. How do you interpret all those numbers? The most satisfying answer is graphics. If it is a trajectory, then draw it. If not a trajectory, then find some way to plot the results: Number of radioactive particles as a function of time, position as a function of time, slope as a function of displacement, number of rabbits as a function of the number of foxes, and so on. A thousand pixels of graphics are worth a million numbers.
There are a variety of computer languages that are capable of generating graphics. As of this writing, Logo is the most flexible in this application and the easiest to learn. It is the most flexible because of its extensive graphics vocabulary. It is the easiest to learn because it allows one to communicate with the computer in much the same way people communicate with each other. Logo is a procedural (or modular) language. It permits one to define new words which then have the same status as the fundamental commands built into the language. These new words may be executed by themselves or as part of another procedure. “Words” are not bound irrevocably into “sentences” as in BASIC.
Language has an important effect on our powers of reason. Sometime when you are mulling over a problem, stop for a moment and notice how your thought process is carried by language. Try to think without using words. The fact that the Logo language structure is close to conventional language structure is a considerable asset.
It must be pointed out that this is neither a Logo manual nor a physics text. Rather, some previous exposure to Logo is presumed and, at the very least, a physics text shoud be available for consultation. While an effort has been made to include in each chapter a discussion of the relevant physics, there are wide gaps which can only be filled by a more comprehensive treatment.
The prerequisites for this book are minimal. Some trigonometry and a little algebra are necessary. However, the chapters are graded in order of mathematical difficulty. The earlier chapters assume less sophisticated programming skills and mathematical background. In fact, much of this early material was used in a pilot program with fifth- and sixth- grade school children.
Since their invention, high-speed computers have been used by scientists to solve problems that are too difficult to solve by normal methods. The languages that have been used to communicate with the computer were not easy to learn or use. With the advent of Logo it is now possible for anyone to speak with the computer. I would not like to represent this book as a revolutionary approach to the study of physics; it is quite traditional. In each chapter some physical principles are described and a problem is presented for solution. A method for its solution is then demonstrated, and, in conclusion, additional problems are posed to allow the student to test and expand his or her grasp of the subject. The only novelty in our presentation is the level at which the student is introduced to the computer. Ten years ago, only graduate students of physics were taught programming techniques. Today, every engineering student is introduced to computer programming. With the advent of simpler computer languages, it is time to offer this capability at an even earlier level. I recall when I was a lad in high school, hearing of some very bright person who had studied “THE CALCULUS” in college. Knowing “THE CALCULUS” was the trademark of great intellect. Today we teach calculus in high school.
We do not want to replace analytical methods with numerical methods. Analytical methods must remain the primary goal in any scientific discipline. Certainly the subject of free fall in a gravitational field is easily dealt with by algebraic means. If the student has the algebraic background, this should be the tool of choice. The purpose of introducing free fall in this book is to allow students without the necessary algebraic skills to solve such problems and to use this simple problem as an illustration of the basic algorithms for generating a solution in Logo. These same techniques will later apply to more complicated problems where algebraic methods are inadequate.
The primary objective of this book is to demonstrate that any person with a curiousity about the laws of nature may solve a wide variety of problems using Logo to overcome the mathematical hurdles. Logo can be for the novice what FORTRAN is for the professional. The subjects dealt with here only scratch the surface and we hope our readers will be inspired to explore the laws of nature with the reassurance of having a turtle at their command.