Often textbooks in physics start with the question, "Why study physics?", and then proceed to expound some high-sounding arguments why the study of physics should be an incredibly exhilarating experience. (Any sense of exhilaration is usually hosed down upon turning a page and seeing - horrors! - the first equation in the book.) Despite these attempts to fill the student with excited anticipation, most students study physics because of one main reason: They have to! You are probably one of these. Sure, physics is arguably the most fundamental and quantitative of all the sciences. True, too, an understanding of physics helps you see the world in an entirely new and more comprehending light. But if it weren't on your degree plan, chances are you would be taking something else - one of the "easier" sciences or, possibly, drugs.

Face it. Physics is tough. Physics teachers have been trying to figure out for decades why it is so difficult for students and how to better teach it. Answers to these questions, it seems, are harder to come by than the physics itself. Through my experience, I have identified several reasons why physics is a difficult row to hoe. For what they are worth, here they are.

1. Students are not prepared for the rigor of thinking that comes with physics.

   Physics is heavy into word problems, and students hate those things. According to a "Far Side" cartoon, the library of hell is filled with nothing but books on word problems. (This implies Satan could be a physicist. There are thousands of students who would not be surprised.) Word problems require you to size up a situation, extract the pertinent information, arrange the information in some sort of logical structure (a model), and then work toward the solution. There is enough involved here to require an entire section (much more than that, really), which will, in fact, appear later in this chapter.

   The point is that students, even those with a strong math background, seem to be much more comfortable with problems that set up a puzzle to solve, such as an algebraic equation solving for x or the integration of a function in calculus. In other words the model is already constructed for you. You have to apply known rules to solve the model. In word problems you have to formulate your own model, a significantly greater intellectual challenge. I have asked students what experience they have had in middle and high school solving word problems, and the answer invariably is "not much". So students come to the college physics classroom unprepared by their previous academic experience.

2. Students are burdened by various misconceptions about how the physical world works.

   This problem is hard to get a handle on, because it is not easy trying to figure out what misconceptions students bring with them to the lecture room. I have had the experience numerous times of explaining something over and over again to a student, trying to recast the explanation into different words, without getting through. Then, when we are both at about the end of our patience, suddenly a light comes on in the student's head, and he realizes that he had been viewing something I had been saying in a way I did not mean at all.

   Often, when this comes up, I can, in retrospect, readily grasp how the student had been confused in the first place. Sometimes I wish I had a device at the entrance to the lecture room that would erase students' memories upon entering. Unfortunately, you actually need some of that stuff, so even if I had such a device it would be a challenge to somehow only erase the dead weight. (It would also most likely be unconstitutional.)

3. Students are scared stiff.

   Now, fear can be a powerful motivation. However, a lot of people avoid things that scare them. Avoiding studying because you are afraid you can't understand the material is a sure way to get into deep doodoo. One way to combat this is to get into a study group. Study groups are an excellent way to find out that the other students are just as clueless as you are. That should bolster your confidence (or, possibly, your sense of futility). Seriously, study groups can be a terrific help in giving form to your questions and increasing your understanding. (One of the tenets of good science is to be able to ask good questions.) Often students can help each other understand material and solve problems better than the instructor. Here is a testimonial I found on the Internet. Read and heed!

   Agilent Technologies
   www.FutureEngineers.com
   Jennifer Tittjung:
   

   A study group is a great way to gain depth of knowledge in your field and it also lets you make friends at your school and in your major. I formed my study group in my first electrical engineering class my freshman year. It was the first day of class and I didn't really know anyone in the lecture hall. The professor passed out the first homework assignment, and I asked the girl sitting next to me if she wanted to work together. Not only did she want to, but the girl sitting in front of us also overheard and asked if she could also work with us. So we worked on the problem set together, but we weren't able to answer a couple of the questions. We went to our professor's office during office hours the next day. There were two guys there who knew the answer to our questions, and we happened to know the answers to their questions. The five of us began working together and the rest is history.

   People would find out about our study group and see that our working together led to great results, i.e., we all got good grades in our classes. So, today (three years later) we have 17 people in the study group. We don't ALL have ALL the same classes, but we always have someone to study with, we always have someone to ask questions of, we always have a lab partner and we always have friends in lecture.

   I wouldn't be where I am today without my study group. Electrical engineering can be a pretty tough field and it can be scary sitting there by yourself. A study group makes everything easier.

4. Students have poor study habits.

   What can I say? This complaint is high on the list of just about every college professor. "Set a time to study and stick with it." "Study at least two hours outside of class for every hour in class." "Find a suitable place to study where there are no distractions, such as the library or a quiet room in your home." Yadayadayada. I doubt if one student in twenty follows any of this advice. Anyhow, those college profs forget how they studied when they were in college. ("No more beer for me tonight, I got a chemistry final in the morning.") Exacerbating the problem these days is the plight of the "non-traditional" student. If you have a spouse, children, job, mortgage payment, etc., this is you. Again, I'd recommend trying to form or get into a study group. The library is full of students studying by themselves, snoring away. If nothing else, a study group will help you stay awake.

5. Physics is not a "quick study".

   There are a lot of smart students who, when confronted with a physics course, find themselves doing poorly. They are not used to this and may (a) panic, (b) blame the instructor, (c) drop the course, or (d) all of the above. My theory is that they are used to memorizing facts for multiple-choice tests, possibly at the last minute before the test is given. This will not work for physics. Physics is, in many ways, a skill. This is probably why there are no prodigies in physics such as there are in music and mathematics. Not that these disciplines do not require skill, but some endeavors, such as forging a fine blade or chipping out of a bunker, really require a lot of practice.

   Physics is like that. You cannot wait, as a novice golfer, until right before the tournament to go out and see if you can actually hit a golf ball with a club and make it go where you want. Similarly, you have to work at mastering physics from the beginning of the course. You must be steady. You must consider yourself to be in training, realizing that, if you are to make the team, you have to put in so many hours of practice a day. Once again, being part of a dedicated study group is an excellent way to introduce discipline into your academic routine as well as to have fun while doing so. Like many other sciences, physics is like a high-rise building. You start with a foundation of knowledge and skills and build to higher and higher levels. The higher floors will not stand without those below. As my prof in graduate school maintained, succeeding in physics has more to do with endurance than brilliance.

   A related matter is the "rob Peter to pay Paul" strategy of approaching course work. Every student, I'm sure, has skipped one class to study for another. This is a self-defeating strategy. I can say this from personal experience as a college student who occasionally did this. It is especially bad in physics. Because physics deals with such difficult concepts, it is hard to come from behind - it's like you are the 2003 Detroit Tigers down by seven runs in the ninth inning to the Atlanta Braves, and John Smoltz is on the mound. Also it is much harder to try to understand the material from someone else's notes or on your own than from a class lecture where, hopefully, you pay at least some attention and benefit either by asking questions on your own or from a braver student who has the temerity to ask the "dumb" question you were afraid to. I gar-own-tee (as they say over the state line in Loo-easy-ann) you will spend far more time trying to catch up in physics than you saved by skipping class. This applies to other college courses as well - the harder the course the more severe the consequences.

6. College is a math-free zone.

   Or it might as well be. Large percentages of students entering college these days are not ready for college-level mathematics. Not to brag or anything, but, after walking five miles to college in a blizzard (OK, OK, this is Texas - it was actually in wind, dust, hail, and tornadoes), the first math course I took was college algebra. In fact, this is definitely not bragging, because, in those days, you were expected to start with calculus, not college algebra. College algebra was considered remedial math!

   These days, however, students starting college often don't have math skills (adding and subtracting fractions, finding an average, using percentages, solving an equation like 2x - 5 = -3x, etc.) that you would think middle schoolers would be comfortable with. Some blame the schools, some blame the parents, some blame the students themselves. At the community college level I see a couple of other issues. First, we get many students who, in my day, probably would not have attended college at all due to lack of interest in academic work. (In rural Texas, where I graduated from high school, there was always work - in cattle, oil, or on the farm.) But we also get students who have worked for years, have a family, and haven't given algebra a second thought for longer than they would like to admit. Unfortunately for these students, physics needs math to quantify its concepts, and there is a lot of math in physics. So if you need physics, you are going to have to "do the math", especially if you are really ambitious and plan to major in a technical field.

   Students don't like to do this, but if you haven't had algebra in a while you might consider retaking it, even a remedial course, before tackling physics. The problem is, if you struggle with the math, the concepts are going to be that much more difficult. Even if your math is top notch, the conceptual content of physics is a significant challenge.

7. Technical students are poor cheaters.

   Ha! I'll bet you're expecting me to give you hints on how to improve your cheating skills. Traditionally, technically-inclined students tend to be more honest than the norm. This no doubt impacts their grades - for the better. That's because cheaters usually cheat off each other. Now, if there were ever a study in stupidity, this would have to be it. Neither of them knows squat, but they hope they can pool their ignorance and pass.

   Academic cheating has a way of catching up to you. I have known several students who essentially destroyed their academic careers by cheating. Getting away with it once often emboldens the student, and he/she expects to be able to pull it off again and again. I once had this student who pulled what he thought was a slick scam. After "taking" a test he left the lecture room without turning his test in, being careful I didn't notice. Next class period I passed the tests back. He came up to me after class, holding his test, complaining that I had failed to grade it. This was a great strategy except for two items. (1) I knew his test was not turned in with the others, because I carefully checked the tests after they were handed in. (2) The test he brought "back" was flat as a pancake; the tests I returned were all curled up like cinammon rolls, because I had rolled them all up in rubber bands. He had to drop not only my class but college altogether.

   It continually amazes me that students will cheat off the tests of others, plagerize material on their reports, drag their feet while their lab partners do all the work, etc., and believe the professor is too stupid to realize what's going on. Wise up, those of you guilty of these things. It will catch up to you. (On the other hand, you may make millions, like the top executives of Enron, before it does catch up to you. When you've soared to heights like that in our society, you can probably expect no more than a few years in a posh Federal pen and then retire in style. Just don't get caught while you're poor.)

As mentioned above, most of you would not be taking physics if it weren't required in your degree plan. You have probably wondered why it is in your degree plan in the first place. In the same vein, this question comes up again and again: "Why do we have to study this stuff if we're never going to use it?" (Even my son, now in high school, has taken up this lament.) Long ago, probably in the time of Socrates, some student came up with this question, and it has not been worn out yet. If there were a Pulitzer Prize for annoying questions, this student should have won it.

It doesn't matter what subject you are studying or why. When you get frustrated with a course (and probably more students, percentage-wise, get frustrated over physics than any other subject), you can always pull this question out of your arsenal to put the instructor on the defensive. Physics instructors absolutely hate this question, and often respond with something like, "It keeps the morons out of medical school." Now, I don't subscribe to this line of argument, but, on the other hand, would you feel comfortable with your life in the hands of a doctor who couldn't at least pass freshman physics?

To be honest (with great internal discomfort), I have to admit that I know of no research that shows physics is necessary for anyone other than physics majors. Even engineering students could probably make fine engineers without ever taking freshman physics (although they would have to learn the physics embedded in their various engineering courses). So, what good is it? Perhaps freshman physics at least serves as an introduction to some of the material in engineering, medical, and technical courses, or maybe it really is, just as you suspect, merely a rite of passage into the technical world. However, I think there are some valid reasons for learning some physics. They are:

• Studying physics sharpens your analytical skills. It gives you practice in sorting out what is relevant and what is not when solving a problem.
• Some knowledge of physics combats the widespread scientific illiteracy extant in our country. Perhaps with better physics education telephone "psychics" would make smaller fortunes and a majority of Americans would understand that the Earth is much older than 10,000 years.
• It introduces you to real intellectual effort. What? You think once you make it into medical school you can just relax?
• More than any other science, physics has an impact on how we perceive reality. Mechanics shows us that nature works by cause and effect, not by magic or divine intervention. Relativity reveals that perceptions of time and space depend on the observer and that mass and energy are interchangeable. Electromagnetic theory is indispensible to modern technology and gives us the tools to expand our knowlege and to communicate and present ideas. Studying the electromagnetic radiation emitted by celestial objects, coupled with relativity, reveals that the universe is unimaginably vast, old, and expanding. Quantum theory tells us that the strict cause and effect we see at ordinary scales (the proverbial "size of a breadbasket" scale) breaks down at the level of elementary particles, a counter-intuitive world where particles behave like waves and waves like particles - where an electron, for example, can go through two separate openings simultaneously, or penetrate an impenetrable barrier. We could not understand atoms and molecules, and thus chemistry, without quantum mechanics.

Actually, the word "supermodel" was just to try and get your attention. What we have to discuss here are models in the sense of SCIENCE. Scientists use other words that essentially mean "model", such as hypothesis or theory. A model, or theory, is really just an explanation for a set of natural phenomena. Although there are all different kinds of models: geometrical, mathematical, phenomenological, and even physical scale models built to mimic some system or other (such as the Bay Model in Sausalito, CA, housed in a warehouse, which mimics the hydraulics of the Bay Area), all models have conceptual content, the collection of ideas of what is germaine to the phenomena the model applies to. The history of science is littered with abandoned models. Other models have survived with minor tinkerings to major overhauls. Models that survive tend to be expanded and refined.

Models (theories) all have one thing in common. They are all generalizations. Humans love to generalize, and it is both a weakness and a strength. You hear of African-American males committing crimes and conclude African-American males are mostly criminal. Muslim extremists commit acts of terrorism, and all Middle Eastern-looking people are viewed with suspicion. The north pole of a magnet repels the north pole of another magnet but attracts the south pole. Therefore all magnets behave in this way.

The last conclusion is likely to be true but not the previous two. For one thing, people are a lot more complex than magnets. Even so, you really need to investigate the behavior of a large number of magnets before you can be confident of the north-south pole generalization. However, you can never guarantee that all magnets will behave in the way you theorize. Generalization is therefore a leap of faith that nature is consistent. Whether this faith is justified or not depends on experience.

Science is in the business of trying to explain how the universe works. Unfortunately, none of us knows how it really works. Even if we fully understood the nature of every type of particle and force in the universe (which we don't), it would be far too complicated to keep track of all of them. Theories and models therefore have to be simplifications and approximations, such as the joke "consider a spherical cow", which never fails to get hoots and gufaws from physicists. It's amazing that so much good science and understanding of nature can come from this simple-minded approach. Einstein is quoted as saying that the most incomprehnesible thing about the universe is that it is comprehensible. If Albert will forgive me, I would like to paraphrase this and say that it seems incomprehensible that the universe has produced beings that have the capability to understand it.

In science our models must correspond in some degree to what nature actually does. This means theories are validated or falsified by experiment. You never know if a theory is really "true" or not (works under all conditions covered by the theory) no matter how many successful experiments you perform. However, just ONE unsuccessful experiment that can be repeated with the same outcome can scrap the best of theories (or at least send theorists scrambling to make suitable modifications to encompass the new experimental results). Consider the following quotes.

No amount of experimentation can prove me right; a single experiment can prove me wrong.
- Albert Einstein

In theory there is no difference between theory and practice. In practice there is.
- Yogi Berra

Don't be fooled by reading somewhere that 3000 scientists have signed a petition claiming global warming is incorrect or that the evolution of life on Earth has not occurred. Scientific theories are not decided by vote or petition. If that were the case we'd still be teaching the caloric theory in thermodynamics classes. Scientists are a highly varied group, and many of them have strong personal beliefs. Like others in a democracy, they have every right to present and argue their opinions. When it comes to the correctness of theories, however, there is no democracy. Mother Nature is dictator.

Not all scientists are blessed with being able to design and control an experiment in his or her lab. Physics is so blessed, but some important sciences, such as astronomy and geology, must rely more on observation and analysis (thinking based on known laws of physics and chemistry) than on experiment. As one astronomer said, you can't just go up to a star, kick it, and see what happens. However, observation combined with analysis can often be as effective a tool in understanding phenomena as experimentation.

You can observe a lot by just looking. - Yogi Berra

We will examine, later in this text, a scientific model of the phenomenon of gravity that worked extremely well for hundreds of years, and still does in some 99%+ of the situations we have encountered in the universe (so far). This model, Isaac Newton's theory of gravity, is applicable where the gravitational forces are not extreme (such as near a black hole). His model contains both novel (for his time) and borrowed features.

One important concept was that of gravitational mass. He proposed that all objects have a property called mass that is independent of the actual chemical composition of the body. That is, as they say, one pound of feathers equals one pound of lead (although the pound is not a unit of mass, actually).

Further, he guessed that the force of gravity between two objects is proportional to the product of the masses of the two objects. However, the Moon is much less massive than Jupiter, yet exerts a much greater gravitational influence on the Earth than Jupiter does. So the distance between the objects is important, and Newton borrowed the idea of a force acting on the planets and centered in the Sun that was proportional to one divided by the distance squared. This is the so-called inverse-square law, except he argued that this inverse-square relationship works between all bodies with mass, not just between the Sun and the planets. This model allows us to compute the orbits of celestial objects and guide spacecraft to their destinations.

But, like I indicated above, it only takes one experiment or observation to shoot down an otherwise good theory, and in Newton's case it was the anomalous precession of the orbit of the planet Mercury. That is, its oval-shaped orbit gradually changes its orientation around the Sun in a way that Newton's theory of gravity could not explain. When Albert Einstein hammered out his General Theory of Relativity, which was his theory of gravity, the problem of the precession of Mercury was solved.

Einstein's theory is considered to be superior to Newton's, because it not only explains what Newton's theory explains but also other phenomena. However, the story is not over, because Einstein's theory breaks down when we try to apply it to understand what happens near the center of a black hole. For this we seem to need a "quantum theory of gravity", but no such theory exists yet, even though progress continues to be made in that direction. Science marches on!

It is important to understand what is a theory and what is not. There is much confusion over this. A scientific theory must be capable of being proven wrong, or, as philosphers say, of being "falsified". Consider the following example. Some religious people, not liking the theory of evolution or the apparent fact the universe is very old (around 14 billion years), pose the theory that the Earth and the rest of the universe were made by God to look old, but they are really only a few thousand years of age. Therefore, for example, the light from a galaxy that is millions of light years from Earth was created already on its way, so that when you observe it, you observe light that was not really emitted by that galaxy; it just appears that way.

Whether this is true or not, and whatever this might say about the character of God, one thing is sure. This is not a scientific theory. Any cosmic scam perpetrated by God, akin to a used-car salesman running the odometers backward, is not going to be exposed, which means this theory is not falsifiable. (Since the universe appears to be ancient from many different sets of observations and experiments, I think it is more reasonable to take this at face value than believe God is a con artist.)

To make this point in class, I occasionally pose this theory, which I call the "Five Minutes Ago Creation Theory". I maintain that the Earth, the universe, and all of us were created just five minutes ago, completely intact with all our memories. We did not exist last year; we did not exist yesterday; we did not even exist six minutes ago. All our memories were created, five minutes ago, by fiat, and none of that stuff we remember older than five minutes actually happened. I challenge the students to prove me wrong. Do they believe this theory? Of course not. Can they prove it isn't true? Not at all. The theory of creation discussed in the last paragraph is just like this theory. You just have to substitute "6000 years ago" or "10 000 years ago" (depending on which creationists you consult) for "five minutes ago".

The theory of "Intelligent Design", being pushed onto school districts by neo-creationists, is similar. In this theory God intervenes from time to time on the Earth; for example, to create life in the first place. After that, God continues to intervene so that life can develop from the forms seen as fossils in ancient rocks into later forms and ultimately into the forms seen today. This is the Five Minutes Ago Creation Theory broken up into countless steps and therefore has no greater scientific validity.

I want to point out that any idea of divine intervention, including Intelligent Design, may be true. But it can't be proven unless, I guess, God somehow reveals it. You could, and people have, come up with unending ideas of God and creation and divine miracles. These have not proven useful when it comes to the exploitation of nature for human purposes. Science has. On the other hand, science is not the place to go for spiritual awareness. Questions of purpose and destiny are (almost certainly) outside the purview of science.

In physics there are a number of theories that have been so successful in describing nature that they can be considered to have transcended "theory" in some sense. These theories have become more than theories. They have become "tools" in the physicist's toolbox. At this point the designation "law of physics" seems particularly appropriate. This does not mean these theories amount to absolute truth. Rather, they have been found to be so reliable that you can "set your clock" by them, so to speak.

The Special Theory of Relativity is one of these tools. It is routinely used by physicists to get results that are in step with nature. Try to compute the energy levels of electrons deep in heavy atoms without it, for example, and you will fail miserably. Or, try to design a particle accelerator based on Newtonian physics instead of Special Relativity. Or, try to understand an energetic astrophysical phenomenon. You will fail completely if you don't take relativity into consideration.

Jumping disciplines for a second, even the much-maligned theory of biological evolution is used as a tool by scientists. Geologists working for oil companies routinely use the idea that organisms evolved over time to correlate subsurface rock strata and predict where oil and gas might be found. It is incredible that a theory which has risen to the status of a tool of science is still derided in some quarters by those who neither understand nature nor how science works.

Much to the grief of physics students the world over, mastering physics at this level means that you have to develop some facility in working out word problems. There is no magic formula I know of - no quick way to become a physics word-problem guru or to lose twenty pounds of ugly fat. You have to have the right (intellectual) diet and exercise, exercise, exercise! However, understanding the nature of word problems in physics might help somewhat. One key is to understand the simplifications that are necessary to make a problem solvable. Remember, you need some sort of model, or picture, of the situation, and you are not going to be able to take into account all of the things going on in the real world. For example, when we first discuss motion, we are going to pretend that all objects are just points of mass. Now, this is patently untrue; however, much can be learned about motion by making this simplifying assumption. That is, we theorize that we can treat bodies as point particles to get useful answers, then compare our findings to the real world.

You can find out what simplifications you need to make from carefully examining the examples in the text and paying attention to problems worked out in lecture and study sessions. If you can learn what is important to consider and what is not, you have made great strides. You have an abundance of concepts to draw upon, but you yourself need to decide what physics applies to the situation and what either does not or is "small potatoes". Is weight important? Are forces acting between surfaces involved? Etc.

There is a hierarchy of models, from a major theory on down to particular implementations of that theory. For that matter there is often a sweeping, fundamental idea (I hesitate to use the overworked word "paradigm") that empowers a whole class of theories. Gravitation is one such idea. Evolution is another (which is much broader than what quickly comes to mind to many people: the evolution of life on Earth - it also encompasses changes in the Earth, the solar system, stars, and the universe itself).

The primary model based on the idea of gravitation may be, for example, Newton's theory of gravity. Your particular physics problem, say computing the height of a geosynchronous satellite, would require a secondary model, based on Newton's theory. Another example of model hierarchy, from general to specific is the following. Newton's theory of motion and gravity can be developed into the study of orbital dynamics. This, in turn, can be used to study the interaction of asteroids with the planet Jupiter. While studying this interaction you are confronted with the phenomenon of asteroidal orbital resonances (which explain why there are gaps in the asteroid belt between Mars and Jupiter).

Take still another example. The geological theory of plate tectonics explains why there are large belts of active geology on the planet and why the continents are moving, among many other things. The study of an active belt, say the Himalaya mountain range, will generate theories specific to that part of the globe, but they are likely to be particular implementations of plate tectonic processes.

Theories are also interconnected and overlapping. For example, the theory of the convection of rock in the Earth's interior (which acts plastically over large time periods) is seen as the driving force behind plate tectonics, and, in fact, plate tectonics is viewed as the expression of Earth's internal convection at its surface.

If there is a big, magic secret to working out physics problems , it is what I am about to tell you now, so listen carefully! Astoundingly, like the extras in the sci-fi movie that fail to heed the warnings of the eccentric scientist and are gobbled up by radioactive monsters, most students seem to thumb their noses at this profound piece of advice. DRAW A PICTURE! Yes! Draw a picture indicating the motions, forces, etc., involved in the problem! A good diagram will put you more than halfway to the solution. What you shouldn't do (but probably will anyway) is to try solving a physics problem by grabbing an equation, plugging in numbers, and grinding out an answer you hope is right. This is what I call the "plug and pray" technique. The student makes a stab at the answer and prays to God it's right. This is the most efficient way to fail physics ever invented. OK, now go and ignore this advice like everyone else. You have been warned!! (Gobble, gobble, munch, munch, ...)

This is, for people other than physicists at NIST (National Institute of Standards and Technology), a pretty boring subject. Nevertheless, in physics we measure stuff, and in order to measure stuff we need stuff to measure with. In particular, we need standards of all the units of measurement we use.

To make a measurement you need a couple of things: a device that responds to the property being measured and a standard by which to calibrate that device. The simplest example is undoubtedly a ruler. A ruler has the property of length and "responds" by indicating a value when the length of an object is measured. A thermometer responds to temperature changes. A speedometer responds to speed. Etc. To put tick marks on your device, however, you need a standard: a standard length, a standard temperature, a standard unit of time, and so forth.

Standards are arbitrary, but ideally they should be reproducible in a reasonably well-equipped laboratory with as little effort as possible. By reproducible I imply that different labs should get the same result with the greatest precision possible. This has driven standards toward "high-tech" definitions, with the notable exception of mass, whose standard is still a hunk of metal lying around in a lab in France. (However, we're working on that.) As of this writing, we have the following standards for quantities used in mechanics. (Other standards will be defined, for example in thermodynamics, as the need arises.) The following were taken directly from NIST's web site.

1. Distance: meter - "length of path traveled by light in a vacuum during a time interval of 1/299 792 458 of a second". (This establishes the speed of light at 299,792,458 m/s. Note the definition of the meter depends on that of the second.)


2. Time: second - "duration of 9 192 631 770 periods [oscillations] of the radiation corresponding to the transition between the two hyperfine levels of the ground state of the cesium 133 atom". (In other words, a cesium atomic clock.) This standard is likely to be refined in the next few years. Physicists are working on an optical clock that can theoretically be much more accurate than the microwave-based cesium clock. (How about a clock that is accurate to within one second over a billion or more years?)


3. Mass: kilogram - "mass of the international prototype". [In other words that hunk of metal (actually an iridium-platinum alloy bar called "Le Grand K") mentioned above.] As I'm writing this, efforts are underway to base the kilogram on the laws of physics rather than on a single object, which, it is suspected, has actually lost a tiny bit of mass since its fabrication. One major effort seeks to tie the kilogram to the electromagnetic force via what is called a "Watt balance". The other is trying to define the kilogram based on a correct number of silicon atoms. Currently the former effort appears more promising.

Also note that I list the standards in terms of metric units. Actually the units are more than just metric, they are the Systeme International (SI), or International System of units. It is an unfortunate fact of life that there are more systems of measurement than you can shake a yardstick at, the Big Two being SI and the British system.

Now, there would only be the Big One (SI) if it weren't for the U.S. True, a couple of other countries, Liberia and Myanmar (formerly Burma), apparently have yet to go metric. However, my latest information is that they are too embarrassed to admit it. (If you ask an official of either country whether or not the metric system is the standard, he will plug his ears and begin to sing "One Hundred Bottles of Beer on the Wall" at the top of his lungs while running off to the bathroom.) Not even the British use it anymore.

Despite our nation's recalcitrance in the world of units, even the British system is based on SI. That is, the foot is defined in terms of the meter and the "slug" is defined in terms of the kilogram. Fortunately, the second is the common time unit in both systems. (Otherwise, we would really have a mess. Not only would distance signs on the highway be expressed in miles and kilometers, but your speedometer would be in miles per hour and kilometers per ETU (European Time Unit). You would have to wear two watches if you had any dealings with the rest of the world.)

The fundamental units of the British system are the foot (ft), the slug, and the second (s). The second is defined exactly as in SI. The foot and slug are related to SI by means of conversion factors, given below.

• 1 foot = 0.3048 m
• 1 slug = 14.59 kg

Finally, I need to say a few words about "going metric". All physics students need to do this. Unfortunately, many of you enter the world of physics with not an iota of an idea what a meter or kilogram is. That is why I usually pass around a kilogram weight and a meter stick at the beginning of the semester. In case you sleep in and miss that lecture, let me describe these SI measurements to you, and I would urge you to develop a "feel" for what they mean.

A meter is a little over a yard. A centimeter, used quite often for smaller lengths and equal to 1/100th of a meter, is about 0.4 of an inch. (In other words, 10 cm approximately equals 4 in.) A millimeter is one tenth of a centimeter (or 1/1000th of a meter) and is about the thickness of the nail on your big toe. A kilometer, 1000 meters, is about sixth tenths of a mile. Although the kilogram is a unit of mass (a measure of how much matter there is), it is best experienced in how much it weighs (which is the amount of force the Earth's gravity exerts on it). A kilogram weighs about two and one fifth pounds at mean sea level. A gram is 1/1000th of a kilogram and is used for small amounts of mass (what you might handle easily with one hand). A milligram is 1/1000th of a gram. This is what prescription drug doses and vitamins are measured in, for example. Finally, getting to an SI unit that is really appropriate for measuring weight, the newton, one newton is a little less than one quarter of a pound, so it takes about 45 newtons to equal 10 pounds of force.

The use of metric prefixes such as "kilo" and "milli" in the above paragraph may have made you nervous if you are not a drug addict or pusher and therefore not acquainted with these terms. See the link above for a more complete listing of the prefixes. Here, I will only mention the more common ones: "kilo" means "1000", so that a kilogram is 1000 grams; "mega" means one million, one megahertz is a million hertz (cycles per second); "giga" means one thousand million (an American billion), a 50 gigabyte hard drive contains 50 billion bytes; "centi" means one hundredth and is usually encountered in the "centimeter", one hundredth of a meter; "milli", this girl I once dated that was really hot - she and I used to... uh, sorry,that was Millie..., "milli" means one thousandth, one millisecond is one thousandth of a second; "micro" means one millionth as in Microsoft, one millionth of a fully functional software system; finally, "nano" means one billionth as in my son's nanosecond attention span.

There is a very good reason for learning metric (if the fact that almost six billion other people on the planet use it is not enough), and that is that if you don't know what the units in your answers to the problems mean, you won't be able to understand whether the answers make sense or not. When I worked at Johnson Space Center I was introduced to the phrase "sanity test". When someone presented numbers at a meeting, for example, and the engineers present weren't sure what they meant, they wondered out loud if the numbers passed the sanity test.

You should always give your answers to physics problems the old sanity test. If you are calculating the orbital speed of a satellite in low-Earth orbit and come up with 1.5 cm/s, you might not realize how ridiculous that answer is unless you know what a centimeter is. Even some physics textbooks are guilty of violating the sanity test. One physics book I know repeatedly talks about such things as tables over 2 m high and guys throwing stones straight upward at 40 m/s, among other things. I'll let you figure out why a 2-m tall table might be a little far-fetched, even if your name were Shaq, and why even Roger Clemens would be hard pressed to throw something upward at 40 m/s. I would like to repeat this. SANITY TEST!! Don't leave homework without it.

Although we defeated the British in the Revolutionary War, they have somehow exacted their revenge by saddling us with their mind-numbing, intellect-crippling, commerce-inhibiting, and logic-challenged system of units. Of course, it could be worse. It could be totally incomprehensible, like shoe and garment sizes, but it is definitely bad enough. As physics or engineering types naturally drawn to the metric system because of its simplicity and utility, we have to learn to live peacefully side-by-side with all these idiots who insist on the British system of units.

One way is to develop conversion factors, which are used as translation aids from the British language to the metric language and vice versa. A conversion factor can be constructed from a relationship between units rather simply and, believe it or not, it always equals one. Don't believe me? Observe.

Say you know that 1 in = 2.54 cm, and you want to convert a quantity in feet to meters. You need a conversion factor with the units of m/ft. This is because units can be treated like algebraic quantities, such that feet times meter/foot equals meters, like this: (ft)(m/ft) = m. The "ft" cancels out just as if it were "x" in an algebraic expression. Divide both sides of your inch-centimeter relationship by 1 in. You get

1 in ÷ 1 in = 2.54 cm ÷ 1 in.

Since (1 in) / (1 in) = 1, you see that 2.54 cm/in = 1. This illustrates another property of conversion factors. They are unitless as well as equaling one! We are not done yet. We need to convert cm to m and in to ft. From the relationships

12 in = 1 ft and 1 m = 100 cm, we can construct

12 in/ft = 1 ft / 1 ft = 1 and 1 m/100 cm (= 0.01 m/cm) = 100 cm / 100 cm = 1.

Since 1 × 1 × 1 = 1, we can multiply all the conversion factors together and the result will still be equal to one.

2.54 cm/in × 12 in/ft × 0.01 m/cm = 0.3048 m/ft  ( = 1).

Note that "cm" cancels "cm" and "in" cancels "in", treating these algebraically. So, there is 0.3048 meter in a foot. If the quantity to be converted is 5280 ft, then 5280 ft is converted to meters thusly.

5280 ft × 0.3048 m/ft = 1609.344 m

I chose 5280 ft because that's how many feet there are in a mile. That means there are exactly 1609.344 meters in a mile. I can say "exactly" because, by definition, there are exactly 2.54 cm per inch and exactly 5280 feet per mile. Then, since there are exactly 1000 meters per kilometer, a mile is exactly 1.609344 km.

A pitfall to avoid in unit conversion occurs whenever you deal with quantities raised to some power other than one. For example, if your aim were to find the number of square meters per square foot, you would have to do this.

(0.3048 m/ft)² = 0.09290304 m²/ft²  ( = 1).

Then to convert a measured area of 45 square feet to square meters, you do this.

45 ft² × 0.09290304 m²/ft² = 4.2 m²,

where the ft² in numerator and denominator cancel. The lesson is, "DON'T FORGET TO RAISE YOUR UNITS TO THE PROPER POWER!" In this example you must have square feet in the conversion factor to cancel the square feet in the measurement, AND your result must be in square meters.

OK, so why did I write only two digits in my answer? That is the joyful topic of the next two sections: accuracy, precision, and the number of significant figures. (The quick answer is that presumably 45 square feet was measured to the nearest square foot, therefore it makes no sense to express the answer to better than the nearest 0.1 square meter.)

These three subjects are all intimately related. My aim here is to try to keep the engineering students in my class from going on to create a man-made disaster. The physics students in my class don't have to pay attention unless they plan to work for NIST. (Just kidding. You too might do something terrible unless you pay attention, such as design a hydrogen bomb that won't explode.)

First let's look at what is meant by accuracy and precision. These two terms may seem to be identical, and that is pretty much the way they are used in day-to-day language. However, we technical nerds have to make a distinction. It's part of the way we keep our activities incomprehensible to the general public. (Insert evil laughter here.)

The best simile I have heard that makes this distinction is that of an archery contest. Two contestants shoot five arrows at a target. The arrows of archer Albert all strike very close to each other. In fact they are all within a circle with a radius of 10 cm, and the center of the circle is 35 cm from the center of the bullseye. (Fig. 1.1)

On the other hand, archer Barry shoots his arrows all over the target. Although not one of them falls as close as 35 cm to the center of the bullseye, the average position of the five arrows is only 15 cm from the center. Who is the winner? Well, it depends on whether the contest is judged on accuracy or precision. If on precision, then Albert wins, because his arrows are all close together. If on accuracy, then Barry wins, since his average position is closest to the bullseye.

You may smell statistics here, and you are right. Getting the measurement of a quantity to nearly the same number time after time is an example of precision. It doesn't mean, however, that the measurement is correct. A great example of this is the Hubble Space Telescope. The fabrication of the mirror took several years of hard work by many top optical engineers. A sophisticated manufacturing technique was employed and stringent optical tests were performed on the mirror. The manufacturer, Perkin Elmer, boasted that if the mirror were as large as the U.S., the undulations in the surface of the mirror would only amount to small, gently sloping hills and valleys with a topographical relief of less than 30 feet on average. Imagine that! The entire USA resembling the lower Rio Grande Valley! The surface of the mirror, they claimed, was honed to an almost perfect shape!

It was, unfortunately, the wrong shape. Precision, yes. Accuracy, no. What had happened, apparently, is that a small fleck of paint came off one of the measuring rods during manufacture, causing the mirror to be flatter than it should have been. This is known as a systematic error, an error that throws measurements off the same way for each measurement affected. The undulations, on the other hand, are examples of random errors. These errors tend to cancel each other out upon taking an average of the measurements. In the case of a mirror, however, you don't want a surface that on average is the correct shape, because each tiny part of the mirror surface contributes to the image, not the average. Now, if the Hubble mirror had been polished to the correct shape but with large undulations, the problem would have been one of precision. This actually would have been much worse. It was possible to correct the systematic error with corrective optics: "eyeglasses for Hubble". Not so with errors of precision in the construction of the mirror. Being random it would be impossible to figure out an optical system that would "unrandomize" such errors.

I'm sure you know by now that you will need a calculator to take physics and other technical courses, and unless you are very unusual you already have one in your possession. If your calculator is like mine, it allows up to 10 digits to be displayed for an answer. If you are like a lot of students, when you get an answer to a problem you will copy down every last one of those digits. Is there anything wrong with that? Well, consider the following story.

In the small country of Igzakhtistan the numbers on every legal document have to be perfectly precise. Young Misty Mark, a citizen of Igzakhtistan, decided to go into surveying. During her training, however, she slept through the lecture on significant figures. Her fellow students reassured her that it was a dull subject and she didn't miss a whole lot. Her first job was to survey a small rectangular plot of valuable land for a prominent client. She dutifully measured the length as 85.3 m. (Igzakhtistan would never stoop to using British units.) Then she found the width to be 44.1 m. When she filled out the official form for the area of the land, she entered 3761.73 square meters. She got this by using her electronic calculator to multiply 85.3 times 44.1. She entered the number and thought no more of it.

It turns out the land was to be sold to the king. The king's lawyer, Phineas T. Pettifogger, being a shrewd (if somewhat unscrupulous) practitioner of Igzakhtistan law, demanded the land owner prove that the king was getting exactly 3761.73 square meters. Otherwise, by Igzakhtistan law, the owner was guilty of misrepresentation and had to pay a stiff fine (to the king, who else). The owner stormed into Misty's office and demanded she verify her figures. Poor Misty did not know what the problem was. She took out her calculator, multiplied 85.3 times 44.1, showed it to the land owner, then looked at her as if she were some kind of idiot. The land owner glared back at Misty as if she were some kind of idiot. Worse than that, the judge at her trial looked down his long nose at her and exclaimed, "What kind of idiot are you!" and took away her license to survey.

What did Misty do wrong? At the trial, the judge asked her if she would stake her life on the validity of every last digit in the land's area. "Are you willing to face execution if the area is not 3761.74 square meters, not 3761.72 square meters, not any other number, but exactly 3761.73 square meters?" (The law in Igzakhtistan is cold.) At this point Misty was not sure and decided to plea bargain to avoid the death penalty.

What Misty subsequently learned is the following. Both 85.3 and 44.1 are measurements that are only good to the nearest tenth of a meter. For example, the width, given as 44.1 m, could, without any other information, be anywhere between 44.05 m and 44.15 m. This is a range of 0.1 m. Similarly, the length can be anywhere between 85.25 m and 85.35 m. If both measurements were too short, the real area could conceivably be close to (44.15 m)(85.35 m) ≅ 3768 m². On the other hand, if the measurements were too long, you might have an actual area of (44.05 m)(85.25 m) ≅ 3755 m². Since Misty's measurements give about 3762 m², you see that there is a range of possible values for the area within about ±6 m² of what Misty's numbers give.

However, things are even worse than this. You don't know the range of error precisely either. You only estimate it as being about ±0.05 m for each measurement. Assuming no human error, the range might be between ±0.045 m and 0.055 m, which is going to bump up your error for the area, ±6 m², a bit more. In light of these difficulties, it seems prudent to estimate your error to the nearest power of ten and just let it go at ±10 m², which means you only know the area to about the nearest 10 square meters. Therefore you conclude that Misty should have entered 3760 square meters in the document, rounding off to the nearest ten square meters. (Or should she have? More on this below.)

The number of digits written down for a measurement are called significant figures. The rightmost digit (so long as it is not a "place holder" zero, see below) is the least significant figure and represents the limit of certainty in the measurement (assuming some human or systematic error isn't involved). Misty's area, for example, was not certain beyond the nearest ten square meters. What we need are rules that tell us how many significant figures to write down in an answer. From considerations similar to those of the previous paragraph, they are as follows.

1. When multiplying or dividing a measurement with an exact quantity, such as π, the answer should have as many significant figures as the measurement. A corollary to this is that the number used for the exact quantity should have at least one more significant figure than the measurement. (You hopefully wouldn't measure the radius of a circle to three significant figures then use 3 for π in computing the circumference.)

Right: A = πr² = (3.142)(1.22 m)² = 4.68 m²
Wrong: A = (3.14)(1.22 m)² = 4.67 m²
Very wrong: A = (3.1)(1.22 m)² = 4.61 m²
Criminally wrong (esp. in Igzakhtistan): A = (3)(1.22 m)² = 4.47 m²

2. When multiplying or dividing two measurements, the answer should have as many significant figures as the less precise measurement. For example, the acceleration of an object can be found by dividing the net force acting on it by its mass. Say the force has been measured as 5.4 newtons and the mass as 3.35 kilograms. The acceleration would be
   
   a = 5.4 N / 3.35 kg = 1.6 m/s²,
   
   Not 1.61 m/s², and most assuredly not 1.611940298 m/s² (unless you want to get the "what kind of idiot are you" look from an Igzakhtistan judge).



3. When adding or subtracting measurements, the least accurate measurement rules the precision. For example, add 4.10 and 3.228. Since 4.10 only goes to the hundredths place, you must round 3.228 to 3.23 before adding, to get 7.33 as your answer. Note that by this rule 2.78 - 0.003 = 2.78, NOT 2.777! The 0.003 has to be rounded to the nearest hundredth before subtraction, which is zero in this case.



4. What about rounding? The rule is you round up when the digit just to the right of the least significant figure is greater than five, down when it is less than five, and pretty much do what you want if it is five. Most people round up for five. In a long calculation, many alternate rounding up and down to try to avoid a round-off bias in the answer. An even better strategy is to round off to one better than the least significant figure, then round off your final answer to the correct number of significant figures. However, check out the calculator rule just below.



5. The final rule is the "calculator rule". In the olden days, when you had to do arithmetic by hand, rounding off properly was necessary to preserve precision without having to do laboriously long calculations. Today we have calculators to do the labor, so why not just carry out the calculation to the precision of the calculator then round off the final answer to the correct number of significant figures? This is the easiest way to go. If you do have to round off as you make your way through a long calculation, you should round off to one better than the least significant figure for maximum avoidance of round-off bias.

What if Misty had only slept through the last part of that important lecture and entered 3760 square meters for the area of the land? Would she be OK? Well, not exactly in Igzakhtistan. The king's lawyer would have tried to argue that the land should be exactly 3760 square meters, not 3761, not 3759. Poor Misty would still be in trouble. She only meant that the area was good to the "6" in the number, not to the final "0", since she realized that her measurement was only good to the nearest 10 square meters. The part of the lecture she sadly missed was the one on expressing significant figures properly. She needed a way to write the area unambiguously as three significant figures, not the four that the lawyer would try to hold her to.

A good way to write a number down clearly defined in significant figures is to use scientific notation. In scientific notation you write the leading digit, decimal point, the fractional part to the least significant figure, then the power of ten that will give it the correct magnitude. Had Misty written the area as 3.76×10³ square meters, the king's lawyer would have nothing to sue over. The zero in 3760, which of course only indicates the position of the decimal in this case, is dropped in scientific notation since it is neither significant nor is needed to show decimal position. The power of ten does that. As another example, take the case of the number 0.0045. You might think this number has four significant figures, but if you think about it you realize that the two zeros to the right of the decimal only show the decimal's position. Written in scientific notation as 4.5×10^-3, the fact that only two figures are significant is apparent.

When you were just a tad, you were taught your numbers. Your parents probably taught you to count to ten before you had any idea what you were doing. You thought it was a fun game. Then you became old enough to go to school and the fun was over. In physics we go a step or two farther to spoil things for you. We go beyond pure numbers by modifying a number with further information. For example, we have already mentioned physical units. These modify numbers in ways that go beyond the actual numerical value. That is, 8 feet is not the same as 8 miles per hour is not the same as 8 apples is not the same as 8 oranges. This, really, is just a view toward the original way humans used numbers until some Greek philospher with not much to do invented mathematics. This led to the invention of mathematicians, which mankind has regretted ever since.

Mathematics may be defined as the subject in which we never know what we are talking about, nor whether what we are saying is true. - Bertrand Russell

A quantity like 8 feet or 8 oranges or -15 degrees celsius is called a scalar. That is, it is a number on a scale that covers a range of values, potentially from minus to plus infinity. Physicists were not satisfied with this situation and for good reason. In physics and engineering it is important whether your jet plane is going 400 mph north or 400 mph east. In fact, it's also quite important to the passengers, whether or not they are engineers or physicists. Is it possible to tag a direction on a number, in addition to the physical units, in order to further specify its meaning? Why not? Such a quantity, one that has a direction associated with it, is called a vector. 400 mph is a scalar; 400 mph to the north is a vector.

Perhaps the simplest vector to visualize is position. To specify a position you need a starting point, called an origin, which, mathematically, is the zero point of your coordinate system. A two-dimensional coordinate system has two axes, often called "x" and "y". (Fig. 1.2) Where both x and y are zero is the origin of this two-dimensional system.

A three-dimensional system would have a third axis, say "z". (Fig. 1.3)

If the axes are mutually perpendicular, the coordinate system is said to be Cartesian.

One of the first users of the vector concept was Weird Beard the Pirate, who had a fortune in gold dubloons to hide. In a deserted field he spied a lone oak tree, which he decided would be the origin of his coordinate system. He then paced 40 steps east, using his compass and choosing east as the positive x axis, then 30 steps north, in the direction of the positive y axis (Fig. 1.4).

W. B., whose friends called him Weirdo (in fact, you had to be a very good friend to call him Weirdo.), knew the Pythagorean theorem, which says that if you square the two sides of a right triangle and add them together, you get the square of the longest side (hypoteneuse). Since 40 squared plus 30 squared equals 50 squared, W. B. knew that his treasure would be buried 50 paces from the oak tree. And since the angle toward the treasure, measured north of east, had a tangent of 30/40, he knew that the dubloons were about 37 degrees north of east from the tree. That is, using the inverse tangent trig function,

tan^-1(0.75) = 37 degrees.

Not wanting to forget the place of burial he recorded the position two ways: (1) 40 paces E, 30 paces N, and (2) 50 paces at 37 degrees N of E. These are equivalent methods of recording position.

Let's look at W. B.'s two position measurements a little more closely.

Method (1): You use the axis positions to specify the location. The treasure according to this method would be at x = 40 paces, y = 30 paces.

Method (2): You use the distance from the origin and an angle to specify location. (This is pretty much the same as polar coordinates.) With this method the gold is 50 paces from the tree at an angle of 37 degrees north of east.

Both methods describe a position vector. The position of the treasure is a vector because it has both magnitude (50 paces) and direction (37 degrees north of east). Method no. 2 is more clearly a vector, but method no. 1 is equivalent because you can get magnitude and direction from the x and y coordinates of the treasure with the formulas,

(1.1) r = (x² + y²)^1/2,

(1.2) θ = tan^-1(y/x).

When you take the square root of x² + y² to get r, you have two choices of sign (as you always do when we take a square root). The obvious choice for r is the positive sign so that r is just the distance to the treasure. "r" is called the magnitude of the position vector. The magnitude of a vector is always positive and tells you how large the vector is. Position is a vector. Distance is a scalar.

Both magnitude and direction are necessary to define a vector. One hundred years after W. B.'s untimely death at the hands of his crew (who finally decided he really was weird), two explorers independently set out to find his treasure. Unfortunately, one had only a scrap of paper saying it was 50 paces from the oak tree. The other had the corresponding scrap with the direction 37 degrees north of east on it. Amazingly, the two arrived at the field the same time. However, neither trusted the other, so they worked independently. Mr. Magnitude began digging holes in a circle 50 paces from the tree. Mr. Direction started at the tree and began digging holes in a straight line, heading 37 degrees north of east (Fig. 1.4) At some point the line of Mr. Direction's holes crossed the circle of Mr. Magnitude's holes, but, as luck would have it, neither dug at exactly the intersection. Boy, were they stupid. However, an old farmer, watching this activity from the shade, waited until they got exhausted and left, then dug at the right spot, sold his farm, and lived uproariously for six months until his ticker gave out.

Actually, I lied at the end of the previous paragraph. The old farmer didn't run off to the flesh pots of the city. Instead, he decided to rebury the treasure at a new spot and come for it later. He paced off 20 steps south then 20 steps west to locate a new burial spot. He recorded what he did and then his ticker gave out. Which proves you should never pass up a good flesh pot.

Fast forward to many decades later when a physics student buys the farm and happens to find the directions to the treasure. Fortunately, the old farmer also recorded the original location, so it was a matter of putting the two together and locating the gold. The student was about to do this, when she became aware of an old state law which stated that if treasure is buried in relation to an oak tree, you must locate the treasure directly from the tree or else it becomes the property of the state. This meant that she couldn't just go out, pace to the original burial spot, then pace from there to the new spot. Rather, she had to pace directly to the treasure starting at the (now very old) oak tree. This would stymie most people, but this was a physics student that paid attention in class. Here is how she found the treasure from the position of the oak tree (Fig. 1.5).

She knew that if the pirate paced 40 to the east and the farmer 20 to the west, the combination would the the same as pacing 20 east. Similarly, 30 paces north and 20 paces south would amount to a net of 10 paces north. The equations she used were

(1.3) x = x₁ + x₂,

(1.4) y = y₁ + y₂,

where x₁ = +40 paces, x₂ = -20 paces, y₁ = +30 paces, and y₂ = -20 paces, taking north positive and east positive. The coordinates of the new location were x = 20 paces, y = 10 paces. However, by law she couldn't pace 20 east then 10 north. She had to proceed directly to the treasure or risk losing it. Once again, physics class had given her the tools to find the answer. She realized she could compute the magnitude and the direction of the treasure from the coordinates (20,10). The formulas are

(1.1) r = (x² + y²)^1/2  ,

(1.2) θ = tan^-1(y/x).

as before. She quickly found that the distance, r, to the treasure was 22 paces, and that it lay in the direction 26.5 degrees north of east. She dug up the gold, dropped out of college, bought the Atlanta Braves, and lived happily ever after. This could happen to you if you pay attention in class.

If the student had reburied the treasure, that would be no problem to any vector-savvy person. To find the new location all you have to do is add all the x components to get the resulting x component, then all the y components to get the resulting y component. (For our purposes, "components" are just x and y parts of the vectors. See Section 1.10 below.) For three two-dimensional vectors (two components each) the equations of addition are

(1.5) x = x₁ + x₂ + x₃,

(1.6) y = y₁ + y₂ + y₃,

etc. If there is a z direction then just insert a z equation above. The magnitude and direction are found from Eqs. (1.1) and (1.2).

Subtracting one vector from another is no sweat if you know how to add. Just change the signs of the coordinates of the vector to be subtracted and add them to the coordinates of the other vector.

The vector addition done above was performed on vectors that appeared in a time order. For example, Weird Beard first paced 40 to the east then 30 to the north. These two pacings occurred one after the other, so in finding the total displacement from the origin, you add the 30 paces north to the 40 paces east. There is a difference in pacing procedure from the case where Weird Beard might have paced north first and then east. He would have walked over different ground, but, as far as the final position of the treasure is concerned, it didn't matter which of the two trips he took. In fact, he could have paced out any number of paths, all different, that would have landed him at the same spot. So, as far as adding these vectors is concerned, the time order is of no consequence.

The vectors displacements of an object being moved occur in a time order but their addition can be in any order if all you are interested in is their sum (the resultant vector). Other vectors may act at the same time, such as in the situation to be covered below, where three forces act on the same object simultaneously. Force has magnitude and direction, so it is a vector, and more than one force can act on an object at the same time. The vector sum of these forces is found exactly the same way as for position vectors, and, once again, the order in which the vectors are added is immaterial as far as the sum is concerned.

Let me make myself clear. You DO NOT want to use graphical methods to add and subtract vectors unless you just absolutely adore drawing lines on graph paper with protractor and ruler. Nevertheless, for the sake of making a picture, it is valuable to know how graphical addition and subtraction is done. Arrows are used to stand for vectors in this method. An arrow is a great way to represent a vector, because an arrow has both magnitude and direction. If you look at examples of physics problems, you will probably see what seems to be thousands of arrows pointed in every conceivable direction. It is so scary that only the most intrepid students will resist the temptation to bolt for the Add/Drop counter. However, there is a reason for all those arrows, and that is that many of the quantities dealt with in physics are vectors, and we instructors absolutely love to draw arrows to represent them. The more the better.

For example, if Larry, Curly, and Moe are trying to get a stubborn mule to move, you might have the following situation. Moe is pulling on the bridle trying to get the mule to go forward. Curly is inexplicabley pushing sideways, and, worse, Larry is pulling on the tail. Figure 1.6 illustrates the situation, as seen from above and with arrows standing for the forces.

Forces, as you might guess, are vectors because they have a magnitude (how strong the force is) and direction (which way the force is being applied). What is the total force that the trio is applying to the mule? Graphically, you can apply the "tail-to-tip" addition method, as illustrated in Fig. 1.7.

The rule is that you place the tail of the second vector on the tip of the first, then the tail of the third on the tip of the second, and so forth, until all vectors are used. The vectors can be added in any order you like, but the length and direction of the arrows must be maintained, according to the mathematical rules for adding and subtracting vectors. This requires you construct a scale in order to draw the vectors to the correct length. For example, your scale might be 10 lbs = 1 in. A force of 50 lb would be an arrow 5 in long. Finally, you draw the resultant vector from the tail of the first vector to the tip of the last. This vector is what results from the addition. You can get the length and direction of this arrow off the graph, then use your scale to convert the length to the magnitude of the vector the arrow represents. Fun, huh.

Now what if you "want" to subtract vectors graphically? For each vector to be subtracted, just draw the arrowhead on the opposite side to make it a negative vector. Use the tail-to-tip method as before. The vector to be subtracted is simply added as the negative of itself.

Say Moe is pulling with 50 lb of force, Larry with 40 lb, and Curly is pushing with 60 lb. According to the scale mentioned above, these vectors would be represented by arrows of 5 in, 4 in, and 6 in in length, respectively. Careful protractor and ruler work will result in an arrow 6.1 in long at an angle of 9.5° east of north. According to your scale, the resultant force has a magnitude of 61 lb (Fig 1.7). Note that order of addition does not matter.

Now see how much easier it is using coordinates. In the notation that follows, Moe = 1, Curly = 2, and Larry = 3. The resultant force in the x direction is, from the modification of Eq. (1.3), replacing position vector with force vector,

F_x = F_1x + F_3x
     = 50 lb + (-40 lb) = 10 lb.

In the y direction it is just

F_y = F_2y = 60 lb.

The magnitude of the force is, again adapting a position vector equation (Eq. (1.1)) to use with force vectors,

F = (F_x² + F_y²)^1/2,
    = ((10 lb)² + (60 lb)²)^1/2 = 61 lb,

and its direction, from Eq. (2.2),

θ = tan^-1(F_y / F_x)
    = tan^-1(60 lb / 10 lb) = 80.5°

or 9.5° to the right of the positive y axis.

Like aliens in a bad sci-fi movie, trig functions occupy several (actually, infinite) parallel universes. This is because if you add a multiple of 360° (2π radians) to the argument of a trig function, it does not change the value of the function. For example,

cos[θ + n(2π)] = cosθ,

where n is any integer (positive, negative, or zero). Fortunately, we can ignore all the universes where n is not equal to one and confine ourselves to 360 degrees. In a cartesian coordinate system this means we only have to worry about four universes (which trig instructors like to call quadrants) (Fig. 1.8). Now, we have to measure an angle with respect to something, and in the discussion that follows I will take the standard way of measuring angles in a cartesian coordinate system, which is from the positive x axis. Counterclockwise is chosen to be positive and clockwise negative in this standard scheme.

Only two of these universes (quadrants) are "parallel" for a given trig function. For example, the cosine function, defined as the ratio x/r, has the same value for an angle θ in the first quadrant and the angle 2π - θ (or, equivalently, the angle -θ) in the fourth quadrant. This is because x is positive in both these quadrants (and r is always positive). Therefore cosθ with θ in the first quadrant has a twin in the fourth quadrant. Similarly, if θ is in the second quadrant, there is a twin, cos(2π - θ), in the third quadrant, and both are negative (x < 0 in those two quadrants). Sine is positive in quadrants I and II, and negative in III and IV, so sine has twins in those respective quadrants. Tangent is positive in quadrants I and III and negative in II and IV.

The following is a technique you may wish to consider for finding the angle describing the direction of a vector. First, identify the quadrant the vector is in by the signs of its x and y components. To calculate the angle θ between the vector and the x axis, just take the absolute value of the ratio of the y component to the x component and take the inverse tangent. This gives you the angle between the x axis and the vector. If the quadrant originally identified was I or IV, the angle is measured from the positive x axis either up (positive y component) or down (negative y component). If the quadrant was II or III, then angle is measured from the negative x axis, either up or down (+y or -y again). For example, say the vector A has components A_x = -4 and A_y = 2. This vector is in quadrant II. The angle is θ = tan^-1|2/(-4)| = 26.6°. Hence the vector is 26.6° above the negative x axis, putting it in the correct quadrant (II).

There is a slight detail missing from the story of the lucky physics student related above. The old farmer had indeed recorded both the original position of the treasure and directions to the reburial; however, not being adept at vectors, he merely recorded the pirate's burial spot according to what he knew, which was 50 paces from the tree at an angle of 37 degrees north of east. The student's dilemma was that the two position vectors were recorded in different ways. The pirate's were in magnitude and direction. The farmer's were in x and y coordinates.

This was no problem for our student. She knew that you can always break up a vector into its components, which is basically the same thing as specifying the vector in terms of x and y (and z for 3D) coordinates. The component vectors of the pirate's original burial spot add, by vector addition, to get the resultant vector. One vector is 40 paces due east, the other is 30 paces due north. Add these tail-to-tip and you get the resultant vector which is 50 paces long at 37 degrees north of east (Fig. 1.9). Therefore the two component vectors are equivalent to the resultant.

Breaking up a vector like this is called "resolving the vector into its components". The student had equations from her lecture notes to do this, too. They were

(1.7) x = r cos θ,

(1.8) y = r sin θ,

where r is the magnitude of the vector (50 paces) and θ the direction. She found the x and y components as follows.

40 paces = (50 paces)cos(37°),

30 paces = (50 paces)sin(37°).

This decomposition (means "resolving into components") will work for any vector, whether position, force, or whatever.

Eq. (1.7) leads to a more general concept, that of finding the component of a vector along some given direction. This component can be thought of as that "part" of the vector than exists in that direction. This is a geometry thing. All you have to do is to find the angle between the vector and the desired direction. This angle will lie between 0° and 180°, depending on the direction chosen. Multiply the magnitude of the vector by the cosine of this angle. The result is the component lying along the desired direction. (This component will be positive if the angle is less than 90°, negative if the angle is between 90° and 180°, and zero if the angle is exactly 90°.)

Here I discuss two common vector addition scenarios that the General Physics folk need to master and that should be a slam dunk for Techies.

You will undoubtedly encounter problems where the vectors to be added are all either parallel or antiparallel. When this is the case the vectors are simply positive and negative numbers that add like scalars. For example, what if you are a flight paramedic and your helicopter flies 10 miles north to pick up a patient then flies 15 miles south to the hospital. This involves two displacement vectors. If you take north positive, you have a vector +10 mi and a vector -15 mi. To add these vectors to get the net displacement of the helicopter you would add

+10 mi + (-15 mi) = - 5 mi,

to get a net 5 mile negative (south) displacement. Note that this is not your travel distance, which is 25 miles. Your average speed, as you will see in more detail in the next chapter, is the 25 mile total distance traveled divided by the travel time. However, your average velocity (again, see Chapter Two) is the net displacement, -5 miles, divided by the travel time.

Another quite common problem requires adding two vectors that are at right angles to one another. Figure 1.9 above shows two such vectors, the two paced paths Weird Beard used to locate the point where he was to bury his treasure. The path 40 paces east and the path 30 paces north are perpendicular to each other and therefore form a right triangle. The sum of these two vectors is the triangle's hypoteneuse, so the magnitude of the resultant vector is found simply by the Pythagorean Theorem. The direction is found exactly as described in the subsection Finding the Angle the Easy Way above.

Looking again at the Three Stooges and the Mule example, you see that you could approach the problem using the advice above, because two of the force vectors are antiparallel and the third is perpendicular to these. Hence add the two antiparallel vectors,

+50 lb + (-40 lb) = +10 lb,

and then use the Pythagorean Theorem to add this result to the third vector,

Resultant force = [(10 lb)² + (60 lb)²]^1/2 = 61 lb.

Observing that the resultant force is in the first quadrant, you find the angle which it makes with the positive x axis is

tan^-1(60 lb / 10 lb) = 80.5°,

same as before.

Right now it's time for a pop quiz. Which of the following quantities are scalars? Which ones are vectors? See how you do on this. The answers will follow. Remember, vectors must have a direction associated with them.

The quantities to consider are:

• velocity
• speed
• area
• volume
• acceleration
• time
• length
• pressure

The following quantities are scalars.

• speed - more about this below
• area - there might be a direction on how to get to the land (like the "north forty"), but how much land there is has no direction associated with it (nevertheless, see below)
• volume - ten gallons of gas has no direction associated with it, although it can be used to propel a vehicle in a given direction
• time - the "arrow" of time does not refer to a spatial direction
• length - remember the Aggie joke where two Aggies were having the devil of a time measuring the height of a flag pole they were about to install? One held the pole vertically and the other went up a ladder with a measuing tape, but the ladder was not tall enough. An engineering student passing by suggested they lay the pole down and measure it that way. After he left the Aggies thought the student was pretty stupid, since they wanted to know the pole's height, not its length. (If you chose "vector" for length, you should do well at A&M.)
• pressure - tricky. Pressure is really neither a vector nor a scalar, but part of something called a tensor. (If you have been introduced to matrices, then you might get an idea of what a tensor is from the fact that it can be expressed as a matrix.) However, for our purposes, it is good enough to treat pressure as a scalar.

The following quantities are vectors. Note I did not include force, position, or displacement, since we have already decided they are vectors. Sorry, no easy points.

• velocity - "OK, what's the deal? Speed is NOT a vector but velocity IS. What gives?" This was not really fair, but many quizzes aren't so get used to it. Physicists use speed and velocity in a different way than they are used in polite conversation. Velocity is a vector because it is speed plus a direction. 16 m/s is a speed. 16 m/s with a compass heading of 104° is a velocity. In physics usage, speed is the magnitude of the velocity vector. If you got these correct, give yourself ESP points.
• acceleration - equals the rate of change of velocity in physics. This makes acceleration a vector since it is proportional to the change of a vector (velocity) and the change of a vector must also be a vector. This makes acceleration a difficult concept for students, but fortunately you do not have to face this quite yet. If you got this one, award yourself some more ESP points.
• area - "What!!?? Area is a scalar! You just said so above, you moron!" Ha, ha, ha. I love to spring this one on unsuspecting physics students. Normally area is indeed a scalar. However, there are instances (which don't arise until we cover electromagnetic theory; lucky you) where you can consider area to be a vector. The simplest case is a flat surface. You can define a vector whose magnitude is equal to the amount of area of the surface and whose direction is perpendicular to the surface. (You have two choices here.) Don't worry about it - yet.

We return to our friend, Weird Beard, the pirate. He is a formidable-looking man at six foot, five inches, wearing a bright red and gold coat, black sash, and eye patch with a duck perched on his shoulder. (That is, he used to have a duck on his shoulder, but it kept trying to sell him health insurance, so it became duck a l'orange). He has set up a base of operations on Yohoho Island and plans to carry out raids in the vicinity. He travels due west (negative x direction) 100 knots and sacks the island of Manhatten. (His use of the "knot", or "nautical mile", finally led to mutiny by his crew. The British knot is 6080 ft = 1853.2 m, also called the "Admiralty mile", whereas the U.S knot is 6080.20 ft = 1853.248 m, which, however, is no longer in use, having been superceded by the international knot, which is 6076.115 ft = 1852 m. This is where the expression "tied up in knots" comes from. You can easily see why his men eventually made him walk the plank and thereafter reckoned distance using the kilometer.)

After looting Manhatten, he sails in a direction 120° E of N (30° S of E) for 70.0 knots until he reaches Coney Island, where he is run off by the savage natives. Finally, he navigates 45° E of N 52.2 knots to Grand Island, Nebraska, where he hears rumors he is being pursued by the Texas Navy. (He doesn't know the Texas Navy was decommissioned upon the signing of NAFTA and is no longer operational.) He prepares to flee to his home base. The question is, how far and in which direction is Yohoho Island (Fig. 1.10)?

Mr. Beard has the following task. He has three position vectors:

• 100 knots at an angle of 0° to the -x axis.
• 70.0 knots at an angle of -30° to the positive x axis.
• 52.2 knots at an angle of +45° to the positive x axis.

You and two friends decide to have a laser-tag dual in a pitch-black warehouse. You all start from a common point and walk in straight lines. You can walk in any direction as far as you want, but you must keep going straight (admittedly hard to do if it is pitch black). You decide to use your knowledge of vectors gained in physics class to nail these two dudes. Taking the common starting point as the origin of your coordinate system, you let your direction define the x axis. You know you are traveling in a straight line 60° to the left of Joe, so that Joe is at an angle of -60° from the positive x axis. Howard is moving at an angle of 80° from your direction to your left. Therefore he is moving at an angle of +80° with respect to the x axis. You count the paces of your two friends, not easy to do even though their shoes make distinctive sounds. You also have to keep up with how many paces you walk. You are definitely beginning to think about just taking random shots in the dark but stick with your plan. Finally, when you have all stopped, you know you have gone 20 paces; Joe has gone 28 paces; and Howard is 16 paces from the origin. Quickly you get out your calculator and make the necessary computations (Fig. 1.11).

The equations you need are (1.1), (1.2), (1.5), (1.6), (1.7), and (1.8). Using Eqs. (1.7) and (1.8) you get the components of the three vectors, where you are 1, Joe is 2, and Howard is 3.

x₁ = 20 paces,
y₁ = 0,

x₂ = 28 cos(-60°) = 14 paces,
y₂ = 28 sin(-60°) = -24.3 paces,

x₃ = 16 cos(80°) = 2.78 paces,
y₃ = 16 sin(80°) = 15.8 paces.

By vector addition, your position vector plus the position vector from you to Joe equals Joe's position vector. Therefore, the position vector from you to Joe is Joe's vector minus yours and similarly for Howard's position. You must subtract their components from yours in turn. For example, you modify Eqs. (1.5) and (1.6) for Joe by dropping the unnecessary third vector, since only two vectors are involved. Also you make your vector negative to effect subtraction. That is, the position vector from you to Joe, x_J, is

x_J = x₂ - x₁ = 14 - 20 = -6 paces,
y_J = y₂ - y₁ = -24.3 - 0 = -24.3 paces.

Hence Joe is at the coordinates (-6, -24.3) from your perspective, taking yourself as the new origin, which means his distance from you is

((-6.00)² + (-24.3)²)^1/2 = 25.0 paces.

Since, for Joe, both x and y are negative, he is in the third quadrant (with respect to you). You compute the angle as

tan^-1|-24.3/-6| = 76.1°.

This is 76.1° below the negative x axis, since it is in the third quadrant with respect to you. Similarly, for Howard you get

x_H = x₂ - x₁ = 2.78 - 20 = -17.2 paces,
y_H = y₂ - y₁ = 15.8 - 0 = 15.8 paces,

which is in the second quadrant with respect to you. Hence Howard is at

((-17.2)² + (15.8)²)^1/2 = 23.4 paces,
tan^-1|15.8/-17.2| = 42.6°,

where the 42.6° is above the -x axis. Unfortunately, just as you get the final numbers, Howard and Joe have figured out that dim green light they see is your calculator display, and you are zapped from both directions at once.