Expert: Daniel Mazur - 11/9/2006

Question
Physics is, from what I can tell about it from my 11th grade high school course, more than simply memorizing formulas, however this is a big part of it. Although I get A's on most of my tests, I was wondering if you had any strategies for this sort of thing? Its not like memorizing vocab words or the like, because you have to understand the formula's implications and uses in physics problems.

Answer
Hello, Shawn,

you are perfectly right in saying that physics is about understanding. I also agree that a large part of physics education (!) is about learning formulas and practice their usage. As long as teachers give you enough ideas of what lies behind each formula, there is nothing out of order with it, but...

Let me clarify one thing, just for comprehension's sake. I hope we both distinguish between memorizing and learning: memorizing is just storing the exact formula in your memory, the way it is written; learning means you just need to remember the principal laws governing it, so that you can get it approximately correct at run-time.
Example: You can memorize Newton's law of gravity, F=-G*m*M/(r^2), or you can just remember that
- it is an attractive force (for the correct sign)
- there is a flux-conservation law for distance (an inverse- squared dependence on distance, I'll explain later)
- the mass of the objects involved is both, the source of the field and the affected thing, so that resulting acceleration of a body is proportional to mass of the other (!) body (this gives you the two mass values in numerator)
When a test comes and you remember the laws governing the formula, you can get rebuild the formula except for the universal constant G. For G there is no other way of learning it than memorizing it:-). Remembering the "laws governing the formula" seems to take more effort at the beginning than memorizing and that's why the latter is so popular. However, after you've learned some number of formulas the hard way, you realize that you learn much faster, because many formulas have the same "shape", although they apply to different phenomena (shining example: Newton's grav. law and Coulomb's electrostatic law). One learns by experience.

Yes, I disliked memorizing formulas as well, I always did, until I mastered them. You see, there are (roughly) three stages of learning:
1) I don't know the formula
2) I know the formula
3) I master the formula
Now, by only memorizing a formula, one gets only half way between 1 and 2. Add the awareness of other formulas and the ability to "cross-breed" them, so that the result is mathematically correct (including dimensions, units), that's stage 2. Normally, all you need for this is in the lectures and textbooks and "memorizing" can just get you there, which secures you prime grades.

However, you can do better as you well know. You can "play" with the formula until you get its meaning and use in your blood. Test the formula on everything in sight and see, what happens. One might say it's what homeworks are for, but that's not quite right. The homework "quality" varies, and most textbook problems are "multiple and shallow". They give you enough practice with the formula, but little or no understanding of what it means. When I say "play" with the formula, I mean making real and thought experiments about "what will happen if...?" or "is this a demonstration of the law I learned?". Example: You've got the Newton's law and you see an attractive girl and you ask: Is she attractive to me, because she is heavy? What part of the attractive force I experience comes from gravity? ... Well, I hope you enjoy yourself, but the essence is this: You look around and see other boys are attracted to girl A as well, but other girls are not at all. Implication-> unless the other girls are mass-less(and we can rule that out by a number of other interesting experiments:-)), gravity cannot be the active force here. Or, you observe that girl A looks very attractive from twenty paces, but when you get to see her from up close, she does not look as attractive anymore. In other words, the attraction does not have the dependence on distance 1/R^2 predicted by the Newton's law. Implication: unless another, repulsive force cancels the attraction at short distances, gravity cannot be the phenomenon responsible for the attraction. And so on, with more serious questions too.

My strategy on learning the laws by other means than just memorizing them is to think about them a lot, think what they mean and how they impact your everyday life experience. Most things you learn at high school do have an impact. Gravitation, motion, light, colors, heat, ... everything is around. If you "play" like this, you will find your brain working harder than ever on more nonsensical ideas than ever, but it is THE way. I had a load of classmates, who were learning by making up a non-existent entity (called "knyava") at the start and then applying the studied laws on if in virtual experiments, and then giving each other reports on, how the "knyava" violated the laws of the real world. It was fun, it lasted them good two years of college, and I am sure in many of their minds "knyava" lives on...:-)

However, you need to realize that this takes REAL effort and time. You won't stay long on this path thinking "Why am I doing this? Why would I want to know, if an Everest mountaineer burns more calories that my coffee maker? I am too tired of it and it is boring, I better go pick up some chicks..." Learning physics properly means one needs to learn hard and continuously focused mental work, and this is the thing that balances our the fun you can have with physics. The choice is ours...:-)

Let me just give an explanation of "flux conservation law" I promised to explain above. I am using it as a label, which I took from another formula, which contains the 1/R^2 dependence. You can use anything else instead, you can even take the Newton's law as the label for 1/R^2. My naming example originates from the following: It so happens that if you shine a bulb and encase it in a container, the total radiation energy absorbed every second (radiative power) by the container is the same for all sizes of the container. If we describe the container as a spherical shell of radius R, the radiative power absorbed by the shell is constant - it's simply the radiative power originally emitted by the bulb. Because the shell's area rises as 4pi R^2 with R, the radiative power absorbed PER UNIT AREA of the shell goes as inverse square of radius, 1/(R^2). Assuming I learned this law before I learned Newton's law, I have just labeled the 1/(R^2) dependence as "flux conservation" law and wherever I see the dependence next time, I am going to label it in my head like this. Yes, it may be easier to just remember "1/(R^2)", but when you have dozens of "little laws" in your head, all in the form of mathematical formulae, it is easy to mistake them, use 1/R^3 in your exam instead, for example. If you remember 1/R^2 as "flux conservation" and 1/R^3 as "volume density", you won't mistake them.

I hope I was at least a tad helpful, and I wish you success in your science education.
Daniel