Define Energy

The great stumbling block for any perceptive student is kinetic energy. Implicit in all the formulas is the assumption that the surface of the earth is absolutely still and in an absolute reference frame. The earth's own rotation and its revolution around the sun, the sun's motion through the galaxy, the galaxy's motion through the universe are ignored, even though these velocities are much larger than that of a bullet, automobile etc. One get's the queazy thought that Newtonian mechanics are bogus. Of course, physicists have a way out, don't they?

I think Richard Feynman had the best definition of energy.

We don't know what energy is. It's just a number we assign to a system (using a consistent set of rules) that doesn't change after you fiddle with that system (assuming it's isolated).

Now, that's probably not sufficiently tangible for most people. The problem with physics is getting students to have realistic expectations. We don't understand the universe very well. Mathematics is the only way it makes sense to us in any deep sense. Sometimes there just is no intuitive understanding to be had. Or rather, there's a degree of intuitiveness, and for some things that degree is lessened, and for others it's heightened.

The best you can do is think up some analogies that more-or-less explain energy, while emphasizing that it's not the whole story.

And that's a long way of saying I WISH I KNEW.

Noether's theorem demands every symmetry evolves a conserved property and every conserved property is coupled to a symmetry. One need not be Newtonian about it. Indeed, one should not be Newtonian about any of it.

Do the math, get the physics. If the math is not predictive or not empirically validated, then the math is unphysical. Taking out the trash not a big deal - unless you don't.

Discontinuous symmetries escape Noether's theorem. There is only one external (coupled to rotation and translation) discontinuous symmetry, parity. Physics routinely chokes on parity. Take the hint, do macroscopic parity-breaking experiments, take out the trash.

I like Uncle Al's suggestion that Noether's Theorem can be helpful here. Energy is time asymmetry.

Why not just give examples then explain the concept?

"Heat is a form of energy. Electricity is a form of energy. Anything that moves has kinetic energy, and anything under the influence of a field force has potential energy."

I haven't taken physics yet, so this might not be entirely accurate. Don't slaughter my comment. :(

On some level, the energy of a system is best understood simply as "a thing that is conserved." To illustrate this point, why not use the excerpt from the Feynman lectures about the child hiding his blocks?

In principle, the equations of motion are all you need. As it turns out, some equations of motion lead to conserved quantities, and some don't. If these quantities exist, they can be very useful. Given the equations of motion, there exists a procedure (Noether's) for finding the corresponding conserved quantities. Energy is just one of these quantities. If you could use calculus, you could show them the procedure, but I don't think there is any more to the story of energy, conceptually (until you get to quantum mechanics and general relativity).

Granted, I took physics with calculus in college, so I don't know the rules, but would it be so terrible to just briefly touch on some of the math required to understand a concept. Of course, you can't hold the students accountable for knowing that math later, but covering the concepts to help cover the physics seems reasonable.

I'm a physics tutor and have run into this issue myself. U = mgh is much more intuitive to understand than K = mv^2/2. This is because there are no coefficients or exponents. Double the mass, strength of gravity, or height and the energy doubles as well. Makes sense. I've found that students can get a feel for the KE equation by thinking about a ball dropping from a height, h. By the time the ball falls to h=0, its PE is zero. They can figure out using the 1-D equations of motion they just learned that the ball has v=sqrt(2gh). If all that PE was converted to KE, which is captured in the balls velocity, what do we have to do with this expression for v to get it to equal the original mgh? The answer, of course, is to square it, multiply by m and divide by 2. This isn't a perfect approach, but it gets at mv^2/2 using an approach (1-D motion) they're already familiar with.

For discussing energy more generally, I like the Feynman discussion mentioned in #2.

Energy can be stated as the "ability to do work". If an object has kinetic or potential energy then it has the ability to do work.

Kinetic energy is just the area under the curve of work done to get it up to speed.

You just have to think of a good working analogy. My daughter, who is an astrophysics TA, does the heavy tennis ball throwing trick on a rolling chair to demonstrate reaction propulsion of a rocket.

For students that have never done physics before, it's probably useful to dispell popular misconceptions about energy. Tell them there is no such thing as "pure energy", that only "things" (whatever the hell those are) can have it. And that if something exists, it has it, and if it doesn't have it, it can't exist. And that the energy of something should, in principle, always be detectable, so that if somebody defines something to be undetectable by all possible scientific experiments, what they're really saying is it doesn't exist.

I would also comment that what we mean by mass is how much energy something has when it's sitting still. And when something moves it gains energy, and that the energy depends on how fast it's moving.

Of course, there are fundamental physics problems with this way of putting it, like what really gives things "mass", what it means when things "move", and according to HUP, there is some energy that you can't directly observe, but these won't defeat the intuitive understanding you're trying to convey to them.

And defining energy as the ability to do work is sort of circular, isn't it? If we define work as "the amount of energy gained by something when we apply a force to it", then wouldn't we be defining energy as "the ability to give something energy"? This is maybe the least useful definition possible.

Chris P, I see where you are coming from, but you are using "work" in a physics idiom, which itself then needs careful explanation. Its circular.

This is a damn tricky one.

Ya know, i remember in HS physics, we hadn't had calc yet, but he still taught us dot products and showed us how to work with vectors properly. Also, even a little bit of calc seemed to make things roll better, but then, when i took college physics many years later, i did the one without calc since i really didn't remember enough, so maybe im not the best person to pontificate.

I think Godfrey's on to something with the whole Noether discussion. On the one hand, it's a really cool theorem that might give students the bigger picture. On the other hand, there's no way they're going to know enough mathematics to understand why it's true. And you'll probably have to explain symmetries, too, which could further confuse.

Here's an idea, you could gather up a bunch of different text books and Internet articles, splice out the jist of what they say about energy, format everything and give it to the students as a hand out. That way they can find a definition that works for them.

Then again, that might be too much work.

When I was younger, my biggest question about energy was not what it was, but how we knew that its various forms were equivalent. I wanted to know what heat had to do with the square of velocity had to do with E=mc^2. I also wondered why a mathematical construct like potential energy was on an equal or greater threshold in equations to something like kinetic energy which could be measured directly.

Still, the thing that would have satisfied me most, I think, would have been a basic discussion of why the concept of kinetic energy can be used to describe thrown balls as well as hot frying pans and laser beams. Of course, that would probably involve introducing the Einsteinian model of a solid and the virial theorem for the frying pan part, and then electromagnetism for the laser.

Maybe you could just tell them that it will make sense when they're older?

" In college there's just too much that has to be said, so such a strategy would be impossible even were it not a waste of student and professor time."
I do not believe this strategy is impossible and there are folks who have found it to not be a waste of time.
http://modeling.asu.edu/rup_workshop/

A huge challenge I've found in my teaching is that it feels like if we could just find the right way of saying it, the students will get it. The research I've read and my experience teaching HS physics tells me that the students need to do the work of making sense of things. If I'm doing all the work, I understand it a lot better, but they don't.

The full apparatus of Noether's theorem isn't necessary in this special case. The clearest derivation (as usual) is in Landau and Lifshitz volume 1; the clearest intuition is in the Feynman lectures, vol.1 (as others have noted).

The only problem with L&L is that it's from the Lagrangian. It's easy from Newton's third law as well (I posted a derivation some time back), but requires integrals, derivatives, and Taylor expansion.

Beyond Feynman's brief thoughts and Landau's lucid proof, the best exposition I know is in Hatsapoulos and Keenan's thermodynamics book (which is, incidentally, the only clear thermodynamics book I know). They take the time to very carefully set up everything explicitly, which is actually a much longer path than you might think.

I suspect the best place to begin is where it historically began: mechanical equivalent of heat in Count Rumford's cannon boring experiment. Something's flowing from our rotating borer to the canon, and from the canon to the surrounding air.

Then hook the borer up to a dropping weight. Now we get a finite amount of boring corresponding to a particular change of position of the weight, so whatever is flowing, it also exists in arrangements of particles.

Then use the dropping weight to spin up a flywheel. A really good flywheel can keep spinning for a long time, so idealize it so that it just keeps spinning. We can use it to lift the weight again, or we can hook it up to the borer. So motion is also equivalent to heat or arrangement of particles. And you can also point out that you can use it to charge car batteries or drive chemical reactions, so there are yet other bizarre forms of "energy."

But you can hook the flywheel up to another system any time you want. It doesn't matter when. So the amount of energy in the system is measuring something about how the flywheel DOESN'T change in time. That segues into Noether's theorem.

It's kind of like trying to explain why long division works as opposed to teaching someone how to do it.

I agree with your students, force is more basic than energy. In general my feeling about the "trouble with physics" is that it is due to excessive worship of symmetry principles and the first place this happens, in the education of a student, is energy.

Explaining Noether's theorem to your students is not going to happen. Most of them cannot understand theorems. When I taught calculus, I faced the same issue. The method I used was examples.

For a ball following y = -0.5 a t^2, you can compute E = 0.5 m v^2 + may and show that the result is a constant =0. Redo the problem with y = -0.5 a t^2 + bt + c. Still, E is a constant. The important fact is that it does not depend on time.

Then say, "a theorem by Noether, a brilliant mathematician from the early 20th century, says that this is a general feature of physics problems. You can write down something called the "energy" which doesn't change with time. And we will use the concept of energy to solve problems even if we don't have the math to understand her theorem."

From a pedagogical standpoint, I agree with DGs statement (@10): For students that have never done physics before, it's probably useful to dispell popular misconceptions about energy. This has some semblance of a constructivist approach, which is likely very important for students entering a learning situation for which they may possess implicit knowledge of concepts that are opposite to formally taught concepts. As an example, one study showed that even after participants demonstrated an explicit understanding that regardless of size, all objects fall to earth with the same acceleration, an implicit memory phenomenon known as representational momentum, showed a bias toward impetus theory, which is the belief that heavier objects fall faster (or rise more slowly).

Based on this, my very general suggestion would be to get an idea of what your students' initial understanding of what energy is, and then build on that. For example, it's been awhile since undergraduate physics for me. The first thought that comes to my mind concerning the concept of energy is my recollection of matter as being described as frozen energy (probably ironically from some television program). The next thing that comes to my mind is of a snowball, with the snowball representing matter. The third thing that comes to mind is of that same snowball rolling down a hill. When it starts rolling, then that snowball becomes matter in motion. And based on that analogy, matter in motion would be my definition of energy. It seems consistent with Einstein's equation, given that if you remove c^2 (a velocity measure), you end up with E=m (i.e., a non-moving mass).

From this concept of energy as matter in motion, it should be easy to introduce various other descriptions of this such as potential energy and kinetic energy (for which, interestingly,the equation looks very much like Einstein's equation, only with the 1/2 removed).

I'm with Jon- I like Feynman's explanation. I usually read aloud the "Dennis and the blocks" story to my students that precedes the definition Jon cites. I've been using it since I learned the modeling method that Damon advocates, about 10 years ago.

Randall Knight has a similar story involving accounting in his textbook, which is also useful.

Finally, I highly recommend the series of articles in The Physics Teacher from earlier this year by John Jewett titled "Energy and the confused student".

I hope that I can get a little better every time I teach about energy.

Energy is like pornography - I can't define it for you but I know when I see it.

Sorry to add to this late, but I also have been thinking about energy. Really, it is confusing. The standard definition of "the ability to do work" is not very good. Anyway, this is a link from my previous thoughts (I just moved it from my old blog to my new blog)

http://blog.dotphys.net/2008/10/what-is-energy/

Motion: look at an object. Now look at it again. Is it in a different place? It's in Motion.

Check its Motion now. Check its Motion later. Is it moving faster, slower or in a different direction? Its Motion has changed over time.

What caused the change in Motion? The causal agent is always called "Force". That's what Force is - the proximate cause of a change in Motion.

When Force changes Motion, Energy is transferred from one system to another. When the Energy is depleted, the Force stops acting and the Motion stops changing.

Energy is Force's bank account. It is a measure of the capacity of one system to apply a Force to another system.

Chris was on the right track, but instead of saying "the ability to do work" make it "the ability to make things move". All forms of energy can make mass move in some fashion; hands push on bricks, hammers drive in nails, photons make electrons dance (er, at least, dance to a different tune).

And how can second-year students (I guess sophomores in the States) not know calculus?! This is absolutely absurd! Calc should be a co-req for first-year physics and a pre-req for second year.

Is there something to be learned from looking at LIVING SYSTEMS for clues about how to teach concepts? Understanding the generation, storage, and utilization of energy is key to understanding nutrition. In turn, nutrition takes principals from physics to explain things. Using familiar concepts may be helpful in teaching students with limited math and science background.

Second the motions on Rumford boring cannon (as opposed to teacher boring student); on my mentor and co-author Feynman; and on giving enough different examples and saying that it takes Calculus to show why these are really the same.

Emmy Noether is too far beyond these students, but, yes, I agree there too. Finally, "The Physics Teacher" is the best journal of it's kind, though I admit to bias because my wife has an article and experiment there.

In September I started out my 9th, 10th, and 11th grade classes with this and other basics. I started with conservation of mass. I did use Einstein and E = mc^2. I borrowed a nickel from a student. I said that it had a mass of 5 grams. I said that the stomic bomb that wiped out Hiroshima converted 5 grams of fissionable material into lighter elements plus energy. The energy in 5 grams kills thousands of people. Then I dropped the nickel on the floor. They watched it fall, heard it drop. "Boom!" I said.

I'd forgotten that that the tactic mentioned in #2 was introduced by Feynman.

One of the nice teaching techniques, especially for your kind of class, is an analogy to money. It even works in higher-level classes, since even the most complicated problem is basically one of keeping the books straight. I saw a really nice version of this in a book about teaching physics by Randy Knight (see post #19), whose textbook also adopts the "new mechanics" described below. Jewett, also mentioned in #19, is one of the authors of what must be the best basic "physical science" book yet written. It also uses the "new mechanics" approach.

The problem with energy is precisely that it is not apparent to an observer. You can see velocity. You can sense mass and force. All of the things in Newton's mechanics are pretty close to our senses.

Contrary to the claim made in #20, the problem is that you cannot "see" energy. You cannot see that the big truck next to you has more kinetic energy than your car does until both of you run into a wall and see what happens. You cannot see, just by looking at a moving car, that going 78 mph means it will take more than twice the distance to stop compared to whey you are going 55 mph. Most people don't even believe it until they do an experiment.

It is no accident that it took centuries before the energy principle shows up in Hamilton's mechanics, and even longer before the idea took hold that thermal energy is just the kinetic energy of atoms and molecules. (We still use U for internal energy, as it if were a potential energy rather than kinetic energy.) The fact that we create a new kind of energy every time we run into a conservation problem -- energy appears? ah, that is just mass energy -- only makes it messier.

The "new mechanics", where momentum is introduced before energy, is a pedagogical strategy to introduce conservation ideas in a case where (a) the entity is a bit closer to direct observation, since it is closely tied to sensing inertia, and (b) it is a purely mechanical concept without all the Hydra heads of chemical, nuclear, mass, thermal, ... to deal with.

There's another reason high school classes include a lot of time where students work on their own. High school students take one class after another after another. Lecture after lecture after lecture would quickly lose them. In fact, unless you are an exceptionally good lecturer, or they are exceptionally interested in the subject, most 15-year-olds will have tuned out after half an hour. Thus, high school teachers are instructed to vary "teacher-centered" and "student-centered" activity.

Lecturing is a very efficient way of getting information from a teacher's brain to 200 students' ears. It can be a very inefficient way of getting information into a high school student's brain.

Reading some of the comments, I feel like Moliere's character who was pleased to discover that all his life he'd been speaking prose; he thought he had just been talking. I teach 11th grade physics and a few years ago decided to teach chapter 6 of Serway and Faugn (momentum) before chapter 5 (energy).

When I introduce Newton's Third Law, I and a student sit in two rolling office chairs facing each other. We pull up our feet, put them together and push off. We go in opposite directions. When I do it with a big student, I go farther. When I do it with a small student, the student goes farther. They see that we go in opposite directions and I assert that we are exerting equal forces on each others. They do some calculations showing that the smaller person will accelerate more, reach a higher velocity, and go further--just what they saw.

When we do momentum, I have them pull out their calculations and calculate this new concept mv. Wow, the mv's are equal. Conservation of momentum stares them in the face.

I wish I had a wonderfully successful way to introduce energy. But I just tell them momentum was one way of looking at motion and now we'll do a second (the idea of explicitly using Noether's Theorem is funny; these kids are definitely pre-calculus. Though hopefully they learn that a Very Important Thing about both concepts is that they are conserved.)

"But before we can do energy, you have to know the physicists' concept of "work" It's not what normal people mean ..." I define work as exerting a force over a distance, and we do a day of W=Fd problems. I then say energy is the ability to do work, to exert a force over a distance. "This won't mean much to you now but it turns out to be an incredibly powerful concept, because energy can change from one form to another, but the total amount remains the same."

A little qualitative back and forth about different forms of energy and how they change into each other and then they calculate the kinetic and gravitational potential energies of an object falling after 1,2,3, etc seconds. It's largely a review of kinematics until I have them add the kinetic and potential to get mechanical energy. Stays the same. How about that?

Rereading that last paragraph, I see that there's a certain amount of "just trust me on this" going on. E.g., why is v squared in kinetic energy and why the half? I wish I could do better.

I don't see that Noether's theorem is much use to a class that doesn't know calculus. According to Wikipedia (yes, I know) ...

"any differentiable symmetry of the action of a physical system has a corresponding conservation law. The action of a physical system is an integral of a so-called Lagrangian function"

Right away you have a lagrangian (whatever that is), and an integral (whatever that is).

#27 "I wish I had a wonderfully successful way to introduce energy."

Well, gravitational PE is easiest to introduce. Lift a brick - it takes physical effort. Each increment you lift it by takes another increment of effort. Drop it. Where did that effort go? Of course - this sneaks in the notion that energy is conserved, the notion that it must still be around somewhere (rather like the idea that we have to "go" somewhere after death sneaks in the idea of conservation of conciousness).

In any case, at that level you can get away with simply declaring that it *is* conserverd, that it *does* have to "go" somewhere. if a clever student should ask why this must be the case, you can point Mr Clever-bottom at the relevant text.

Once that's out of the way, energy becomes "that which can lift a brick up by a certain distance". You can demonstrate that heat can lift things (when a piston expands, the gas in it cools down), that a moving object can lift things (balistic pendulum) - heck, if you can work out a way to propel a thing exactly twice as fast as another thing (meccano comes to mind), you could even demonstrate that it will lift a pendulum four times as far. Or, for that matter, that it will lift itself four times as far.

Do the integration with them, using areas. eg: Suppose a constant force of 10 Newtons is applied to 1 kg for 1 seconds. That's a rectangle, 1 wide by 10 tall. To stop the kilo (back to the lab frame of reference), apply 10 Newtons for one second the other way. Alternatively, apply 100 Newtons for 1/10th of a second. Slow it down fast, but it's the same amount of energy. To know how much the kilo has changed from rest, we need to know how much force has been applied for how long. Or, 1 Newton for 10 seconds. Same area=same change (specifically, the kind of change called energy). You can apply 1 Newton for 5 seconds, then 10 Newtons for 1/2 a second. The area of triangles should be accessible to undergraduates who dared take a non-calculus thread (?), so variable forces can be covered. One could perhaps do an experiment using two springs (one each for speed-up and slow-down) and an appropriately chosen weight, measuring and graphing the extension of the springs over time, so that the relative areas can be worked out?

I think you can do these rectangle (and triangle) graphs, then say in passing that integration is just working out areas under fancier curves than triangles. That's fancy mathematics, but Real Experimental Data is about line segments, there are no curves.