Tuesday, 23 September 2014

A little science lesson -- for me.

During my high school lecture tour perpetuating myths about Galileo, I met a teacher at a school in the remote Far North of New Zealand, who taught me a nice lesson about science.

Actually, he just frustrated the hell out of me, and his memory has been irritating me for the last 17 years, like a mosquito bite on the brain. But now I have worked out how to interpret his maddeningly obstinate confusion as a "teachable moment". I have found some brain ointment.

Over lunch after the lecture, this teacher complained how difficult it was to explain basic physics to his students. For example, the simple concepts of force and acceleration. Even the textbooks got them confused.

I nodded along to everything he said, but the last bit made me suspicious. Exactly how were the textbooks wrong? And which sloppy textbooks was he talking about?

"When you throw a ball in the air, it goes up and then it comes down, right?"

So far, so good. "Uh huh."

"And somewhere in between it stops, right?"

"If you throw it exactly straight up, yes," I said, watching out for the catch.

"So on the way up, there's a force slowing it down. Right?"

"Yes. Gravity."

"And on the way down, there's a force speeding it up?"

"Yes. Gravity again."

After a week of talking about Galileo, and dropping hammers and feathers from the top of a ladder, and about to embark on a PhD studying gravity, I felt I was on especially familiar territory.

"So," he continued, "there's a point in the middle where it's not slowing down, and it's not speeding up. It's stopped. And at that moment there is no force on it. Right?"

Aha! Wrong.

I tried to explain. There were two other physicists travelling with me, and they tried to explain, too. There is a constant force, acting downwards. He was not convinced. The arguments got increasingly complicated. At some point we imagined careful video recordings of the experiment, and complex measurement apparatus strapped to the ball, and observers suspended by hot-air balloon at the apex of the ball's flight. None of it helped. Nothing would convince him, and eventually it was time for us to leave.

What was wrong with this guy? Having completely failed to explain one of the most elementary concepts in physics, I have spent years cursing this simpleton teacher with Kauri gum for brains, bumbling about in some hick school in the back of beyond.

But that doesn't wash. The place was not that remote! Getting there did not require three days of treacherous travel by mule through mountain passes. It was not like the only book in town was a battered copy of Gone with the Wind. It was not Macondo, for God's sake, beset by travelling ice salesmen and run by deranged ex-soldiers and populated by feverish fabulists. A major highway passed through. There was an airport. There was electricity, television and radio, and a public library. You could access the internet. The school certainly possessed the same basic science textbooks as any school in London or New York or, for that matter, my own high school.

And he was not stupid. He was very clever -- he countered all of our arguments with inventive and thoughtful reasoning, and was likely equally or more intelligent than many who teach the correct explanation of accelerated motion.

All these years later, I will make another attempt to clobber this demon from my past, and explain it again. Let's look at some numbers. Imagine that you perform this experiment with a tennis ball, or a peanut, or a brick; any object that you can throw without too much concern for air resistance. (Poor choices include a sheet of paper, a living bug, or a handful of self-raising flour.) You throw your object upwards at 20 metres per second. That is about 70 km/hr, so I suggest you do it outdoors, and I suggest you do not use the brick.

Every second the force of gravity will cause this object's speed to change by 10 m/s. The object is going up, and gravity is directed downwards, so each second the object's speed drops by 10 m/s. It starts at 20 m/s. One second later, it is rising at only 10 m/s. Another second later it is at 0 m/s -- it has stopped. If we look at the speed of the object every second, starting at the moment it was thrown, we have 20, 10, 0, -10, -20, and then it is back at ground level. The force of gravity just keeps doing its thing, in the same way, in the same direction. When we look at it this way, there is nothing very special about the moment when the speed is zero; when the numbers become negative, that means the ball is going down.

And this is where I have finally learned a lesson. If this is clear, it is because we used some numbers. In other words: mathematics. It was not very complicated mathematics -- my phone has a calculator app to do the subtraction for me.

The point is that before Galileo the idea of using any mathematics at all was unheard of.

Without mathematics you have to be extremely careful and precise with the words you choose, and you need to make your reasoning especially sharp to distinguish one idea from another, to refine opposing ideas, and to decide between them. Clever people could spend hours, or weeks, or years, or centuries, arguing over this sort of thing and not get it straight. And they did: that's why it was not clear until the time of Galileo, and it was not until yet hundreds of years later that it was clear enough to be put into a high-school textbook.

There are not any numbers in Aristotle, or diagrams or equations. Just arguments. One line of flawless reasoning after another -- leading, in the case of motion, to the wrong answer.

There are no experiments, either.

We need mathematics to make even the simplest ideas clear and precise, and we need precise measurements and experiments to test our ideas. Without them we get lost.

If I had tried to argue with that teacher with a few numbers, or a diagram, perhaps I would have been more successful.

He had misunderstood one of the most basic concepts in science, but so had I.


  1. Did the teacher recognize that there was a non-zero acceleration at the highest point?

    I teach that acceleration is a measure of change in motion. Force is that which causes motion to change. If there is a net force, motion changes. If motion is changing, there is a net force.

    In your example, I would ask the teacher to describe the motion. The answer I get from students is usually something like, "it's going up and getting slower, then slower, then it stops, then it goes down, and gets faster, and faster."

    "At which point," I ask, "did the motion stop *changing*?"

    It is important to break students of the concept that force relates to motion. I spend a good chunk of a term working on this. I'll show a free-body diagram, and ask students what the net force is. When students reply with "zero," I'll ask how they know. At least one student will answer, enthusiastically, "because it's not moving."

    I always reply, with equal enthusiasm, "and it's still not moving, and it's still not moving, and it's still not moving...." I continue until someone blurts out, "it's not accelerating," or "its motion is not changing."

    It takes a while, but most eventually get it.

    Of course, it is easiest with math. If you asked that teacher to draw a velocity vs. time graph, he would probably be able to draw a correct one. He would probably also recognize acceleration as the slope. For my more mathematically inclined students, recognizing that there is a non-zero slope at v=0 is the spark that makes everything clear.

  2. A diagram would also have helped.

    I was mostly just amazed that it was a *teacher* who was asking this question.

  3. "So," he continued, "there's a point in the middle where it's not slowing down, and it's not speeding up. It's stopped. And at that moment there's no force on it. Right?"

    Right! There's no force acting on it at that moment. The force was acting when you threw the ball upwards. You did work on that ball. F=ma and E=Fd. You gave that ball kinetic energy. Gravity merely converted this into potential energy in the ball*, then into kinetic energy again as the ball descended. We call it the force of gravity, and yes, it alters motion. But at no point did gravity do any work. Gravity isn't a force in the Newtonian sense, conservation of energy applies.

    As for misunderstanding basic concepts, try this: gravity is said to cause a descending photon to blueshift. But it doesn't. Gravity doesn't do any work, conservation of energy applies. You measure a blueshift because when you descend you shed the kinetic energy which used to be potential energy which was mass-energy. That photon appears to have gained energy because you lost it.

    * People talk about the system, but whilst p=mv momentum is equal and opposite there's no detectable motion of the Earth, so none of the KE=½mv² can be assigned to the Earth.

    1. Reading comprehension, John, that's your problem.

    2. The problem, Miranda, is that when you send a 511keV into a black hole, the black hole mass increases by 511keV/c².

    3. Tsk. Photon. When you send a 511keV photon into a black hole, the black hole mass increases by 511keV/c². And when you repeat with a 511keV/c² electron, you're left with Friedwardt Winterberg's firewall, and it's all downhill from there, because the coordinate speed of light can't go less than zero so you can kiss your black hole dynamics goodbye.

    4. Before we worry about black holes, let's try and get basic Newtonian dynamics straight. If the ball is accelerating, there is a force on it. From the moment when we start observing this experiment (the ball is moving directly upwards at 20m/s), until the moment we stop (the ball is moving downwards at 20m/s), there is a constant acceleration, and therefore a constant force. Can we agree on that?

    5. Agreed. Although if the measurements are being done in a non-inertial frame, the measured acceleration may have to be attributed to an inertial force such as centrifugal force or coriollis force or (if you believe in GR) gravity.

      Teaching physics is hard, because what you are really trying to do is change someone's world view. I agree with you that the only way you can do this is through using math to make precise statements and incorporating experiments (bleghhh) and testing into your approach.

    6. You're right -- I should have added that we are in an inertial frame.

  4. Mark: yes, I agree that in terms of Newtonian dynamics, there's a constant force.

    1. Ok. I'm confused, because you said, "There's no force acting on it at that moment", which I took to mean the moment when the ball is at the top of its flight.

      You do add the qualifier, "in terms of Newtonian dynamics". In this case that means accepting the relationship between acceleration, mass and force, F = ma. I assume you're fine with that?

    2. No, because the principle of equivalence relates the force of gravity to an accelerating rocket when you're standing on the ground, not when you're in free-fall. Gravity isn't a force in the F=ma or E=Fd Newtonian sense. It doesn't do any work on the falling ball. It doesn't add any energy to the ball. The ball's kinetic energy comes from its potential energy. From its mass-energy. Gravity merely converts internal kinetic energy into external kinetic energy, because there's a local gradient in the coordinate speed of light. The "invariant" mass of the ball is reducing as it falls, hence the mass deficit. You cannot keep doing this without consequences. And once the coordinate speed of light is zero, it can't go any lower, and there is no more gradient, no more gravity, and black holes don't fly true.

    3. The principle of equivalence has nothing to do with it. Viewed by anyone moving at a constant velocity, the ball is seen to accelerate. That means there is a force on it.

      If there were no force, the ball would continue upwards at a constant speed, and disappear off into space. To cause the ball to slow down and stop, a force must be applied, and that force does some work. You can think of it in terms of a change from kinetic energy to potential energy and back again, but either way, along each step of the way some work is done.

      We cannot move on to special relativity, the equivalence principle, or general relativity until all these points are straightened out.

      To connect back to the blog post: most of this discussion is again with clunky words and no mathematics. But we have some hope (yes, some!) of making progress because in the background there are all the calculations that reassure us that acceleration, force and energy and meaningful and well-defined concepts.

    4. Mark, it’s no good ignoring general relativity in favour of the physics it replaced. When you lift a brick, you do work on it. You add energy to it, you increase its mass because the mass of a body is a measure of its energy content. When you drop the brick, gravity converts some of the internal kinetic energy into external kinetic energy. Simplify the brick to one electron. You know about pair production and electron spin and electron diffraction and the wave nature of matter, so simplify the electron to light going round and round, then simplify it further to light going round a square path, like this. You know that light bends in a gravitational field, so draw the square repeatedly with the horizontals bending down a little, like this. See how the electron falls down? There’s no actual force acting on it. See Baez, and note that a curvature of rays of light can only occur when the propagation speed of light varies with position. The “force” of gravity at some location depends on the local gradient in the "coordinate" speed of light. And the important thing for you is this: light can't go slower than stopped, and if you stop the light going round that square, it doesn't fall down.

  5. General relativity reduces to the Newtonian picture for weak gravitational fields, low speeds, etc. Throwing a tennis ball in your back yard counts as just such a situation.

    But for the record: how can you sensibly include the gravitational potential energy in an object's mass? Are deep space satellites, lifted extremely high in the Earth's gravitational field, now incredibly massive?

  6. You sensibly include it because you know about the mass deficit and binding energy and about mass in general relativity. You start with two planets of mass m. They fall towards one another, they collide at say 22km/s, the kinetic energy gets radiated away into space, and the mass of the combined planet is a little less than 2m. Conservation of energy applies.

    Deep space satellites aren't incredibly massive. The mass increase is small. Compare the E=mc² mass-energy of a body to the mass-equivalence of the KE=½mv² kinetic energy of that body when it's doing 11km/s.

    1. Sorry for diverting the discussion with the second part of my comment.

      Before discussing general relativity, we still need to agree that the Newtonian picture of forces and accelerations is entirely self-consistent and reasonable. If you want to argue that the ball at the apex of its flight has no force on it, then you have no place discussing physics of any kind.

      This is very very simple. Some people still don't get it, and that's fine -- but they don't pretend to pontificate on science.

    2. Mark, general relativity is different to the Newtonian picture. Go find a relativist, somebody you trust and respect, and get him to check what I've said. When he verifies what I've said, then maybe we can talk some more about black holes.

    3. As I said before: general relativity reduces to Newtonian gravity in the weak-field limit. That was one of Einstein's checks that his theory made sense, and every GR text makes this very clear. Wald, Schutz, Weinberg, MTW, Carroll, Hartle, whoever you want. Also all of the relativists I know, which is quite a few, being one myself.

    4. And in Schutz section 7.3 you can read this:

      "Although in the appropriate limit our curved-spacetime picture of gravity predicts the same things as Newtonian theory predicts, it is very different from Newton’s theory in concept."

      That's right. That's what I've been trying to tell you. And the selfsame section relates to something else I've been trying to tell you:

      "It turns out that for any system whose spatial extent is bounded (i.e. an isolated system), a total energy and momentum can be defined, in a manner which we will discuss later. One way to see that the total mass energy of a system should not be the sum of the energies of the particles is that this neglects what in Newtonian language is called its gravitational self-energy, a negative quantity which is the work we gain by assembling the system from isolated particles at infinity. This energy, if it is to be included, cannot be assigned to any particular particle but resides in the geometry itself. The notion of gravitational potential energy, however, is itself not well defined in the new picture: it must in some sense represent the difference between the sum of the energies of the particles and the total mass of the system, but since the sum of the energies of the particles is not well defined, neither is the gravitational potential energy. Only the total energy–momentum of a system is, in general, definable, in addition to the four-momentum of individual particles."

      That isn't right. Gravitational field energy is positive. "The energy of the gravitational field shal act gravitatively in the same way as any other kind of energy". And if I may reiterate: when you lift a brick you do work on it. You add energy to it. You increase its mass. The work you do does not end up as energy that is "in the geometry". The kinetic energy of a falling brick does not come from "the geometry". It comes from the brick. Internal kinetic energy is converted into external kinetic energy as per my electron example above. Again, please check this with a relativist you respect and trust.

    5. On the first quote: yes, of course GR is different in concept from Newtonian's gravity. But forces and accelerations are still meaningful, and neither Einstein nor any of his illustrious successors would claim that a ball thrown in the air ever has zero force on it. To claim such a thing would betray an extreme (and frankly perplexing) ignorance of basic physics concepts.

      On the second quote: just because potential energy (or linear momentum, or angular momentum, etc, etc) are not well defined in GR doesn't mean the concepts don't exist, or are not meaningful in themselves. It just means that you cannot produce coordinate-independent quasi-local definitions. It certainly doesn't mean any of the kooky stuff you claim.

      I'm afraid that reading these books is not equivalent to understanding what they say. You might want to re-read the posts I wrote about scientific expertise. Don't comment -- just read them.

  7. Huh? See free fall on wiki where you can read this: "In Newtonian physics, free fall is any motion of a body where its weight is the only force acting upon it. In the context of general relativity, where gravitation is reduced to a space-time curvature, a body in free fall has no force acting on it ..." Now go and check what I've told you.

    And do note that conservation of energy is not "kooky", and nor is the mass deficit. Send a 511keV photon into a black hole and its mass increases by 511keV/c². Send a 511keV electron into a black hole and its mass increases by 511keV/c². Not a zillion tonnes. There is no magic, and there is no mystery.

    What posts about scientific expertise? There's your Experts in a Haystack which was dripping with intellectual arrogance and words like crackpot and moron. That didn't teach us about expertise, that was a lesson in hubris.

  8. Yes, in GR the "force of gravity" is replaced by motion through curved spacetime. But we are still free to talk in terms of forces if we wish, and in the Newtonian limit (like a ball thrown in the back yard), it is needlessly pedantic to say, "There is no force." GR is to a *fantastic* approximation Newtonian gravity, and there is no need to throw out our introductory physics texts.

    In particular, all calculations that we perform in this context using Newtonian gravity (including forces, potential energy, and all the rest of it), will produce the correct answers. There is no contradiction.

    As for expertise: beneath the tone of the post you cited, there is a serious point. Perhaps you find some of the words objectionable because you hear them more often than I do.

    It is not arrogance, but a simple (and arguably unfortunate) fact that some people are experts in a given scientific field, and most people are not, and are unqualified to speak on it. For example, I am not an expert on light rays for which a square is a simpler path than a circle. You are an expert -- perhaps the only expert. And so I will leave that subject to you.

    This discussion is closed.


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