Posts Tagged ‘Physics’

One day in the summer of 2011, while mowing the lawn, I saw a strange creature flying through the air.

Actually, “flying” is too generous a term.  The creature was lilting through the air.  Lurching.  It appeared to have ten legs, and was about the size of a silver dollar.  I was puzzled, to say the least, but that lawn wasn’t going to mow itself so I went back to work.


The mystery bug…

I live in a rural area in the mountains of North Carolina, only 30 minutes away from the entrance to the Great Smoky Mountains National Park.  There’s a lot of wildlife here: I’ve seen bear, elk, deer, raccoons, opossums, groundhogs, voles, and squirrels; our bird feeders are always full of cardinals, chickadees, towhees, finches, and titmice; and I once came home to a 4-foot black snake inside my house.  As for arthropods, I’m very familiar with flies, no-see-ums (family Ceratopogonidae), moths, wasps, honeybees, crickets, ants, beetles, and spiders of all kinds.  I’ve had close encounters with black widow spiders no less than 3 times in my life.

But this thing?  With 10 legs?  Lurching through the air like a drunken hang glider?  Incomprehensible.

Over the course of that summer, I saw such creatures on numerous occasions.  I gradually came to realize that they were insects, since subsequent sightings showed 6 legs, not 10.  My working hypothesis was that what I saw that first day was a mating pair: two of these things stuck together.  But I still had no idea what the confounded creatures were.

In appearance, the insects were bizarre to say the least.  They were striped, like zebras, and their legs appeared  to have at least 3 joints each, so that the legs took on a zigzag character.  They didn’t appear to use their wings, which I guessed were vestigial; rather, picture a 6-legged starfish up on one end, clawing and grasping its way forward.  As a physicist, it looked very much like the creatures were literally swimming through the air.  And so I resolved, with the help of the internet, to positively identify them.

Rutherford said that “all science is either physics or stamp collecting.”  A lot of people take this to be a disparaging comment about sciences other than physics, but I don’t.  I kind of like stamp collecting.  I like being meticulous, and being detailed.  That’s why I like pastimes such as putting together 1000-piece puzzles.

But my search for the identity of the “mystery bug” took stamp collecting to a whole new level.  It literally took me a month of sleuthing to identify the things.  I tried the obvious first: I googled things like “strange zebra striped bug” and “bug that swims in the air” but had no luck.  I posted a question on an entomology bulletin board.  I looked at websites dedicated to “insects of the Appalachians.”

Finally, I had a breakthrough: I saw one of the bugs hitting up against a window in our house.  For the first time, I could see the creature close up and for more than just a second or two.  I verified that the creature did have six legs; I verified that it did have wings, although they seemed useless.  I realized that my mystery bug was a crane fly.  Here’s a more typical, run-of-the-mill crane fly:


A typical crane fly

Regular crane flies are common where I live; kids often mistake them for gigantic mosquitoes (which they are not).

Once I realized that the mystery bug was a type of crane fly, my task was eased enormously.  And eventually I found this assortment of photographs.  Eureka!  I had done it!  They were phantom crane flies, of the family Ptychopteridae.  Specifically, they were the species Bittacomorpha clavipeswhich, according to this Wikipedia article, are “known for the odd habit of spreading out [their] legs while flying, using expanded, trachea-rich tarsi to waft along on air currents.”

It turns out that the phantom crane fly is one of the very, very few creatures on Earth that fly without using their wings.  They are literally swimming, somehow taking advantage of a high Reynolds number (let’s say, 265?) to sludge through the atmosphere without those wings that evolution gave them.  Consequently they look more like seed pods drifting on the wind than they do insects.

What is my point?  I don’t have one.  I just think these bugs are cool, and you should try to find them if you ever visit the Eastern United States.  They hang out in marshy areas in late summer.  Oh, and if you’re a physicist or an entomologist, think about studying these little guys.  The field’s wide open as far as I can tell.  Somebody needs to video the flight of the phantom crane fly, so get on it!  [Note added later: I did find this video which shows the weird flight, are there more?]


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Very slightly more green than blue, “Tropical Rain Forest” can be thought of as a dark cyan.

My wife called me the other day and asked what my favorite color was.

“Hold on one second,” I said.  “I have it written down.”

She explained that she just needed a color in the most general terms, because she was buying me a case for my new iPhone.  So I said “blue.”  But I was disappointed that I didn’t get to be more specific.

You see, my actual favorite color is (currently) Tropical Rain Forest, formulated by Crayola in 1993.  Its RGB color code is (0, 117, 94).  If you want to read about the color, it’s the first variation on jungle green in the Wikipedia article of the same name.

But what’s an RGB color code?  Anyone familiar with computer graphics will recognize RGB as standing for Red/Green/Blue, which are taken to be the three primaries.  And therein lies a tale: for didn’t we all learn in kindergarten that red, blue, and yellow (not green) were the primary colors?  What’s going on?

Light comes in different wavelengths, or more commonly, combinations of multiple wavelengths.  Color is a purely biological phenomenon having to do with what we perceive with our eyes.  So when a kindergarten teacher says that “mixing red and blue make purple”, there’s really a whole lot of physics and biology that’s being glossed over.

In our retinas, we (generally) have three kinds of cones that react to incoming light.  These cones can detect many wavelengths of light, but each peaks in a different part of the spectrum.  Very simplistically, we can say that one peaks in the “red” part of the visible spectrum, one peaks in the “green” part of the spectrum, and one peaks in the “blue”.

Now, the “red” cones don’t just react to red light—it’s just that they react most strongly to red light.  But light in the “green” part of the spectrum might also stimulate a “red” cone to some degree.  The colors that we see depend on how our brains interpret three signals: how much each of the three kinds of cones is stimulated by incoming wavelengths of light.  For example, if a “red” cone and a “green” cone were stimulated about equally, your brain would interpret this as seeing yellow.  If all three cones were stimulated strongly, you’d “see” white.  (It’s weird to note that different combinations of wavelengths can actually cause the same sensation in your brain: there’s not necessarily a unique combination of wavelengths for any given color perceived.)

Here’s a chart to help you out (note that this is very simplistic and glosses over many issues which I will address later):

Kind(s) of cone stimulated            What you perceive

“Red”                                                                Red

“Green”                                                            Green

“Blue”                                                               Blue

Red & Green                                                    Yellow

Red & Blue                                                       Magenta

Green & Blue                                                   Cyan

Looking at this chart makes the notion of an “additive” primary easy to understand.  We declare red, green, and blue (RGB) to be the additive primary colors.  We can then build (most) other colors by adding these colors together.  This corresponds to multiple wavelengths of light stimulating one or more cones in the retina to varying degrees.  If you want an applet to play around with this kind of additive color mixing, try this.  Input (0, 117, 94) if you want to see Tropical Rain Forest.


Additive color mixing

One caveat: the RGB scheme arbitrarily chooses three exact wavelengths of light to be “the” additive primaries, but this represents a judgment call on our part.  The degree to which different wavelengths of light stimulate the three kinds of cones is messy; the graphs of intensity (of cone response) vs. wavelength are not perfect bell curves, and have bumps and ridges.  Furthermore, it has long been known that if you try to limit yourself to only three “primary” additive colors then you cannot reproduce every possible color that humans can perceive.  We would say that the gamut of possible colors you can make with an RGB scheme does not encompass all possible perceived colors.  (For example, true violet as seen in the rainbow cannot be reproduced with RGB—it can only be approximated.  You can’t see true violet on a computer monitor!)

Now, tell a 6-year-old that Red + Green = Yellow, and they will look at you like you’ve grown a second head.  That’s because most experience we have with “color mixing” doesn’t involve mixing different kinds of light; it involves mixing different kinds of pigments.  And that’s a totally different ball of (crayon?) wax.

Suppose I have a flashlight that shines red light.  I have another flashlight that shines green light.  If I shine both flashlights into your eyes, you will see yellow, as we just discussed.  With two flashlights (two colors), more light has reached your eyes than would have with just one flashlight.

Pigments (such as crayons or paint) work in the opposite way.  “Red” paint is paint that takes white light (a combination of R,G, and B) and subtracts some of the light away, so that only the R reaches your eyes.  Green paint takes RGB light and lets only the G reach your eyes.  In other words, red paint “blocks” G and B, whereas green paint “blocks” R and B.

Can you guess what happens if we mix red and green paint?

The 6-year-old knows you get black.  That’s because two successive blockers have filtered out all the light, and nothing reaches your eyes at all.  And when no cones are stimulated, we perceive that as black.

When a teacher says that the “primary” colors are red, blue, and yellow, they are referring to so-called subtractive primaries.  By mixing those three kinds of pigments, you can make many of the colors we can see.  But not all the colors.  Try mixing red, blue, and yellow to make pink.  It cannot be done.  Like the additive primaries, the gamut of the subtractive primaries is limited.  And, like the choice of RGB as additive primaries, the choice of red, blue, and yellow as the subtractive primaries is arbitrary.  Arbitrary, and inferior.  It turns out that using yellow, magenta, and cyan as the subtractive primaries expands the gamut and increases the number of colors you can make by subtraction.

Why yellow, magenta, and cyan?  Well, those choices make sense if you’ve already picked RGB as your additive primaries.  Consider the chart above.  It’s clear that a paint that looks magenta must be blocking green, since you’re seeing an (additive) combination of red and blue.  Similarly, yellow paint blocks blue, and cyan paint blocks red.  So what happens if we mix, say, yellow and cyan?  Well, the mixture will block blue, and then block red, so what is left is green.  You can try this here.

Anyway, I spent some time at this site trying to determine exactly my favorite color.  I finally chose Tropical Rain Forest, RGB=(0, 117, 94).  I think it’s peaceful and organic.  I also enjoy Tyrian purple, RGB = (102, 2, 60).  Let me know which colors you favor.

Coming soon: some thoughts about the extra-spectral colors!

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A recent Toyota commercial begins, “In space, the shuttle Endeavor is practically weightless.”

Do we really have to go over this again?

The fact that the word “practically” is in there indicates that the copy writers don’t have a clue about physics at all.

If they had just said that Endeavor is weightless, I’d be more forgiving.  Such a statement could mean that Endeavor was millions of miles from the solar system, in deep space, and therefore (almost) weightless.  Or it could (more plausibly) mean that Endeavor was in orbit, and that its apparent weight was zero, and they were just confusing weight with apparent weight (like most non-physicists do).

But the Madison Avenue geniuses said Endeavor was “practically weightless.”


This implies, of course, that in space you have weight, but it has been reduced—by being in space, apparently.  The acceleration due to gravity, g, does decrease as you leave the Earth, but as I’ve already discussed, it doesn’t go down enough to approach zero—not unless you go ridiculously far from any other massive object.

Now, a commercial with stupid physics wouldn’t normally get me to reblog about a topic I’ve already covered.  But it gets worse: the Toyota people double-down on their ignorance, and pile BS onto their BS.  The whole point of the commercial is that their truck can pull the space shuttle.

Gee, really?  Well guess what—a mini-Cooper could have pulled the space shuttle, too, given enough time.  So could I.  So could Mr. Burns.  You see, Newton’s 2nd Law says that a net force causes an acceleration, so any net force will cause (some) acceleration.  Sure, it might be small, but in the absence of friction it will eventually get the job done.


…and so, ad infinitum.

I once saw a video of a flea pulling a hockey puck along the ice, even though the puck (around 160 g) had a mass over 700,000 times bigger than the flea (around 220 μg).  It took some time, but the puck eventually moved noticeably.  (Sorry, I couldn’t find the video on the internet.)

Well, what about friction?  Maybe there’s some horizontal friction between the shuttle and the ground, and a Toyota Tundra is forceful enough to “overcome” that friction whereas a mini-Cooper is not.  This is a valid point, but the writers of the commercial were definitely not thinking of this.  How do I know they were not thinking of this?  Well, because they say (as if it is important), “that bad boy weighed 292,000 pounds.”  If that’s all the information we are to be given, then we can’t conclude anything about the merits of their truck: if friction is zero, then the feat is less than impressive.  If instead the coefficient of friction is tremendous, and the normal force between the shuttle and the ground is truly 292,000 pounds, then the feat is amazing, in particular because I would wonder why the truck doesn’t subsequently pull itself back towards the shuttle à la Newton’s 3rd law.  But they don’t mention friction, and therefore they don’t get to play that card.  Occam’s razor suggests that the copy writers just don’t know squat about physics.

Anyway, I have nothing to say about the merits of the Toyota Tundra.  Maybe it’s a good truck, maybe it’s not.  But as for Toyota Truck commercials…please turn the channel.  You’d do better to watch a roadrunner cartoon.  The physics isn’t any better, but at least it’s entertaining.

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CNN has no love for the room-temperature maser.

It’s almost a new year, which of course is the time that everyone writes retrospectives of the year before.  I don’t really want to write a retrospective; I’d rather start a new tradition: criticizing someone else’s retrospective.  So to begin: I draw your attention to CNN’s Top 10 science stories of 2012.

Let me start by emphasizing that this list was not written by a scientist.  It was written by the CNN Health beat reporter.  So maybe I shouldn’t be so critical: maybe I should give that reporter a pass.  But come on, shouldn’t we expect at least half of the “top 10 science stories” to actually be science?  Is that so much to ask for?  Ideally, such a list should be written by several scientists, or at the very least one scientist.  Having the CNN Health reporter compile a list of the top 10 science stories is a kind of near miss—like having Bob Vila comment on the top 10 advances in mechanical engineering, or having Tiger Woods list the best Cricket players in Australia (be sure to mention Michael Clarke, Tiger).

I am a scientist, so I feel qualified to comment on CNN’s list.  Therefore, in the spirit of new year snarkiness, let’s evaluate each “top 10 science story” for import, for scientific value, and for “wow” factor.  And let’s see how the health reporter did.  Remember, that reporter got paid for their work (and I am not getting paid).  Go figure.

1. Curiosity lands, performs science on Mars

OK, this is cool, and maybe some science will be done eventually—I am not aware of any actual results published yet in a peer-reviewed journal.  But the Curiosity landing on Mars is in itself not science; it’s a remarkable feat of technology and engineering.  So it shouldn’t be on the list.

2. Higgs boson — it’s real

I don’t have a problem with this being on the list.  This is big, and important, and exciting to most physicists.  The one thing it is not is surprising: most physicists had faith in the Standard Model, and most expected the Higgs to be found in the 125 GeV/c2 range.  Now the real work begins: determining all the properties of the Higgs, and all the interactions that it might participate in.

3. James Cameron’s deep dive

Seriously?  How is this science?  Avatar-boy goes on a vanity jaunt to the heart of the ocean, and we pay attention why?

4. Felix Baumgartner’s record-breaking jump

This is an even more embarrassing entry than the previous one.  An idiot pushes the envelope, and we call it science?  Does the CNN Health reporter even know what science is?

5. Planet with four suns

Planethunters.org discovered a quadruple star system with a planet in a (somehow) stable orbit.  This is an interesting discovery and an impressive feat by an amateur collective.  Maybe someone will get a journal article out of this someday, but that’s it.  A bigger story is how many extrasolar planets have been discovered so far—854 by Dec. 24, 2012.

6. Nearby star has a planet

So Alpha Centauri B has a planet.  That’s nice.  But didn’t we already cover extrasolar planets in the previous entry in the list?  A good list should vary its entries: if you were listing your top 10 favorite comfort foods, and if #5 were pepperoni pizza, would #6 be sausage pizza?  I didn’t think so.

7. Vesta becomes a ‘protoplanet’

Sigh.  What’s with all the space stuff?  Hey CNN Health reporter: only a small percentage of physicists are astronomers, you know, and there are many other branches of science than just physics.  Did you consider asking a chemist what’s hot in chemistry?  Did you think of calling a geologist, or a neuroscientist, or a paleontologist, or a solid-state physicist?  I didn’t think so.


Hey CNN? Why didn’t you call Dr. Mayim Bialik?

8. Bye-bye, space shuttles

Again?  More space?  And this isn’t even remotely science.  This is about the retirement of a vehicle.  Good riddance, I say: imagine all the real science that could have been done if the space shuttle money had instead been used to send out hundreds of unmanned probes, to Europa, Titan, Callisto, Ganymede…

9. SpaceX gets to the space station, and back

And still more space?

Dear lord, you’d think from this list that space exploration is the only kind of science that anyone does.  And again: not a science story.  It’s a technology story.  Look up the difference, CNN.

10. Baby’s DNA constructed [sic] before birth

“For the first time, researchers at the University of Washington were able to construct a near-total genome sequence of a fetus, using a blood sample from the mother and saliva from the father.  The study suggested this method could be used to detect thousands of genetic diseases in children while they are still in the fetal stage.”

This is interesting, and may be important, but I have an issue with the list item as the CNN Health reporter wrote it.  The baby’s DNA wasn’t constructed before birth.  The DNA was present before birth, sure; fetuses do have DNA.  The baby’s genome was sequenced before birth, which is a completely different thing.  You’d think a health reporter would know better.

Final Score

So what’s the final tally?  By my count, only of 6 of the items on the list are science, and that’s being very generous.  Of those 6, the fields of science represented are astronomy, particle physics, astronomy, astronomy, astronomy, and biology/medicine.  Nice, balanced list there.  Way to go.

So what stories did deserve to be on the list?  I don’t know.  I am a (mostly) solid-state physicist, and I could mention that researchers successfully used neutrinos to communicate, built a room-temperature maser, and found Majorana fermions.  I’ll stop there, and let chemists, neuroscientists (that means you, Mayim), and geologists add items of their own.

Happy New Year, everyone!

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“I am your probability density”

In an earlier post I discussed my philosophy of teaching special relativity.  My main idea was that physics professors should keep the “weird stuff” at bay, and start with non-controversial statements; once students are on board, you can push under the grass and show them the seething Lynchian bugs beneath.

Well, what about quantum mechanics?  Does the same philosophy apply?

My answer is yes, of course.  Don’t start with Schrödinger’s cat.  Don’t mention the Heisenberg uncertainty principle, or wave collapse, or the EPR experiment, or Bell’s theorem, or the double slit experiment, or quantum teleportation, or many worlds, or Einstein’s dice.  Start with the problems of physics, circa 1900, and how those problems were gradually solved.  In working out how physicists were gradually led to quantum mechanics, students will build up the same mental framework for understanding quantum mechanics.  At least, that’s how it works in theory.

Now, my perspective is from the point of view of a professor who teaches only undergraduates.  I only get to teach quantum mechanics once a year: in a course called Modern Physics, which is sort of a survey course of 20th century physics.  (If I were to teach quantum mechanics to graduate students, my approach would be different; I’d probably start with linear algebra and the eigenvalue problem, but that’s a post for another day.)  As it is, my approach is historical, and it seems to work just fine.  I talk about the evidence for quantized matter (i.e. atoms), such as Dalton’s law of multiple proportions, Faraday’s demonstration in 1833 that charge is quantized, Thomson’s experiment, Millikan’s experiment, and so on.  Then I explain the ultraviolet catastrophe, show how Planck was able to “fix” the problem by quantizing energy, and how Einstein “solved” the problematic photoelectric effect with a Planckian argument.  Next is the Compton effect, then the Bohr model and an explanation of the Balmer rule for hydrogen spectra…

We’re not doing quantum mechanics yet.  We’re just setting the stage; teaching the student all the physics that a physicist would know up until, say, 1925.  The big breakthrough from about 1825-1925 is that things are quantized.  Things come in lumps.  Matter is quantized.  Energy is quantized.

The big breakthrough of 1925-1935 is, strangely, the opposite: things are waves.  Matter is waves.  Energy is waves.  Everything is a wave.

So then, quantum mechanics.  You should explain what a wave is (something that is periodic in both space and time, simultaneously).  Here, you will need to teach a little math: partial derivatives, dispersion relations, etc.  And then comes the most important step of all: you will show what happens when two (classical!) wave functions are “averaged”:

ψ1 = cos(k1x – ω1t)

ψ2 = cos(k2x – ω2t)

Ψ(x,t) = (1/2) cos(k1x – ω1t)  + (1/2) cos(k2x – ω2t)

Ψ(x,t) = cos(Δk·x – Δω·t) · cos(k·x – ω·t)

where Δk ≡ (k1 – k2)/2, k ≡ (k1 + k2)/2, etc.

[Here I have skipped some simple algebra.]

This entirely classical result is crucial to understanding quantum mechanics. In words, I would say this: “Real-life waves are usually combinations of waves of different frequencies or wavelengths.  But such ‘combination waves’ can be written simply as the product of two wave functions: one which represents ‘large-scale’ or global oscillations (i.e. cos(Δk·x – Δω·t)) and one which represents ‘small-scale’ or local oscillations (i.e. cos(k·x – ω·t)).

This way of looking at wave functions (remember, we haven’t introduced Schrödinger’s equation yet, nor should we!) makes it much easier to introduce the concept of group velocity vs. phase velocity: group velocity is just the speed of the large-scale wave groups, whereas phase velocity is the speed of an individual wave peak.  They are not necessarily the same.

It is also easy at this point to show that if you combine more and more wave functions, you get something that looks more and more like a wave “packet”.  In the limit as the number of wave functions goes to infinity, the packet becomes localized in space.  And then it’s simple to introduce the classical uncertainty principle: Δk·Δx > ½.  It’s not simple to prove, but it’s simple to make plausible.  And that’s all we want at this point.

We’re still not doing quantum mechanics, but we’re almost there.  Instead, we’ve shown how waves behave, and how uncertainty is inherent in anything with a wave-like nature.  Of course now is the time to strike, while the iron is hot.

What if matter is really made from waves?  What would be the consequences of that?  [Enter de Broglie, stage right]  One immediately gets the Heisenberg relations (really, this is like one line of algebra at the most, starting from the de Broglie relations) and suddenly you’re doing quantum mechanics!  The advantage of this approach is that “uncertainty” seems completely natural, just a consequence of being wave-like.

And whence Schrödinger’s equation?  I make no attempt to “prove” it in any rigorous way in an undergraduate course.  Instead, I just make it imminently plausible, by performing the following trick.  First, introduce complex variables, and how to write wave functions in terms of them.  Next, make it clear that a partial derivative with respect to x or t can be “re-written” in terms of multiplication:

d ψ /dx  →  ik ψ

d ψ /dt  →  –iω ψ

Then “proving” Schrödinger’s equation in a non-rigorous way takes 4 lines of simple algebra:

E = p2/2m

E ψ = (p2/2m)ψ

Now use the de Broglie relations E = ħω and p = ħk…

ħw ψ = (ħ2k 2/2m) ψ

iħ(∂ψ/∂t) = (–ħ2/2m) ∂2ψ/∂x2

There’s time enough for weirdness later.  Right now, armed with the Schrödinger equation, the student will have their hands full doing infinite well problems, learning about superposition, arguing about probability densities.  As George McFly said, “I am your density.”  And as Schrodinger said, probably apocryphally, “Don’t mention my cat till you see the whites of their eyes.”

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If you don’t go sideways,
you will return to Earth

Why is it that astronauts “float around” in space?

If you were to ask Bill O’Reilly or the Insane Clown Posse, the answer would be that it is a mystery.  If you were to ask the average person on the street, the answer would be that there’s no gravity in space.  Both answers are ridiculous, of course.  It’s not a mystery; we have a very firm working knowledge of the physics of orbits.  And there’s plenty of gravity in space: at 230 miles up, where the International Space Station is, the acceleration due to gravity is about 8.8 m/s2, which is only 10% less than its value at sea level.

So why does the general public still not understand this whole “floating astronaut” thing?

I submit that some of us physics professors are teaching it poorly.  Here’s an explanation from a physics book on my desk:

“All objects in the vicinity of, say, the space station are in free fall with the same acceleration, and so, absent nongravitational forces, they remain at rest relative to each other and their freely falling reference frame.” [Rex and Wolfson, Essential College Physics (2010) p. 215]

I don’t find this very helpful.  And many physics instructors teach “weightlessness” in the same non-helpful way: by hand waving and saying that astronauts are in free fall, and that they are only apparently weightless.  Unfortunately, to the novice this brings up a host of new questions: what’s the difference between apparent weightlessness and actual weightlessness?  More importantly, if you’re in free fall, why don’t you crash into the Earth?

Another book on my desk does a better job:

“Why don’t planets crash into the Sun [if they truly are in free fall]?  They don’t because of their tangential velocities.  What would happen if their tangential velocities were reduced to zero?  The answer is simple enough: their falls would be straight toward the Sun, and they would indeed crash into it.”  [Hewitt, Conceptual Physics, 10th edition (2006), p. 193]

Newton himself also got it right:

“We may therefore suppose the velocity to be increased, that it would describe an arc of 1, 2, 5, 10, 100, 1000 miles before it arrived at the Earth, till at last, exceeding the limits of the Earth, it should pass into space without touching it.”  [Isaac Newton, The System of the World, Section 3, translated by Motte, edited by Cajori (1946)]  [Note Isaac’s use of the word “till”!]

The key idea which we physics professors should emphasize is that astronauts are in free fall, but they don’t hit the Earth because they are moving very, very fast horizontally.  That’s it.  That’s the secret.  They are going so fast that they fall “around the curve of the Earth” so to speak.  I don’t think this horizontal motion is emphasized enough.  I’ll say it again: you need to go sideways to get into orbit.  The next time you’re piloting a spaceship, remember the old adage: that which goes (straight) up will surely come back down (unless you reach escape speed).  So don’t aim for infinity and beyond—aim for the horizon.

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As a physics professor, I have certain pet peeves.  For example, I cringe when someone says that “gravity” is 9.8 m/s2 when they mean the acceleration due to gravity.  I’m annoyed if someone says that an object “weighs” 7 kg.  And I stifle a laugh if someone says that a roller coaster is exciting because it goes so “fast”—humans can only detect acceleration, not speed, which is why we don’t notice that we’re traveling something like 67,000 mph right now in our orbit around the sun.


“I feel the need for acceleration!”

But my biggest pet peeve may be students doing algebra with numbers.

Fellow physics professors will know exactly what I’m talking about, but for the uninitiated, here’s an example:

If you drop an object from a height of 20 m, how long will it take to hit the ground?

A student knows that a kinematics equation is needed, hits upon the correct one, Δyvi Δt + (1/2) a Δt2, and then correctly identifies Δy = –20 m, a = –9.8 m/s2, and vi = 0.  So far, so good.  They’ve studied their physics, right?  What happens next is sheer madness:



Over and over again I tell students, “don’t plug numbers in until the end.”  But students love plugging in numbers.  They feel they’re actually getting closer to the answer if they’re manipulating numbers.  On some level, they still feel uncomfortable with letters—as if manipulating letters isn’t really “math”.

How does this problem look in my answer key?  Like this:

algebra 2

You can now plug in values if you like…and get Δt = √[2(-20)/-9.8] = 2.02 s.

Which of these approaches is more beautiful, more powerful?  The approach you pick indicates whether you “get” algebra or not.  If you do algebra with numbers, the answer you get is very narrow and very specific, even if you do it correctly.  That hypothetical student could have gotten 2 seconds as an answer, and I would have given them full credit.  But their answer would have been ugly.

The second approach is beautiful, because it is completely general and applicable to multiple situations.  I try to tell students “Look!  You found the time to fall a certain distance.  You now know the answer no matter what the height is, and even no matter what planet you’re on, since g doesn’t have to be 9.8 m/s2.”  This is usually followed by a blank open-mouthed stare, much like Kristen Stewart in a Twilight movie.

There is a more practical reason to avoid doing algebra with numbers.  It’s simply that when you do algebra with numbers, other people cannot follow your work as easily.  And then, if you make a mistake, it’s harder for someone else to spot.  Quick: what algebra error did the student make above?  It takes a while to find the mistake.

My ultimate point is that students need experience seeing the power of algebra.  It’s all well and good that algebra classes stress real-world applications—else, why teach algebra in the first place?  But real-world doesn’t only mean with numbersE=mc2 is certainly a real-world application of algebra, and it’s a lot more elegant than saying that 378,000,000,000,000 Joules is released when a teaspoon of sugar with mass 4.2 grams  is converted to pure energy, given that the speed of light is 300,000,000 m/s.  The hard part, for us physics professors, is to help this spoonful of algebra go down.

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Poor Einstein.  Is there anyone else who is misquoted more often?  Is there anyone else to whom more nonsense is attributed?

I have no desire to rehash things that Einstein said about “God”.  Einstein was by all accounts an atheist, an agnostic, or a pantheist—depending upon your definitions—and various religious apologists have been trying to co-opt the man for years by misquoting him.  Others have already discussed this at length.

My goal today is to tackle that old chestnut, “Imagination is more important than knowledge,” as seen on T-shirts, bumper stickers, and even on the packaging of the Albert Einstein action figure.  Did Einstein really say this, and if so, what did he mean?

Here’s the quote in context:

“At times I feel certain I am right while not knowing the reason.  When the [solar] eclipse of 1919 confirmed my intuition, I was not in the least surprised.  In fact I would have been astonished had it turned out otherwise.  Imagination is more important than knowledge.  For knowledge is limited, whereas imagination embraces the entire world, stimulating progress, giving birth to evolution. It is, strictly speaking, a real factor in scientific research.”  [From A. Einstein, Cosmic Religion: With Other Opinions and Aphorisms, p. 97 (1931).]

So Einstein did say this.  However, I maintain that the full quote in context has a different feel to it than the quote in isolation.

When I see “Imagination is more important than knowledge” on a bumper sticker, I think this: “Flights of fancy and imagination are more important than learning stuff.  So why should I study?  Einstein didn’t study.  He just sat around and daydreamed and came up with the most remarkable breakthroughs about the workings of our universe.  Imagination is more important than learning all the proofs and figures ranged in columns before me.  So I am going to follow good ol’ Einstein and daydream about being Batman.”

The New Age meaning of the quote is this: “I’d rather daydream than study.”  It’s Walt Whitman’s “learn’d astronomer” nonsense all over again.

In context, it’s clear that Einstein was talking about doing science.  Imagination is more important in making scientific breakthroughs than knowledge, but that doesn’t mean that knowledge is not important.  Einstein worked very, very hard to learn an awful lot of physics.  By all accounts, it took him almost 10 years to flesh out general relativity, during which time he had to acquire a lot of mathematical knowledge about Riemannian geometry and tensor analysis.  The “intuition” that Einstein developed during this time frame is what allowed him to be so confident of the results of Eddington’s expedition.  What Einstein calls “intuition” is just knowledge that has become so ingrained that you are no longer cognizant of it.

Einstein may have been more famous than most of his contemporaries, and it was probably due to his superior imagination.  But take Einstein’s imagination today and give it to a twenty-five year old high school dropout, and he’d be lost in obscurity, stocking shelves at Wal-Mart.  Imagination is more important than knowledge.  But only slightly more.

[Note: my book Why Is There Anything? is now available for download on the Kindle!]

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Einstein circa 1905

There are a lot of people who, to this day, deny the truth of Einstein’s special relativity (SR).  I’m not even referring to the OPERA superluminal neutrino debacle—an anomaly that was eventually found to be caused by a misconnected fiber optic cable.  No, I’m referring to laymen who deny SR because it goes against common sense.

A Google search will find such people readily.  Most of the time, their arguments aren’t even worth refutation, since it’s obvious in most cases that they haven’t mastered even the simplest algebra, much less sophomore-level physics.  (I am planning to use this gem in my Modern Physics class in the spring as a homework problem: for 20 pts., find the elementary flaw in this person’s logic.)  However, as a working physicist, I sometimes find myself dismissing such people too readily: it is easy, and self-gratifying, to call such people cranks.

A person who doubts SR is not necessarily a crank.  After all, relativity is very counter-intuitive, and our brains have been exquisitely fine-tuned by natural selection to perceive the world as inherently classical.  In fact I will go so far as to say that if you accept SR whole-cloth, without any mathematical or scientific background, then you’re basically showing a blind faith in science in the same way that Iotians have a blind faith in “The Book”.  I would rather beginning physics students showed some skepticism; it makes their final “conversion” that much more intellectually pleasing.

I think the main problem with perceptions of SR is the way it is normally presented.  My thesis is this: most physicists are teaching it wrong.  And as a consequence, many people who have studied SR come away with a misguided notion of what SR is all about.

The old way to teach SR begins with Einstein’s two postulates.  The first is that the laws of physics should be the same, in any inertial reference frame.  The second is that the speed of light is the same for all inertial observers.  There is then an obligatory picture of a train and lightning bolts, and talk about how simultaneity isn’t preserved in SR.  This leads (usually after a lengthy derivation) to time dilation and length contraction.  And then, out of the blue, there might be talk of the twin paradox and the obligatory pole vaulter in the barn.

Shudder.  Such a pedagogically confusing approach!  No wonder very few first-time SR students “get it” at all.

The original train picture from Einstein’s 1916 book

This approach has a long history.  In Relativity: The Special and General Theory (1916), by Einstein himself (!) the discussion begins with the two postulates, and there is then a diagram of a train and a discussion of simultaneity (see above).  Seriously?  I’m not blaming Einstein, mind you; I’m blaming the textbook authors today who can’t let go of that stupid train.  It’s been almost 96 years.  Get over it.  Hop off that train, please.  There are more intuitive approaches that are easier for the layman to grasp.

Here’s the approach I use in my classes.  This is not the only approach, of course, nor do I claim it is the best approach.  However in my experience (admittedly, just one data point) this approach is a better way to get students to gradually accept SR.  The trick is to present information one plausible chunk at a time, and then only gradually derive all the weird stuff.  Thus, without realizing it, the students have been convinced of the truth of SR despite themselves.  If you start with simultaneity and time dilation and length contraction then half of the students will get turned off immediately (because their common-sense alarms will be blaring full-force).

  1. Talk about classical (Galilean) relativity.  That is, discuss how the laws of physics should be the same in any (inertial) reference frame you choose.
  2. Talk about coordinate transforms: how you can take the spatial coordinates of an object (x,y,z) and find what the coordinates (x’,y’,z’) would be in a different coordinate system.
  3. Talk about how some coordinate transforms are “good” and some are “bad”.  For example, a translation in space such as x’ = x – L preserves distance, but a rescaling transform like x’ = ax does not.
  4. Mention how the good ol’ Pythagorean theorem s2 = x2 + y2 + z2 gives you an invariant quantity (s2) that is preserved under “good” transforms.
  5. Mention that the (experimental) behavior of light throws a monkey wrench into this analysis.  For whatever reason, all observers measure the same speed c for light, and this actually makes things a little harder.  (Don’t do any math at this point!)
  6. Here you should start talking about time as being a 4th dimension.  The earlier you introduce the idea of an event P as a point P=(x,y,z,ct) in space-time, the better.
  7. State Einstein’s postulate about the speed of light.
  8. Show that the light postulate implies that s2 = x2 + y2 + z2 is no longer an invariant quantity, when talking about transforms as applied to space-time.
  9. If s2 = x2 + y2 + z2 is no longer invariant, can we modify the formula in any way so as to make s2 invariant, while still preserving the light postulate?  The answer is yes; and so you should derive the 4D version of the distance formula, s2 = x2 + y2 + z2c2t2.

To me, this is the core idea of SR.  Everything else follows from the invariant interval s2.  One should no longer think of our existence as being 3D; time represents another “direction”.  And it turns out that the time you perceive depends upon your vantage point (time is “relative”), just like position.

For example, suppose you are looking at a row of trees.  From one location, the trees are lined up in front of you (they all share the same x-coordinate).  From another vantage point, they are separated by 1 m each (x=0, 1m, 2m, 3m, etc.)  No one, not even Galileo, would find this controversial.

But now imagine that you think of time as just another “direction”.  Why is it so hard to believe that your time coordinate could have one value in one reference frame, and another value in a different frame?  Why is it so hard to believe that events that are simultaneous in one frame are not simultaneous in another?

Time dilation and length contraction follow from this in a straightforward way.  And they are much easier to visualize if you buy into the paradigm (I’ll say it again) that time is another “direction”, and therefore relative just like position.

[Caveat: I do know that time is special, in the sense that there’s a minus sign in the s2 = x2 + y2 + z2c2t2 formula.  That minus sign is crucial.  But discussing its importance should be deferred to a later (pun-intended) time.]

If you’re interested, here’s the rest of my SR program:

10.  Discuss space-like, time-like, and light-like intervals, and the ideas of proper length and proper time.
11.   Show how the Galilean boost (which is a “good” transform in classical relativity) must be modified into the Lorentz boost in order to preserve s2.
12.   Show how the Lorentz boost implies length contraction and time dilation.
13.   Discuss relative velocity in SR.
14.   Discuss so-called paradoxes like the twin paradox and the pole vaulter paradox.

The discussion can then go into advanced topics: momentum, energy, E=mc2, forces, etc.  However, with the foundation I have described, I believe these topics are much easier to present.

I’m sure there are some great professors out there who have had great success with the “traditional” program of SR instruction.  I’m sure Feynman could teach circles around me, even with the train and lightning bolts.  But I prefer this different approach, as I have presented it, and I hope others will realize that there’s more than one way to explain special relativity.

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