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Posts Tagged ‘Physics’

Magnus Carlsen is the current world chess champion. He’s the best in the world at something. Not that many people can make that claim, can they?

Magnus Carlsen at FIDE World Chess Championship

Then again, there are lots of things in the world that you could be best at. Whistling, lemur training, lemon-pie-making, juggling, lying, rock climbing, sleepwalking. Somewhere in the world, there is “the best in the world” at each of these pursuits. Maybe my chances of being best at something are not so bad, after all? Maybe I just have to find the right thing…

Consider the modern pentathlon. In this sport, athletes compete in five events—fencing, shooting, swimming, running, and horse jumping—to achieve the overall best combined score. The winner need not be the best at any one specific event, but must have proficiency in all five.

Let’s say I am in the 99th percentile in all five events: very good, but not world class. [Here I am assuming that I’m in the 99th percentile of all humans, not just people who fence.] Taken individually, I wouldn’t have a prayer of making the Olympics. For example, the 99th percentile in épée fencing would still mean that there are

(0.01)^1 * 7,000,000,000 = 70,000,000

people with a similar proficiency around the world. Doesn’t seem that impressive, does yet? But I’m in the 99th percentile in all five events, right? So in reality there are only

(0.01)^5 * 7,000,000,000 = 0.7

people like me. That is, there’s just me. I’m probably the best at this combination of events. I should medal in the modern pentathlon.

And this brings me to my broader point. If you can think of five events in which you are in the 99th percentile individually, then in all likelihood you would be world champion if these events were combined into a single composite event. For those scoring at home, here’s where the number five comes from:

(0.01)^N * 7,000,000,000 = 1 (a single champion)

N ln (0.01) = ln [1/(7 x 10^9)]

N = [–ln (7 x 10^9)] / [ln (0.01)] = 4.9 ≈ 5

Let’s take my own skill set and see how I would do. I am certainly in the 99th percentile when it comes to physics. (Remember, I am comparing myself to the general population, not just physicists. I would never claim to be in the 99th percentile of people with physics PhD’s.) I am probably in the 99th percentile when it comes to chess (considering that I am in the 85th percentile for tournament players based on an 1800 rating). But am I good, really good, at anything else?

I will claim without proof that I am also in the 99th percentile (among the general population) in the following additional skills:

  • Knowledge of classical music
  • Playing the recorder
  • Geometry

Remember, I am not claiming any particularly high proficiency in any of these things. I just claim a 99th percentile rank in the general population. And individually, any one of these skills would only put me in the company of some 70 million others.

But now: make a hybrid event, where competitors have to take a battery of tests on physics, geometry, and classical music, then perform on the recorder, and then play chess… I believe I may do well in such an event. I might even be world champion.

Of course, nothing is that simple. I have ignored the fact that some of these skills may be correlated. Anyone who can play the recorder will probably also know about classical music. And many physicists will also be good at geometry. This means that my competition will be stiffer than I suppose, since if the events aren’t mutually exclusive then I’ve calculated the probabilities incorrectly. But I can improve my chances by making the five events as disparate as possible. I might change “Geometry” to “Movie Trivia”, for example.  My chances of becoming world champion would thereby be increased.

If you think that “99th percentile” is too high a bar, we could lower it to 90th percentile. Most people are in the top 10% at several things. Redoing our calculation, we get N = 9.8 in this case. So if you can find ten things you’re fairly good at and combine them, you too can be a world champion.

Of course, you also have to convince the Olympic governing body that that particular concatenation of events is worthy of a medal. But hey, that’s your problem.

I have some geometry to do.

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I was watching Dr. Who the other day and came across a physics mistake so common I thought I’d address it here.  The mistake is this:

Black holes suck you in like a vacuum cleaner!

The setup: in Dr. Who [2.8] “The Impossible Planet”, the good Doctor and Rose meet the crew of a ship who are on “an expedition [to] the mysterious planet Krop Tor, impossibly in orbit around a black hole.” [Wikipedia]  That phrase “impossibly in orbit” made me almost spit out my drink while watching the show.

Black holes have event horizons.  I get it.  Even light cannot escape.  I get that, too.  But why does that mean I cannot orbit a black hole?

OK, time for a little general relativity.  Einstein figured out, between 1905 and 1915, that gravity is “just” a warping of space-time.  Matter causes the space-time around it to curve; the curvature of space-time determines how matter moves (insofar as objects in the absence of gravitational forces follow geodesics).  The formulas that link the distribution of matter to the curvature of space are Einstein’s equations:

einstein_equation

This expression is compact and might seem relatively simple, but it’s not.  Gαβ and Tαβ are components of tensors, which are like vectors, but worse; they’re really 4×4 matrices.  So this equation is not one equation, but 16 different equations, since α and β can take on any of four values each.

What do all those letters stand for?  Gαβ is a component of the Einstein tensor, which tells you about how space-time is curved; the indices α and β can be any of four values in a 4D space-time.  (If you’re mathematically inclined, the Einstein tensor can be related to the Ricci scalar, the Ricci tensor, and the Riemann tensor.)  Tαβ is a component of the stress-energy tensor, which basically describes how matter/momentum/energy/stress/strain is distributed in a region of space-time.  So here’s another way to visualize Einstein’s equations:

einstein_explained

The cause (mass) is on the right; the effect (the curvature of space-time) is on the left.

So what does this have to do with black holes?

One of the first solutions discovered to the Einstein equations is called the Schwarzschild solution, which applies to a spherically symmetric gravitational source.  The solution gives you a “metric” (essentially, a geometry) that is almost the same as “flat” space-time, except for a pesky (1–2GM/c2r) term.  But that pesky term has a strange implication: when that term equals zero, the solution “blows up” (i.e. becomes infinite).  Space becomes so curved that you essentially have a hole in the fabric of space-time itself.

When does this happen?  It happens when R = 2GM/c2, as one line of algebra will show.  This is called the Schwarzschild radius.  The Einstein equations predict that something weird and horrifying happens when a mass is squeezed down to the size of its Schwarzschild radius.  Current understanding is that the mass would then keep going, and squeeze itself into a point of zero radius.  Literally, zero.  (I did say it was weird and horrifying).

Incidentally, the Schwarzschild radius is exactly the radius you’d get if you set the escape speed for an object equal to the speed of light.  So this means that not even light can escape this super-squeezed object.

And here’s where various misconceptions start to creep in.

Another name for the Schwarzschild radius is the event horizon.  It’s a boundary of no return:  if you cross it, you can never go back.  But that’s all it is: a boundary.  There is not necessarily anything physical at the event horizon.  You might never know that you had crossed it.  Remember, all the mass is at the center.

Here’s how I “picture” a black hole:

black hole

Now, if I am outside the event horizon, what would I see?  Well, nothing from inside the event horizon could reach me (hence the term “black”) but I might see Hawking radiation.  I would certainly see gravitational lensing: the bending of distant light around a black hole.  Here’s a cool picture of gravitational lensing in action (artists conception only!) from Wikipedia:

225px-Black_hole_lensing_web

Let’s say the Sun were a black hole.  Its event horizon would be around 3 km.  As long as we never got closer than 3km, we could do what we like.  We could fly in, fly out, orbit the black hole as we please.

Would the black hole “suck us in”?  Sure, in the same way that the Sun sucks us in already.  There is a strong pull of the Sun on the Earth.  And there would be a strong pull on our hypothetical spaceship.  But change the Sun to a black hole, and the pull would not get any stronger.  That is the key point that most people miss: black hole gravity is not somehow “stronger” than ordinary gravity.  There is just gravity; that’s it.  Change the Sun to a black hole, and the Earth would continue in its orbit, and nothing would be any different.  Except for, maybe, the lack of light.

Why was the planet Krop Tor’s orbit impossible?  Astronomical black holes (created by stellar collapse) have a lot of mass; when there’s a lot of mass hanging around, things tend to orbit them.  That’s what you’d expect.  It would only be impossible if somehow the orbit crossed the event horizon multiple times during its trajectory.  But of course, the show didn’t mention this.

I want to end my rant on GR with a suggestion: that there are two kinds of sci-fi: science fiction, and “sciency” fiction.  The first kind tries to get the science right, and makes an effort to be possible (if not plausible).  The second kind throws sciency words around in an effort to appeal to a certain demographic.  Basically, “sciency” fiction is fantasy, set in outer space.  When seen in this light, Dr. Who has more in common with Lord of the Rings than it does with 2001.

Don’t get me wrong: I love Lord of the Rings, and I love Dr. Who.  Just don’t call it science fiction.

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One of the most common criticisms of the many-worlds interpretation (MWI) of quantum mechanics is that it is absurdly complicated, and therefore violates Occam’s razor.  Most people’s first reaction, on hearing of MWI, think that the theory is (to quote Martin Gardner) “bizarre”, “monstrous”, “fantastic”, “radical”, “appalling”, “nonsense”, “frivolous”, and “low”.  And many people seem to think that theorists who ascribe to MWI have their heads in the clouds to believe such nonsense.  MWI seems to be taken, in fact, as evidence that physics has lost its way—as if (supposedly) blind belief in such frivolity is indicative of a philosophical rot that pervades theorists like me.

rotten-apple

Theoretical physics today, to some

There are so many refutations of such criticisms that I don’t know where to start.  First of all, although MWI is popular, it is by no means canon, and I daresay that a majority of physicists reject it still.  So there!  We’re not all sheep.  Still, MWI has become almost mainstream (especially with cosmologists) so maybe it’s the cosmologists and the ivory-tower theorists who should be singled out for criticism?

People who think this have probably never met a theoretical physicist before in their life.  Getting such people to agree is like herding cats; every theory one puts forth (in a journal article or in a conference talk) is debated, criticized—dare I say, attacked.  And that is as it should be.  There is not, contrary to popular belief, some holy scripture that every theorist quotes verbatim.  We are all different, and have basically come to interpret quantum mechanics in our own personal way…not at the behest of some lord on high.

How do I know this?  Because I was never taught about interpretations of quantum mechanics.  Ever.  Everything I know about such things, I learned on my own since graduation.  Thinking of taking a quantum mechanics class at your local university?  Guess what: they will probably not talk about MWI, or the Copenhagen interpretation, or Schrodinger’s f***ing cat.  Why not?  Because those are philosophy topics, not physics.  You can do quantum mechanics without ever interpreting a single thing.  There’s no crying in baseball, and there’s no philosophy in quantum mechanics.  It is a purely mathematical theory, that undeniably works, and most people just leave it at that.  The idea that thousands of physicists subscribe to one particular world-view just because they constitute a single monolithic conformist society is ludicrous.  Invite a physicist to lunch if you don’t believe me.

But I still haven’t addressed the idea that MWI is obviously absurd.  It is absurd, right?  I mean, come on!

But wait.  Let’s think back to the Copernican revolution.  It’s obvious that the Earth is stationary, no?  I bet people thought that Copernicus and Galileo and their ilk were bizarre, monstrous, fantastic, radical, appalling, nonsensical, frivolous, and low.

And what about the idea that there are billions and billions of galaxies, each with billions and billions of stars?  We forget now, but this idea was radical when first presented and wasn’t settled until the 1920’s.  Why are we OK with a multiplicity of stars, but not a multiplicity of universes?  Why aren’t people complaining about the absurd notion (fact) that there are more stars in the observable universe than there are grains of sand on Earth’s beaches?

PerseusCluster_041008_041214_2000

Whoa.

So, Occam’s razor.  MWI just seems to have too much baggage, right?  For a lot of people MWI is too high a cost to bear to have a mathematically simple interpretation of quantum mechanics.  And let’s be clear: MWI is a simpler theory than (say) the Copenhagen interpretation (CI).  For you can start with three postulates, and add a fourth about wave-function collapse, and you get CI.  Or you can start with just three, and say nothing of wave-function collapse, and you get MWI.  Which interpretation seems simpler now?  MWI is a consequence of accepting the three basic postulates of quantum mechanics.  If you don’t like that, then you must introduce a fourth postulate ex nihilo to make yourself feel better.

But wait! you say.  10100 universes doesn’t seem simpler.  It’s a huge number!  It’s ridiculous!

OK.  You wanna go there?  I’ll turn the argument around.  By that rationale, you probably believe that there are only a finite number of integers, because any finite number is simpler than infinity.  There.  That makes sense, right?

The truth is that an infinite set is often simpler than a single member of that set.  Take the natural numbers.  I can write a computer program in BASIC that writes every natural number.  Here it is:

10           x=1

20           PRINT x

30           x=x+1

40           GOTO 20

On the other hand, if I want to print out the number

5679200359662711389685023885761799

then my computer program is longer:

10           PRINT “5679200359662711389685023885761799”

Count the keystrokes.  The second program requires more typing.  And note that the first (simpler) program will eventually print this number—the long arbitrary program is “contained” within the first.

In information theory, the information “content” of something is related to its algorithmic complexity—roughly speaking, how easy it is to write a computer program that “specifies” the object.  By that measure, “all the natural numbers” is a simpler concept than the number

5679200359662711389685023885761799

Similarly, “all possible universes” is a much simpler concept than one specific arbitrary universe.  You want to recreate this universe?  Good luck…you’ll have to specify the position and momentum of every particle in the universe.  That’s a long computer program.  However, if you just say “create all possible universes” then eventually this one will pop up…

Do I believe in the MWI?  Yes.  Why?  It’s not because I was indoctrinated into such belief; I don’t think a single professor in graduate school ever mentioned MWI.  It’s because I’ve looked at the evidence over a number of years, and (tentatively) decided that it fits the data best.  That is the only reason to ever believe something, ever.  It fits the data best.  But I stress that my conclusion is tentative because, hey, it’s science.  There is no dogma.  There is just stuff that we are 99.44% sure of.

Like evolution by natural selection, or heliocentrism, or the existence of ghosts.  I mean, seeing’s believing, right?

               [Note: more Americans believe in ghosts than evolution.  Sigh.]

If you enjoyed this post, you may also enjoy my book Why Is There Anything? which is available for the Kindle on Amazon.com.

sargasso

I am also currently collaborating on a multi-volume novel of speculative hard science fiction and futuristic deep-space horror called Sargasso Nova.  Publication of the first installment will be January 2015; further details will be released on Facebook, Twitter, or via email: SargassoNova (at) gmail.com.

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I thought I would re-post this excellent discussion of the many-worlds interpretation by David Yerle:

Why I Believe in the Many-Worlds Interpretation

I agree with him 100%, and he says it better than I ever could.  The crux of the argument is this: it depends on the book you’re reading, but as a practical matter there are typically 4 postulates of quantum mechanics (about the primacy of the wavefunction, Schrödinger’s equation, measurements being Hermitian operators, and wave function collapse).  Many worlds is what you get when you reject the unmotivated “wave function collapse” postulate.  It is a simpler theory in terms of axioms, so obeys Occam’s razor.  If multiple universes bother you, think of how much it bothered people in the 1600’s to contemplate multiple suns (much less multiple galaxies!)

Wvf_Anti-Bond

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In my continuing effort to present cutting-edge research, I present here my findings on the 9 kinds of physics undergrad.

First, let’s look at a scatter plot of Ability vs. Effort for a little more than 100 students.  (This data was taken over a span of five years at a major university which will remain unnamed.  Even though it’s Wake Forest University.)

HR diagram

Student ability is normalized so that 1 is equivalent to the 100th percentile, 0 is the 50th percentile, and –1 is the 0th percentile.  [This matches the work of I. Emlion and A. Prilfül, 2007]  Ability scores below –0.5 are not shown (such students more properly belong on the Business Major H-R diagram).

On the x-axis is student effort, given as a spectral effort class [this follows B. Ess, 2010]:

O-class: Obscene

B-class: Beyond awful

A-class: Awful

F-class: Faulty

G-class: Good

K-class: Killer

M-class: Maximal

As you can see, most students fall onto the Main Sequence.

HR typical

The Typical student (effort class G, 50th percentile) has a good amount of effort, and is about average in ability.  They will graduate with a physics degree and eventually end up in sales or marketing with a tech firm somewhere in California.

HR giant

The Giant student (effort class K, 75th percentile) has a killer amount of effort and is above average in ability.  Expect them to switch to engineering for graduate school.

HR smug

The Smug Know-it-all student (effort class O, 100th percentile) is of genius-level intellect but puts forth an obscenely small amount of effort.  They will either win the Nobel prize or end up homeless in Corpus Christi.

HR grad

The Headed to grad school student (effort class B, 75th percentile) is beyond awful when it comes to work, and spends most of his/her time playing MMORPG’s.  However, they score well on GRE’s and typically go to physics graduate schools, where to survive they will travel to the right (off the main sequence).

HR industry

The Headed to industry student (effort class F, 55th percentile) is slightly above average but has a faulty work ethic.  This will change once they start putting in 60-hour weeks at that job in Durham, NC.

HR phobe

The Hard working math-phobe student (effort class M, 30th percentile) is earnest in their desire to do well in physics.  However, their math skills are sub-par.  For example, they say “derivatize” instead of “take the derivative”.  Destination: a local school board near you.

HR super

The Supergiant student (effort class K, 100th percentile) is only rumored to exist.  I think she now teaches at MIT.

HR frat

The Frat boy student (effort class O, 50th percentile) is about average, but skips almost every class.  Their half-life as a physics student is less than one semester.  They will eventually make three times the salary that you do.

HR dwarf

The White dwarf student (effort class B, 30th percentile) is below average in ability and beyond awful when it comes to putting forth even a modicum of effort.  Why they don’t switch to being another major is anyone’s guess.

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If you enjoyed this post, you may also enjoy my book Why Is There Anything? which is available for the Kindle on Amazon.com.  The book is weighty and philosophical, but my sense of humor is still there!

***********************************************************************************

sargasso

I am also currently collaborating on a multi-volume novel of speculative hard science fiction and futuristic deep-space horror called Sargasso Nova.  My partner in this project is Craig Varian – an incredibly talented visual artist (panthan.com) and musician whose dark ambient / experimental musical project 400 Lonely Things released Tonight of the Living Dead to modest critical acclaim a few years back.  Publication of the first installment will be January 2015; further details will be released on our Facebook page, Twitter feed, or via email: SargassoNova (at) gmail.com.

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As a public service, I hereby present my findings on physics seminars in convenient graph form.  In each case, you will see the Understanding of an Audience Member (assumed to be a run-of-the-mill PhD physicist) graphed as a function of Time Elapsed during the seminar.  All talks are normalized to be of length 1 hour, although this might not be the case in reality.

Typical

The “Typical” starts innocently enough: there are a few slides introducing the topic, and the speaker will talk clearly and generally about a field of physics you’re not really familiar with.  Somewhere around the 15 minute mark, though, the wheels will come off the bus.  Without you realizing it, the speaker will have crossed an invisible threshold and you will lose the thread entirely.  Your understanding by the end of the talk will rarely ever recover past 10%.

Ideal

The “Ideal” is what physicists strive for in a seminar talk.  You have to start off easy, and only gradually ramp up the difficulty level.  Never let any PhD in the audience fall below 50%.  You do want their understanding to fall below 100%, though, since that makes you look smarter and justifies the work you’ve done.  It’s always good to end with a few easy slides, bringing the audience up to 80%, say, since this tricks the audience into thinking they’ve learned something.

Theorist

The “Unprepared Theorist” is a talk to avoid if you can.  The theorist starts on slide 1 with a mass of jumbled equations, and the audience never climbs over 10% the entire time.  There may very well be another theorist who understands the whole talk, but interestingly their understanding never climbs above 10% either because they’re not paying attention to the speaker’s mumbling.

Experimentalist

The “Unprepared Experimentalist” is only superficially better.  Baseline understanding is often a little higher (because it’s experimental physics) but still rarely exceeds 25%.  Also, the standard deviation is much higher, and so (unlike the theorist) the experimentalist will quite often take you into 0% territory.  The flip side is that there is often a slide or two that make perfect sense, such as “Here’s a picture of our laboratory facilities in Tennessee.”

Undergrad

You have to root for undergraduates who are willing to give a seminar in front of the faculty and grad student sharks.  That’s why the “Well-meaning Undergrad” isn’t a bad talk to attend.  Because the material is so easy, a PhD physicist in the audience will stay near 100% for most of the talk.  However, there is most always a 10-20 minute stretch in the middle somewhere when the poor undergrad is in over his/her head.  For example, their adviser may have told them to “briefly discuss renormalization group theory as it applies to your project” and gosh darn it, they try.  This is a typical case of what Gary Larson referred to as “physics floundering”.  In any case, if they’re a good student (and they usually are) they will press on and regain the thread before the end.

Guest

The “Guest From Another Department” is an unusual talk.  Let’s say a mathematician from one building over decides to talk to the physics department about manifold theory.  Invariably, an audience member will gradually lose understanding and, before reaching 0%, will start to daydream or doodle.  Technically, the understanding variable U has entered the complex plane.  Most of the time, the imaginary part of U goes back to zero right before the end and the guest speaker ends on a high note.

Nobel

The “Nobel Prize Winner” is a talk to attend only for name-dropping purposes.  For example, you might want to be able to say (as I do) that “I saw Hans Bethe give a talk a year before he died.”  The talk itself is mostly forgettable; it starts off well but approaches 0% almost linearly.  By the end you’ll wonder why you didn’t just go to the Aquarium instead.

Poetry

The “Poetry” physics seminar is a rare beast.  Only Feynman is known to have given such talks regularly.  The talks starts off confusingly, and you may only understand 10% of what is being said, but gradually the light will come on in your head and you’ll “get it” more and more.  By the end, you’ll understand everything, and you’ll get the sense that the speaker has solved a difficult Sudoku problem before your eyes.  Good poetry often works this way; hence the name.

Politician

The less said about “The Politician”, the better.  The hallmark of such a talk is that the relationship between understanding and time isn’t even a function.  After the talk, no one will even agree about what the talk was about, or how good the talk was.  Administrators specialize in this.

If you enjoyed this post, you may also enjoy my book Why Is There Anything? which is available for the Kindle on Amazon.com.  The book is weighty and philosophical, but my sense of humor is still there!

sargasso

I am also currently collaborating on a multi-volume novel of speculative hard science fiction and futuristic deep-space horror called Sargasso Nova.  My partner in this project is Craig Varian – an incredibly talented visual artist (panthan.com) and musician whose dark ambient / experimental musical project 400 Lonely Things released Tonight of the Living Dead to modest critical acclaim a few years back.  Publication of the first installment will be January 2015; further details will be released on our Facebook page, Twitter feed, or via email: SargassoNova (at) gmail.com.

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Formula snobs

I am a formula snob.

We all know about grammar snobs: the ones who complain bitterly about people using who instead of whom.  Many people know how to use whom correctly; only grammar snobs care about it.  I gave up the whom fight long ago (let’s just let whom die) but I am a grammar snob when it comes to certain words.  For example, ‘til is not a word, as I have discussed before.

However, I am almost always a formula snob.

Consider this formula from the text I’m currently using in freshman physics:

x = v0 t + ½ a t2.

Beetlejuice-square

Robin Thicke, c. 2012

To me, looking at this equation is like watching Miley Cyrus twerk with Beetlejuice.  I would much, much rather the equation looked like this:

Δx = v0 Δt + ½ a Δt2.

The difference between these two formulas is profound.  To understand the difference, we need to talk about positions, clock readings, and intervals.

A position is just a number associated with some “distance” reference point.  We use the variable x to denote positions.  For example, I can place a meter stick in front of me, and an ant crawling in front of the meter stick can be at the position x=5 cm, x=17 cm, and so on.

A clock reading is just a number associated with some “time” reference point.  We use the variable t to denote clock readings.  For example, I can start my stopwatch, and events can happen at clock readings t=0 s, t=15 s, and so on.

Here’s the thing: physics doesn’t care about positions and clock readings.  Positions and clock readings are, basically, arbitrary.  A football run from the 10 yard line to the 15 yard line is a 5 yard run; going from the 25 to the 30 is also a 5 yard run.  The physics is the same…I’ve just shifted the coordinate axes.  If I watch a movie from 8pm to 10pm (say, a Matt Damon movie) then I’ve used up 2 hours; the same thing goes for a movie from 9:30pm to 11:30pm.  Because a position x and a clock reading t ultimately depend on a choice for coordinate axes, the actual values of x and t are of little (physical) importance.

Suppose someone asks me how far I can throw a football.  My reply is “I threw a football and it landed on the 40 yard line!”  That’s obviously not very helpful.  A single x value is about as useful as Kim Kardashian at a barn raising.

nebraska_amish_barn_raising

Can you pass that hammer, Kim?

Or suppose someone asks, “How long was that movie?” and my response is “it started at 8pm.”  Again, this doesn’t say much.  Physics, like honey badger, doesn’t care about clock readings.

Most physical problems require two positions, or two clock readings, to say anything useful about the world.  This is where the concept of interval comes in.  Let’s suppose we have a variable Ω.  This variable can stand for anything: space, time, energy, momentum, or the ratio of the number of bad Keanu Reeves movies to the number of good (in this last case, Ω is precisely 18.)  We define an interval this way:

ΔΩ = Ωf – Ωi

So defined, ΔΩ represents the change in quantity Ω.  It is the difference between two numbers.  So Δx = xf – xi is the displacement (how far an object has traveled) and Δt = tf – ti  is the duration (how long something takes to happen).

honey badger

Honey badger doesn’t care.

When evaluating how good a football rush was, you need to know where the player started and where he stopped.  You need two positions.  You need Δx.  Similarly, to evaluate how long a movie is, you need the starting and the stopping times.  You need two clock readings.  You need Δt.

I’ll say it again: most kinematics problems are concerned with Δx and Δt, not x and t.  So it’s natural for a physicist to prefer formulas in terms of intervals (Δx = v0 Δt + ½ a Δt2) instead of positions/clock readings (x = v0 t + ½ a t2).

But, you may ask, is the latter formula wrong?

Technically, no.  But the author of the textbook has made a choice of coordinate systems without telling the reader.  To see this, consider my (preferred) formula again:

Δx = v0 Δt + ½ a Δt2.

The formula says, in English, that if you want to calculate how far something travels Δx, you need to know the object’s initial speed v0, its acceleration a, and the duration of its travel Δt.

From the definition of an interval, this can be rewritten as

xf  – xi = v0 (t– ti) + ½ a (t– ti) 2.

This formula explicitly shows that two positions and two clock readings are required.

At this point, you can simplify the formula if you make two arbitrary choices: let xi = 0, and let ti = 0.  Then, of course, you get the (horrid) expression

x = v0 t + ½ a t2.

I find this horrid because (1) it hides the fact that a particular choice of coordinate system was made; (2) it over-emphasizes the importance of positions/clock readings and undervalues intervals, and (3) it ignores common sense.  Not every run in football starts at the end-zone (i.e. x = 0).  Not every movie starts at noon (i.e. t = 0).  The world is messier than that, and we should strive to have formulas that are as general as possible.  My formula is always true (as long as a is constant).  The horrid formula is only true some of the time.  That is enough of a reason, in my mind, to be a formula snob.

critique-film-the-informant-l-infiltre-matt-damon-steven-soderbergh-texte

A formula snob?

Bonus exercise: show that the product

ΩKeanu Reeves  x  ΩMatt Damon  ≈  3.0

has stayed roughly constant for the past 15 years.

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