Effective today, this site has moved! The good folks at ScienceBlogs asked me if I’d be interested in writing at their site. I was, so I am. Don’t worry, everything will stay exactly the same with the sole exception of the new address. The full story is at the new location. Should you happen to have this page (or its RSS feed) bookmarked, the new location is where all the new stuff will be. All the old stuff will still be here, until eventually the archives get moved. At that point this page will become a redirect to the new one.

The new link: http://scienceblogs.com/builtonfacts

See you there!

**Tags:** Miscellaneous

After writing yesterday’s post I happened across a news article about the Aptera moving closer to production. What’s the Aptera? It’s a car, but not just any car:

The Aptera Typ-1 will be the most efficient passenger vehicle in the world. The first production models are planned to be available in December 2008 with the production rate increasing throughout 2009. With a coefficient of drag literally one-third of a subcompact car and less than half the weight, the all-electric version will get up to 120 miles per charge, while the hybrid version, which will follow in about 12 months, will achieve close to 300 MPG.

So it’s a press release and thus you might wonder just how much of this is hot air. Not much, as far as I can tell. It has a large pool in investors including Google, and the company is taking pre-orders already.

This particular type of vehicle is a little wonky looking, but manages to retain a little sportiness nonetheless. More important than that is the engine. It’s an electric vehicle with a small gasoline engine to fill in when the batteries get low. This overcomes one of the biggest objections to electric cars, which is that they’re fine for driving around town but will leave you stranded if you ever have to drive across the state. The full hybrid Aptera is supposed to have a range of 600+ miles with an average efficiency of a staggering 130 MPG. For $20,000 it’s a pretty tempting deal.

With the Tesla Roadster already in production and the major manufacturers designing their own electric vehicles, the move to actual no-kidding energy independence might finally be taking its first steps.

**Tags:** Physics News

In daily life, there’s pretty much three kinds of frictional forces that you’re likely to encounter if you classify them by how much they are affected by speed. We’ll not worry about the direction of those forces since for our purposes it’s good enough to say that friction acts in the direction opposite the motion. So we can drop the vector notation.

The first one is those frictional forces not affected by speed, like sliding friction. As a good first approximation, this is given by

The frictional force is just equal to some constant depending of what material are producing the friction times the force perpendicular to the surface which your object is sliding across. Heavier objects mean more frictional force. It’s easier to slide a book across the floor than it is to slide a piano across the floor. But it really doesn’t matter how fast you do the pushing, the force per distance stays the same.

The second is viscous resistance. Dip a smooth stick into the still waters of a pond and very slowly and smoothly move it around. You’ll feel a force resisting the motion which is proportional to the speed of the stick.

Also very easy. You have some constant b which depends on the size and shape of the object and the properties of the liquid. You have the velocity. Multiply them together and you get the force.

Now why do you have to move the stick very slowly and smoothly? Because if you start moving it too quickly you’ll get turbulence. What’s turbulence? Basically, it’s lack of smoothness in the fluid flow. Like this picture, from Wikipedia:

Once you get turbulent flow, the frictional force starts to be proportional to the *square* of your velocity.

The constants are unimportant, the v^{2} is the issue. Go twice as fast, and you have to put in four times the work. While aerodynamic drag is definitely not the only thing sapping a car’s forward progress, at highway speeds it’s a very significant effect. A car traveling at a slightly over the limit speed of 75 mph will experience about 1.15 times the drag as a car traveling at the speed limit of 70 mph. While 60 mph is hair-tearingly slow compared to 75 mph, it reduces drag by better than a factor of 50%. Now you definitely shouldn’t drive unsafely slow, but even cutting your speed to 70 from whatever slightly-over speed you usually prefer can save you considerable gas at the cost of only a tiny bit of extra travel time.

**Tags:** Physical Concepts

Tom Hudson was a retired small-town engineer, an avid remote control aircraft hobbyist, and a good friend. When I was a young person living in Slidell between 1989 and 2001, he and his wife Lucy were my family’s next door neighbor. Our families became very close, and in a way they were like another set of grandparents. In fact, my own grandparents became good friends with them during their own visits.

He was a quiet and soft spoken man, skilled at building and tinkering. He’s one of the reasons I enjoy doing the same kinds of things with little motors and electronics, and when he went out to the hobby airfield to fly his planes sometimes my family would go as well to watch him fly, and even take the controls ourselves sometimes. Some other people built speedy dogfighting planes that zipped around aerobatically, but he was a patient man and preferred to fly his elegant planes in long, graceful arcs. He was an inspiration.

Monday he was killed when his car was struck from behind on Interstate 55 by a hit-and-run driver. He was 72. The other driver turned out to be both drunk and possibly on illegal drugs. His wife Lucy was in the car with him, and survived with moderate injuries. It’s hard to describe how infuriating this is. That a person can blithely saturate their brain in chemicals and go charging down the highway is just… He was caught and will go to prison, I imagine, but at best that will keep him from killing someone else. It’s a cold comfort. I don’t really care what you do to entertain yourself chemically, but please don’t impair yourself and then do anything dangerous. It’s not worth it, to risk ending your own life or that of someone else.

His family and friends will carry on his legacy in a much better way than this poor tribute on a personal website, but I would be remiss if I didn’t at least try to commemorate his life in some way. Rest in peace.

**Tags:** Miscellaneous

Have you seen The Dark Knight yet? No? What’s wrong with you, go add to its box-office gross now. It deserves it. No, I’m not kidding. Also, stop reading since there’s about to be some minor spoilers.

This post was inspired by my mom, who wants to know how plausible the various two major cell phone schemes are. Let’s see.

1. Lucius Fox develops a very snazzy phone which can emit sound pulses. The reflections of those pulses are received by the phone and reconstructed into a sonar image of whatever room the phone happens to be in. The idea is not inherently absurd. Submarines use it to find other subs and ultrasound machines use it to examine unborn children, to take just two of the most familiar examples. Could this work to characterize a room? Sure, to an extent. But as with any wave, there’s tradeoffs. Low frequencies can’t see detail, but high frequencies can’t see around corners. If you approach a high school band from a distance, you’ll hear the drums from the greatest distance not because they are louder but because they are lower frequency and can more easily bend around obstacles. Thus without a direct and unobstructed path to whatever surface it’s supposed to measure (as in fetal ultrasound), the phone is going to be hard pressed to see anything at all.

To make matters worse, air is not a continuous fluid. For a lot of purposes it can be treated as one, but as frequency becomes very high the wavelength begins to become comparable to the intermolecular spacing. This tends to cause severe attenuation at high frequencies, limiting the range badly.

Don’t even get me started on the signal processing problems. Real-time inverse Fourier transforms of gigabits per second per phone? Not happening. And how in the world is the phone supposed to pick up direction and phase information in the incoming wave with adequate resolution in the first place?

It’s just not happening. The concept is not absurd, but it is impossible in practice.

2. Batman arranges for his own program code to be installed in nearly every cell phone in Gotham City. On activation, it turns every phone into both a continuous bug, listening in on its surroundings even with the phone turned off. He hooks the millions of signals into a computer and scans for the Joker’s voice in order to find his location.

Now this one is not merely plausible, it’s a reality. Law enforcement has already used cell phones as bugs and tracking devices. I’m not sure if the processing power of the installed phone company hardware is enough to simultaneously monitor every phone to look for a specific voice, but it’s certainly not beyond the realm of possibility. Anybody with Bruce Wayne’s wealth could certainly afford it. Lucius Fox believed this is much too great of a power for Batman to have since it invades the privacy of the good people of Gotham; Batman agreed and told Fox to destroy the hardware once the Joker was found. I think it’s unlikely the FBI would feel the same way.

So if you’re doing something (criminal or not) that requires privacy, don’t bring your cell phone. Or at the very least remove its battery.

**Tags:** Physical Concepts

Picture a railroad spike, held vertically as if to be pounded into a railroad tie. Now picture a 14 pound bowling ball poised 35 feet above the spike, and then let drop straight down to hammer the spike down into the ground.

The energy m*g*h works out to about 600 joules, a typical energy for the .40 S&W bullet. On this picture it’s the third from the right. It’s also the caliber of the pistol I own.

*Image courtesy Wikipedia*

Now the damage the spike would do and the damage the bullet would do are two different things, but the comparison is not a terrible one. The energy of the bullet will be transfered to the body by exerting a force through a distance. While the speed and acceleration will be different, energy is the main quantity of interest and so the bowling ball provides a rather dramatic comparison as to just exactly how much you don’t want to be hit by a shot. The .40 S&W is probably the second most popular pistol cartridge after the 9mm round, which is the second from the right. Typical muzzle energies for the 9mm are around 480 J, or that bowling ball being dropped on the spike from 27 feet.

And you can go the other way up through the calibers to the .44 Magnum of Dirty Harry fame (second on left). Energy ranges widely for this one depending on the actual weight of the bullet, but 2000 J is perfectly plausible. That’s a 116 foot drop for the bowling ball onto the spike. Make my day indeed. It’s not the most powerful commercial pistol cartridge out there, but it’s the most powerful you’re likely to see in a gun for anything other than explicitly firing ridiculous specialty calibers.

Rifles? They’re several steps up, and while there’s many, many calibers out there I’ll only mention two. The .30-06 of WWII and hunting rifle fame is usually in the vicinity of 4000 J, or 230 feet worth of bowling ball drop. This is considerably more powerful than the ammunition currently used by US soldiers in their M16 rifles, as modern tactics generally prefer volume of fire over the strength of one particular powerful but bulky shot.

If you want a truly preposterous round, there’s the .50 BMG. Legal in most states (And why not? It won’t kill you any deader than a .30-06, each shot is about $4, and the guns that fire it are absurdly bulky.), it has a muzzle energy of an astonishing 13,000 J, or about 750 feet worth of bowling ball drop. But if you’re in an enemy APC it could ruin your whole day. Here’s it’s picture. It’s the big one. An M16 fires the second from the right.

*Also from Wikipedia*

The lesson to take away from all this? A Chevy Suburban at interstate speeds has something like a hundred times the kinetic energy of the .50 BMG. Like a car or truck, a bullet is a tool that is dangerous if not treated with respect. When you use them, be safe!

**Tags:** Physical Concepts

This weekend we’ll add a little statistics to our ongoing Sunday tour of the zoo of mathematical functions!

This is the Poisson distribution for λ = 4. To be formally correct, it’s only defined for integer values. This pops up in physics all the time in the context of counting discrete events with independent probabilities. For instance, if you have a sample of a long-lived radioactive isotope and you see on average 4 decays per minute, during any given minute the probability of seeing k decays is given by the formula above.

**Tags:** Miscellaneous

Let there be pi?

Mark Chu-Carroll at Good Math, Bad Math has an interesting post taking down a guy who thinks messages from God are encoded into pi.

Stare at any number, and set of numbers, or any numeric coding of a text. If you try hard enough and long enough, then you *will* find some interesting patterns. Looked at probabilistically, the chances of finding a large sequence of numbers or letters where we *can’t* find any pattern is vanishingly small. So given a pattern, we need to ask, is this pattern just the result of randomness? Just because you found an apparent pattern doesn’t mean that it’s deliberate or meaningful. In fact, it probably isn’t.

Very true, and I don’t think I can improve on his analysis. You should read the whole thing, it’s quite fascinating.

There are two points I’d like to add, one mathematical and one theological. For the mathematical point, let’s take as our jumping-off point this, from Mark’s post

Pi isn’t random. No one who understands what “random” means would say that. In fact, pi is very much *not* random. It’s a highly compressible number: there’s a simple algorithm for computing it, which means that *by definition*, it’s not random.

Absolutely correct. However, there’s an interesting property of numbers like pi with regard to the decimal digits of the number. As you read through the trillions of digits of pi that have been computed, you’ll notice that while the numbers certainly aren’t random (they’re exactly the digits of pi!), in a way they behave as though they were random. If you throw a dart at a printout of the digits of pi, the odds that you’ll hit a 3 are about 1/10. The odds that you’ll hit your 7-digit phone number are about 1/10^{7}. In other words, though the string of digits is not random, the digits themselves are statistically distributed in the same kinds of ways actually random digits would be distributed. If you go far enough, eventually you’ll find any given sequence of digits. A number whose decimal digits exhibit this property is called a* normal number*.

Well, those are the properties of normal numbers and pi gives every indication that it is in fact a normal number. But this hasn’t actually been proved yet. Maybe there’s some subtle statistical irregularity that has not yet been detected in pi. For instance, maybe you won’t ever get Shakespeare’s Hamlet in binary no matter how far you go in pi. Nonetheless, mathematicians have very good reasons to believe pi, e, and most other numbers (in a technical sense) are normal. It’s not a certainty, but it’s pretty darn likely.

So that’s kind of neat. A string of digits that looks random might actually be something completely nonrandom, like the digits of pi. And correspondingly, a very orderly string of digits like those of pi might in some ways seem as though it were random. Numbers are cool like that.

Now the theological point, which Mark also mentions. Omnipotence means the ability to do anything. Unfortunately language uses the same words to describe doable things as it does to generate meaningless syntactical statements. The classic example is “Can God create a rock so big he can’t lift it?” The problem is that while the question looks like sensible English, it isn’t. If one of those things exists, the other can’t. This is no ill reflection on God’s power, it’s an acknowledgment that some sentences just don’t mean anything in the first place. Mathematics works the same way. “What if pi were something else?” is a nonsense sentence, a low-rent Zen koan. The idea of God encoding messages in mathematically defined numbers simply is not coherent. If something were encoded in those numbers, they wouldn’t be those numbers anymore.

We’re thus pretty much assured that the constants of mathematics are message-free, movies and internet speculation notwithstanding.

**Tags:** Looking Beyond

Disclaimer for casual readers: I write posts which vary wildly in technical difficulty, this one is a little more mathematical than most. Don’t let it scare you off! Even if you’re a little lost, it’s good to have seen it.

The various worked problems I’ve been doing recently have mostly been on the intro undergraduate level. This is mainly because we’re between semesters, and I don’t really have the time to write up long and difficult problems from scratch. Once the fall semester starts up I’ll start writing up solutions to my homework problems.

To assuage my guilt over mainly doing easy problems, how about at least something that’s undergraduate physics major difficulty?

Consider a particle in the ground state of a 1-d infinite square well with sides at positions 0 and L. Suddenly the right-hand wall is moved outward so the walls are now at new positions 0 and 2L. What is the probability of observing the particle in the ground state of this new potential?

The main thing to understand with this problem is the idea that for any potential, there’s only certain quantum states that are allowed. If a particle is in an allowed state of one potential and suddenly the potential changes to something else, the particle’s new state *must* be expressible in terms of the new allowed states. But a wavefunction can’t just snap instantly from being one thing to being another, so there’s a problem.

Or there would be a problem if there weren’t the superposition principle. Just like you can expand a function in terms of the polynomials when you use Taylor series or in terms of trig functions when you use Fourier series, you can expand the old state in terms of the new allowed states no matter how different the new states are. It seems like magic, but it’s mathematically sound. The allowed eigenfunctions for a given potential form a complete set, so you can expand just about any square-integrable function in terms of the allowed functions over the appropriate interval.

Shall we try it here? The old ground state wavefunction was

The new allowed functions are

So the old ground state has to be expressible in terms of those new allowed functions. In other words,

With the constants C_{n} to be determined. How can we find those constants? Skipping the derivation (which is in any intro quantum textbook), we use the fact that

Why L for the upper limit of integration? Remember that the original ground state wavefunction is 0 outside of the box. If we forget that and instead calculate as though the sine function keeps going, we’ll get a very wrong answer. The star represents complex conjugation which doesn’t matter here, but it’s good to be accurate.

So plug in the functions with n = 1 (since we’re going from one ground state to another) and integrate. I get

Almost done. Remember it’s not the wavefunction which represents probability, it’s the square of the absolute value of the wavefunction. So the final answer is

It’s actually somewhat more probable that the particle will end up in the n = 2 state. The probability for that happens to be 0.5 exactly. The n = 3 state has a probability of about .1296, the n = 4 state has probability of about 0.00735, and for higher states the probability continues to fall like a stone. Of course as a check you can add up these values to find that it’s very nearly equal to 1, as expected.

Clear as crystal? Probably not. It took me a while to understand this stuff. But don’t worry, there’s tons and tons of good practice problems out there and we’ll end up doing a lot of them.

**Tags:** Undergraduate Physics Major · Worked Problems

I love teaching, but I have to say I am not a fan of teaching the summer session. Everything is way too disorganized. Oh well, at least it’s only for another three weeks or so.

The way undergraduate intro physics is taught is usually in two halves, and right now I’m teaching the second. The first course is an introduction to classical mechanics: Newton’s laws, gravity, friction, momentum, energy, the usual. The second course is an introduction to electricity and magnetism. The second course does build on the first to some extent; you can’t very well find the acceleration of a particle in an electric field without knowing how force and acceleration work in the first place.

But the connection between the first and second courses is not always very obvious. To take a tiny example, problems with charged particles used as projectiles almost always ignore gravity. It’s not always clear to a beginner that this is valid. Sure the book says that the electron is going so fast that gravity doesn’t have time to do much in the context of the problem, but it’s hard to have an intuitive understanding without some concrete examples. How about we try one?

Find the electric potential required to accelerate an electron to escape velocity.

The answer to that question should give us some idea of what potential serves as kind of a lower limit for non-negligibility in comparison to gravity. We could find or look up the escape velocity directly, but I prefer to keep everything in terms of energy for conceptual benefit. The potential energy of an object in a gravitational field is the integral of the force with respect to distance, taken from the surface of the earth to infinity.

Where R and M are the earth’s radius and mass, and little m is the mass of the electron. I’ve played a little fast and loose with some negative signs, but it doesn’t affect anything. Now the potential energy a charge gains through a potential difference V is just qV. So set the two potential energies equal:

And solve that for V.

Plugging the mass and radius of the earth, along with the mass and charge of the electron and I get

A third of a millivolt. Tiny. An AA battery is 1.5 volts, thousands of times stronger than that. An electron accelerated through just about any realistic potential is going to be traveling at speeds which render gravity irrelevant for most practical purposes. An electron accelerated through the 1000 V or so of an old TV tube will experience “bullet drop” in the nanometer regime, compared to an actual sniper rifle drop of several inches over a typical 300 yard shooting distance.

Incidentally, are these speeds fast enough to worry about the effects of relativity? The answer is “not really”. As a general rule of thumb, relativistic speeds become important once the kinetic energy starts to become comparable to the mass of the particle. For the case of an electron this is 511 keV, and so even a few thousand volts is small enough to treat classically except in cases requiring very high precision.

**Tags:** College Physics 101 · Physical Concepts · Worked Problems