Chutes And Landers

From: Robin Feder <>
To: Sy Moire <>
Subj: Questions

Hello again, Mr Moire. Kalif and I have a question. We were talking about falling out of stuff and we wondered how high you have to fall out of to break every bone in your body. We asked our science teacher Mr Higgs and he said it was something that you or Randall Munroe could answer and besides he (Mr Higgs) had to get ready for his next online. Can you tell us? Sincerely, Robin Feder

From: Sy Moire <>
To: Robin Feder <>
Subj: Re: Questions

Hello again, Robin. You do take after your Dad, don’t you? Please give my best to him and to Mr Higgs, who has a massive job. Mr Munroe may already have answered your question somewhere, but I’ll give it a shot.

You’ve assumed that the higher the fall, the harder the hit and the more bones broken. It’s not that simple. Suppose, for instance, that your fall is onto the Moon, whose gravity is 1/6 that of Earth. For any amount of impact, however high the fall would have been on Earth, it’d be six times higher on the Moon. So the answer depends where you’re falling.

But the Moon doesn’t have an atmosphere worth paying attention to. That’s important because atmospheres impose a speed limit, technically known as terminal velocity, that depends on a whole collection of things

  • the Mass of the falling object
  • the local strength of Gravity
  • the Density of the atmosphere
  • the object’s cross‑sectional Area in the direction of fall

The first two produce the downward pull of gravity, the others produce the upward push of air resistance. Fun fact — in Galileo’s “All things fall alike” experiments, he always used spheres in order to cancel the effects of air resistance in his comparisons.

Let’s put some numbers to it. Suppose someone’s at Earth’s “edge of space” 100 kilometers up. From the PE=m·g·h formula for gravitational potential energy and dividing out their mass which I don’t know, they have 9.8×105 joules/kilogram of potential energy relative to Earth’s surface. Now suppose they convert that potential to kinetic energy by falling to the surface with no air resistance. Using KE=m·v² I calculate they’d hit at about 1000 meters/second. But in real life, the terminal velocity of a falling human body is about 55 meters/second.

That Area item is why parachutes work. Make a falling object’s area larger and it’ll have to push aside more air molecules on its way down. Anyone wanting to survive a fall wants as much area as they can get. A parachute’s fabric canopy gives them a huge area and a big help. Parachute drops normally hit at about 5 meters/second. Trained people walk away from that all the time. Mostly.

Which gets to the matter of how you land. Parachute training schools and martial arts dojos give you the same advice — don’t try to stop your fall, just tuck in your chin and twist to convert vertical kinetic energy to rolling motion. Rigid limbs lead to bones breaking, ligaments tearing and joints going out of joint.

So let’s talk bones. Adults have about 210 of them, about 90 fewer than when they were a kid. Bones start out as separate bony patches embedded in cartilage. The patches eventually join together as boney tissue and the cartilage proportion decreases with age. Bottom line — kid bones are bendy, old bones snap more easily. For your question, breaking “every bone in your body” is a bigger challenge if you’re young.

But all bones aren’t equal — some are more vulnerable than others. Sesamoid bones, like the ones at the base of your thumb, are millimeter‑sized and embedded in soft tissue that protects them. The tiny “hammer, anvil and stirrup” ear bones are buried deep in hard bony tissue that protects them, too. Thanks to bones and soft tissues that would absorb nearly all the energy of impact, these small bones are almost invulnerable.

To summarize, no matter how high up from Earth you fall from, you can’t fall fast enough to hit hard enough to break every bone in your body. Be careful anyhow.

Sy Moire.

~~ Rich Olcott

  • Thanks to Xander and Lucas for their input.

Here’s a Different Angle

“OK, Sy, so there’s a bulge on the Moon’s side of the Earth and the Earth rotates but the bulge doesn’t and that makes the Moon’s orbit just a little bigger and you’ve figured out that the energy it took to lift the Moon raised Earth’s temperature by a gazillionth of a degree, I got all that, but you still haven’t told Al and me how the lifting works.”

“You wouldn’t accept it if I just said, ‘The Moon lifts itself by its bootstraps,’ would you?”

“Not for a minute.”

“And you don’t like equations. <sigh> OK, Al, pass over some of those paper napkins.”

“Aw geez, Sy.”

“You guys asked the question and this’ll take diagrams, Al. Ante up. … Thanks. OK, remember the time Cathleen and I caught Vinnie here at Al’s shop playing with a top?”

“Yeah, and he was spraying paper wads all over the place.”

“I wasn’t either, Al, it was the top sending them out with centri–…, some force I can never remember whether it’s centrifugal or centripetal.”

“Centrifugal, Vinnie, –fugal– like fugitive, outward‑escaping force. It’s one of those ‘depends on how you look at itfictitious forces. From where you were sitting, the wads looked like they were flying outward perpendicular to the top’s circle. From a wad’s point of view, it flew in a straight line tangent to the circle. It’s like we have two languages, Room and Rotor. They describe the same phenomena but from different perspectives.”

“Hey, it’s frames again, ain’t it?”

“Newton’s inertial frames? Sort‑of but not quite. Newton’s First Law only holds in the Room frame — no acceleration, motion is measured by distance, objects at rest stay put. Any other object moves in a straight line unless its momentum is changed by a force. You can tackle a problem by considering momentum and force components along separate X and Y axes. Both X and Y components work the same way — push twice as hard in either direction, get twice the acceleration in that direction. Nice rules that the Rotor frame doesn’t play by.”

“I guess not. The middle’s the only place an object can stay put, right?”

“Exactly, Al. Everything else looks like it’s affected by weird, constantly‑varying forces that’re hard to describe in X‑Y terms.”

“So that breaks Newton’s physics?”

“Of course not. We just have to adapt his F=m·a equation (sorry, Vinnie!) to Rotor conditions. For small movements we wind up with two equations. In the strict radial direction it’s still F=m·a where m is mass like we know it, a is acceleration outward or inward, and F is centrifugal or centripetal, depending. Easy. Perpendicular to ‘radial‘ we’ve got ‘angular.’ Things look different there because in that direction motion’s measured by angle but Newton’s Laws are all about distances — speed is distance per time, acceleration is speed change per time and so forth.”

“So what do you do?”

“Use arc length. Distance along an arc is proportional to the angle, and it’s also proportional to the radius of the arc, so just multiply them together.”

“What, like a 45° bend around a 2-foot radius takes 90 feet? That’s just wrong!”

“No question, Al. You have to measure the angle in the right units. Remember the formula for a circle’s circumference?”

“Sure, it’s 2πr.”

“Which tells you that a full turn’s length is times the radius. We can bridge from angle to arc length using rotational units so that a full turn, 360°, is units. We’ll call that unit a radian. Half a circle is π radians. Your 45° angle in radians is π/4 or about ¾ of a radian. You’d need about (¾)×(2) or 1½ feet of whatever to get 45° along that 2-foot arc. Make sense?”

“Gimme a sec … OK, I’m with you.”

“Great. So if angular distance is radius times angle, then angular momentum which is mass times distance per time becomes mass times radius times angle per time.”

“”Hold on, Sy … so if I double the mass I double the momentum just like always, but if something’s spinning I could also double the angular momentum by doubling the radius or spinning it twice as fast?”

“Couldn’t have put it better myself, Vinnie.”

~~ Rich Olcott

Traffic Control

Jeremy Yazzie @jeremyaz
hi @symoire, this is jeremy. ive been reading about the osiris‑rex mission to astrroid bennu and how they’re bringing back a sample – so complicated – fancy robot arm, n2 squirter, air‑cleaner thingy – y not just vacuum the dust or pick up a rock?

Sy Moire @symoire
@jeremyaz – quick answer is that Bennu and OSIRIS-REx are already surrounded by the vacuum of space. Sample collectors can’t suck any harder that that. I’ll email you a more complete answer later

Hi, Sy, can you believe this weather? Temps last week were twice today’s high.

Not to a physicist, Sis.
Those 90s and today’s 45 are just Fahrenheit
scale numbers.
Can’t do ratios between them, “twice” does not compute.
I don’t suppose it would help if we went centigrade and said last week’s highs were around 35 and today it’s 5?

No, that’s worse, today’s down by 85% from last week.

Centigrade’s another scale you can’t do ratio arithmetic in. Kelvins is the way to go.
Temp in K tracks the average molecular kinetic energy.
Starts at zero where nothing’s moving and rises in proportion.
Last week’s highs ran around 308 K, today is 278 K.
Today we’re only 10% cooler than last week.

Physicists! Grrrr. However you measure the weather, it still feels cold. No picnic this weekend ;^(

From: Sy Moire <>
To: Jeremy Yazzie <>

Jeremy –

OK, now I’m back at the office I’ve got better tech for writing long answers.

First, the “grab a rock” idea has several issues

  • If you pick up a rock, you only have that rock, says nothing about any of its neighbors or the subsurface material it might have smacked into. Dust should be a much better representation of the whole asteroid.
  • The rock might not be willing to be picked up. When the scientists and engineers were planning the OSIRIS‑REx mission, they didn’t know Bennu’s texture — could be one solid rock or a bunch of middle‑size rocks firmly cemented together or a loose “rubble pile” of all‑size rocks and dust held together by gravity alone, or anything in between.
  • Have you ever played one of those arcade games where you try to pick up a toy with a suspended claw gadget and all you’ve got is a couple of control knobs and a button? Picking up a specific rock, even a willing one, is hard when you’re a robot operating 15 light‑minutes away from the home office.

So dust it is, but how to plan dust collection in low gravity when you know nothing about the texture? Something like a whisk broom and dust pan would work unless the surface is too uneven. Something like a drill or disk sander would be good, except to use either one you need a solid footing to work from or else you go spinning one way when the tool spins the other. (That was a problem on the International Space Station.) The Hayabusa2 mission to asteroid Ryugu used a high‑velocity impactor to create dust, but a bad ricochet or shrapnel could kill the OSIRIS‑REx mission. The planners decided that best alternative was puff‑and‑grab.

So why not an astronautical Roomba that just sucks in the dust? The thing about vacuum is that it’s a place where gas molecules aren’t. Suppose you’re a gas molecule. You’re surrounded by your buddies, all in motion and bouncing off of each other like on a crowded 3‑D dance floor. You stay more‑or‑less in place because you’re being hit more‑or‑less equally from every direction. Suddenly there’s a vacuum to one side. You’re not hit as much over there so that’s the direction you and a bunch of your buddies move. If you encounter a dust particle, it picks up your momentum and moves toward the emptiness where it could be trapped in somebody’s filter.

The planners decided to capture dust particles by entraining them in a flow of gas molecules through a filter. To make gas flow you need more gas on one side then the other. Gas molecules being few and far between in space, the obvious place to put your pusher gas is inside the filter. Hence the nitrogen squirt technique and the “air‑cleaner thingy.”

— Sy

Diagram of TAGSAM in operation
Adapted from
Credit: University of Arizona

~~ Rich Olcott

Better A Saber Than A Club?

There’s a glass-handled paper-knife on my desk, a reminder of a physics experiment gone very bad back in the day. “Y’know, Vinnie, this knife gives me an idea for another Star Trek weapons technology.”

“What’s that, Sy?”

“Some kinds of wave have another property in addition to frequency, amplitude and phase. What do you know about seismology?”

“Not a whole lot. Uhh … earthquakes … Richter scale … oh, and the Insight lander on Mars has seen a couple dozen marsquakes in the first six months it was looking for them.”

“Cool. Well, where I was going is that earthquakes have three kinds of waves. One’s like a sound wave — it’s called a Pwave or pressure wave and it’s a push-pull motion along the direction the wave is traveling. The second is called an Swave or shear wave. It generates motion in some direction perpendicular to the wave’s path.”

“Not only up-and-down?”

“No, could be any perpendicular direction. Deep in the Earth, rock can slide any which-way. One big difference between the two kinds is that a Pwave travels through both solid and molten rock, but an Swave can’t. Try to apply shearing stress to a fluid and you just stir it around your paddle. The side-to-side shaking isn’t transmitted any further along the wave’s original path. The geophysicists use that difference among other things to map out what’s deep below ground.”

“Parallel and perpendicular should cover all the possibilities. What’s the third kind?”

“It’s about what happens when either kind of deep wave hits the surface. A Pwave will use up most of its energy bouncing things up and down. So will an Swave that’s mostly oriented up-and-down. However, an Swave that’s oriented more-or-less parallel to the surface will shake things side-to-side. That kind’s called a surface wave. It does the most damage and also spreads out more broadly than a P- or Swave that meets the surface with the same energy.”

“This is all very interesting but what does it have to do with Starfleet’s weapons technology? You can’t tell a Romulan captain what direction to come at you from.”

“Of course not, but you can control the polarization angle in your weapon beams.”

“Polarization angle?”

Plane-polarized electromagnetic wave
Electric (E) field is red
Magnetic (B) field is blue
(Image by Loo Kang Wee and
Fu-Kwun Hwang from Wikimedia Commons)

“Yeah. I guess we sort of slid past that point. Any given Swave vibrates in only one direction, but always perpendicular to the wave path. Does that sound familiar?”

“Huh! Yeah, it sounds like polarized light. You still got that light wave movie on Old Reliable?”

“Sure, right here. The red arrow represents the electric part of a light wave. Seismic waves don’t have a magnetic component so the blue arrow’s not a thing for them. The beam is traveling along the y‑axis, and the electric field tries to move electrons up and down in the yz plane. A physicist would say the light beam is planepolarized. Swaves are polarized the same way. See the Enterprise connection?”

“Not yet.”

“Think about the Star Trek force-projection weapons — regular torpedoes, photon torpedoes, ship-mounted phasers, tractor beams, Romulan pulse cannons and the like. They all act like a Pwave, delivering push-pull force along the line of fire. Even if Starfleet’s people develop a shield-shaker that varies a tractor beam’s phase, that’s still just a high-tech version of a club or cannon ball. Beamed Swaves with polarization should be interesting to a Starfleet weapons designer.”

“You may have something. The Bridge crew talks about breaking through someone’s shield. Like you’re using a mace or bludgeon. A polarized wave would be more like an edged knife or saber. Why not rip the shield instead? Those shields are never perfect spheres around a ship. If your beam’s polarization angle happens to match a seam where two shield segments come together — BLOOEY!”

“That’s the idea. And you could jiggle that polarization angle like a jimmy — another way to confuse the opposition’s defense system.”

“I’m picturing a Klingon ship’s butt showing through a rip in its invisibility cloak. Haw!”

~~ Rich Olcott

How To Wave A Camel

“You’re sayin’, Sy, no matter what kind of wave we got, we can break it down by amplitude, frequency and phase?”

“Right, Vinnie. Your ears do that automatically. They grab your attention for the high-amplitude loud sounds and the high-frequency screechy ones. Goes back to when we had to worry about predators, I suppose.”

“I know about music instruments and that, but does it work for other kinds of waves?”

“It works for waves in general. You can match nearly any shape with the right combination of sine waves. There’s a few limitations. The shape has to be single-valued — no zig-zags — and it has to be continuous — no stopping over here and starting over there..”

“Ha! Challenge for you then. Use waves to draw a camel. Better yet– make it a two-humped camel.”

“A Bactrian camel, eh? OK, there’s pizza riding on this, you understand. <keys clicking> All right, image search for Bactrian camel … there’s a good one … scan for its upper profile … got that … tack on some zeroes fore and aft … dump that into my Fourier analysis engine … pull the coefficients … plot out the transform — wait, just for grins, plot it out in stages on top of the original … here you are, Vinnie, you owe me pizza.”

“OK, what it it?”

“Your Bactrian camel.”

“Yeah, I can see that, but what’s with the red line and the numbers?”

“OK, the red line is the sum of a certain number of sine waves with different frequencies but they all start and end at the same places. The number says how many waves were used in the sum. See how the ‘1‘ line is just a single peak, ‘3‘ is more complicated and so on? But I can’t just add sine waves together — that’d give the same curve no matter what data I use. Like in a church choir. The director doesn’t want everyone to sing at top volume all the time. Some passages he wants to bring out the alto voices so he hushes the men and sopranos, darker passages he may want the bases and baritones to dominate. Each section has to come in with its own amplitude.”

“So you give each sine wave an amplitude before you add ’em together. Makes sense, but how do you know what amplitudes to give out?”

“That gets into equations, which I know you don’t like. In practice these days you get all the amplitudes in one run of the Fast Fourier Transform algorithm, but it’s easier to think of it as the stepwise process that they used before the late 1960s. You start with the lowest-frequency sine wave that fits between the start- and end-points of your data.”

“Longest wavelength to match the data length, gotcha.”

“Mm-hm. So you put in that wave with an amplitude near the average value of your data in the middle art of the range. That’s picture number 1.”

“Step 2 is to throw in the next shorter wavelength that fits, right? Half the wavelength, with an amplitude to match the differences between your data and wave 1. And then you keep going.”

“You got the idea. Early physicists and their grad students used up an awful lot of pencils, paper and calculator time following exactly that strategy. Painful. The FFT programs freed them up to do real thinking.”

“So you get a better and better approximation from adding more and more waves. What stopped you from getting it perfect?”

“Two things — first, you can’t use more waves than about half the number of data points. Second, you see the funny business at his nose? Those come from edges and sudden sharp changes, which Fourier doesn’t handle well. That’s why edges look flakey in JPEG images that were saved in high-compression mode.”

“Wait, what does JPEG have to do with this?”

“JPEG and most other kinds of compressed digital image, you can bet that Fourier-type transforms were in play. Transforms are crucial in spectroscopy, astronomy, weather prediction, MP3 music recordings –“

Suddenly Vinnie’s wearing a big grin. “I got a great idea! While that Klingon ship’s clamped in our tractor beam, we can add frequencies that’d make them vibrate to Brahms’ Lullaby.”

“Bad idea. They’d send back Klingon Opera.”

~~ Rich Olcott

How To Phase A Foe

“It’s Starfleet’s beams against Klingon shields, Vinnie. I’m saying both are based on wave phenomena.”

“What kind of wave, Sy?”

“Who knows? They’re in the 24th Century, remember. Probably not waves in the weak or strong nuclear force fields — those’d generate nuclear explosions. Could be electromagnetic waves or gravitational waves, could be some fifth or sixth force we haven’t even discovered yet. Whatever, the Enterprise‘s Bridge crew keeps saying ‘frequency’ so it’s got to have some sort of waveishness.”

“OK, you’re sayin’ whatever’s waving, if it’s got frequency, amplitude and phase then we can talk principles for building a weapon system around it. I can see how Geordi’s upping the amplitude of the Enterprise‘s beam weapons would help Worf’s battle job — hit ’em harder, no problem. Jiggling the frequencies … I sort of see that, it’s what they always talk about doing anyway. But you say messing with beam phase can be the kicker. What difference would it make if a peak hits a few milliseconds earlier or later?”

“There’s more than one wave in play. <keys clicking> Here’s a display of the simplest two-beam interaction.”

“I like pictures, but this one’s complicated. Read it out to me.”

“Sure. The bottom line is our base case, a pure sine wave of some sort. We’re looking at how it’s spread out in space. The middle line is the second wave, traveling parallel to the first one. The top line shows the sum of the bottom two at each point in space. That nets out what something at that point would feel from the combined influence of the two waves. See how the bottom two have the same frequency and amplitude?”

“Sure. They’re going in the same direction, right?”

“Either that or exactly the opposite direction, but it doesn’t matter. Time and velocity aren’t in play here, this is just a series of snapshots. When I built this video I said, ‘What will things look like if the second beam is 30° out of phase with the first one? How about 60°?‘ and so on. The wheel shape just labels how out-of-phase they are, OK?”

“Give me a sec. … OK, so when they’re exactly in sync the angle’s zero and … yup, the top line has twice the amplitude of the bottom one. But what happened to the top wave at 180°? Like it’s not there?”

“It’s there, it’s just zero in the region we’re looking at. The two out-of-phase waves cancel each other in that interval. That’s how your noise-cancelling earphones work — an incoming sound wave hits the earphone’s mic and the electronics generate a new sound wave that’s exactly out-of-phase at your ear and all you hear is quiet.”

“I’ve wondered about that. The incoming sound has energy, right, and my phones are using up energy. I know that because my battery runs down. So how come my head doesn’t fry with all that? Where does the energy go?”

“A common question, but it confuses cause and effect. Yes, it looks like the flatline somehow swallows the energy coming from both sides but that’s not what happens. Instead, one side expends energy to counter the other side’s effect. Flatlines signal success, but you generally get it only in a limited region. Suppose these are sound waves, for example, and think about the molecules. When an outside sound source pushes distant molecules toward your ear, that produces a pressure peak coming at you at the speed of sound, right?”

“Yeah, then…”

“Then just as the pressure peak arrives to push local molecules into your ear, your earphone’s speaker acts to pull those same molecules away from it. No net motion at your ear, so no energy expenditure there. The energy’s burned at either end of the transmission path, not at the middle. Don’t worry about your head being fried.”

“Well that’s a relief, but what does this have to do with the Enterprise?”

“Here’s a sketch where I imagined an unfriendly encounter between a Klingon cruiser and the Enterprise after Geordi upgraded it with some phase-sensitive stuff. Two perpendicular force disks peaked right where the Klingon shield troughed. The Klingon’s starboard shield generator just overloaded.”

“That’ll teach ’em.”

“Probably not.”

~~ Rich Olcott

The Top Choice

Al grabs me as I step into his coffee shop. “Sy, ya gotta stop Vinnie, he’s using up paper napkins again, and he’s making a mess!”

Sure enough, there’s Vinnie at his usual table by the door. He’s got a kid’s top, a big one, spinning on a little stand. He’s methodically dropping crumpled-up paper wads onto it and watching them fly off onto the floor. “Hey, Vinnie, what’s the project?”

“Hi, Sy. I’m trying to figure how come these paper balls are doing a circle but when they fly off they always go in a straight line, at least at first. They got going-around momentum, right, so how come they don’t make a spiral like stars in a galaxy?”

Astronomy professor Cathleen’s standing in the scone line. She never misses an opportunity to correct a misconception. “Galaxy stars don’t spray out of the center in a spiral, Vinnie. Like planets going around a star, stars generally follow elliptical orbits around the galactic center. A star that’s between spiral arms now could be buried in one ten million years from now. The spiral arms appear because of how the orbits work. One theory is that the innermost star orbits rotate their ellipse axes more quickly than the outer ones and the spirals form where the ellipses pile up. Other theories have to do with increased star formation or increased gravitational attraction within the pile-up regions. Probably all three contribute to the structures. Anyhow, spirals don’t form from the center outward.”

My cue for some physics. “What happens in a galaxy is controlled by gravity, Vinnie, and gravity doesn’t enter into what you’re doing. Except for all that paper falling onto Al’s floor. There’s no in-plane gravitational or electromagnetic attraction in play when your paper wads leave the toy. Newton would say there’s no force acting to make them follow anything other than straight lines once they break free.”

“What about momentum? They’ve got going-around momentum, right, shouldn’t that keep them moving spirally?”

I haul out Old Reliable for a diagram. “Thing is, your ‘going-around momentum,’ also known as ‘angular momentum,’ doesn’t exist. Calm down, Vinnie, I mean it’s a ‘fictitious force‘ that depends on how you look at it.”

“Is this gonna be frames again?”

“Yup. Frames are one of our most important analytical tools in Physics. Here’s your toy and just for grins I’ve got it going around counterclockwise. That little white circle is one of your paper wads. In the room’s frame that wad in its path is constantly converting linear momentum between the x-direction and the y-direction, right?”

“East-West to North-South and back, yeah, I get that.”

“Such a mess to calculate. Let’s make it easier. Switch to the perspective of a frame locked to the toy. In that frame the wad can move in two directions. It can fly away along the radial direction I’ve called r, or it can ride along sideways in the s-direction.”

“So why hasn’t it flown away?”

“Because you put some spit on it to make it stick — don’t deny it, I saw you. While it’s stuck, does it travel in the r direction?”

“Nope, only in the s direction. Which should make it spiral like I said.”

“I’m not done yet. One of Newton’s major innovations was the idea of infinitesimal changes, also known as little-bits. The s-direction is straight, not curved, but it shifts around little-bit by little-bit as the top rotates. Newton’s Laws say force is required to alter momentum. What force influences the wad’s s-momentum?”

“Umm … that line you’ve marked c.”

“Which is the your spit’s adhesive force between the paper and the top. The wad stays stuck until the spit dries out and no more adhesion so no more c-force. Then what happens?”

“It flies off.”

“In which direction?”

“Huh! In the r-direction.”

“And in a straight line, just like Newton said. What you called ‘going-around momentum’ becomes ‘radial momentum’ and there’s no spiraling, right?”

“I guess you’re right, but I miss spirals.”

Al comes over with a broom. “Now that’s settled, Vinnie, clean up!”

~~ Rich Olcott

  • Thanks for the question, Jen Keeler. Stay tuned.

The Big Chill

Jeremy gets as far as my office door, then turns back. “Wait, Mr Moire, that was only half my question. OK, I get that when you squeeze on a gas, the outermost molecules pick up kinetic energy from the wall moving in and that heats up the gas because temperature measures average kinetic energy. But what about expansion cooling? Those mist sprayers they set up at the park, they don’t have a moving outer wall but the air around them sure is nice and cool on a hot day.”

“Another classic Jeremy question, so many things packed together — Gas Law, molecular energetics, phase change. One at a time. Gas Law’s not much help, is it?”

“Mmm, guess not. Temperature measures average kinetic energy and the Gas Law equation P·V = n·R·T gives the total kinetic energy for the n amount of gas. Cooling the gas decreases T which should reduce P·V. You can lower the pressure but if the volume expands to compensate you don’t get anywhere. You’ve got to suck energy out of there somehow.”

Illustrations adapted from drawings by Trianna

“The Laws of Thermodynamics say you can’t ‘suck’ heat energy out of anything unless you’ve got a good place to put the heat. The rule is, heat energy travels voluntarily only from warm to cold.”

“But, but, refrigerators and air conditioners do their job! Are they cheating?”

“No, they’re the products of phase change and ingenuity. We need to get down to the molecular level for that. Think back to our helium-filled Mylar balloon, but this time we lower the outside pressure and the plastic moves outward at speed w. Helium atoms hit the membrane at speed v but they’re traveling at only (v-w) when they bounce back into the bulk gas. Each collision reduces the atom’s kinetic energy from ½m·v² down to ½m·(v-w)². Temperature goes down, right?”

“That’s just the backwards of compression heating. The compression energy came from outside, so I suppose the expansion energy goes to the outside?”

“Well done. So there has to be something outside that can accept that heat energy. By the rules of Thermodynamics, that something has to be colder than the balloon.”

“Seriously? Then how do they get those microdegree above absolute zero temperatures in the labs? Do they already have an absolute-zero thingy they can dump the heat to?”

“Nope, they get tricky. Suppose a gas in a researcher’s container has a certain temperature. You can work that back to average molecular speed. Would you expect all the molecules to travel at exactly that speed?”

“No, some of them will go faster and some will go slower.”

“Sure. Now suppose the researcher uses laser technology to remove all the fast-moving molecules but leave the slower ones behind. What happens to the average?”

“Goes down, of course. Oh, I see what they did there. Instead of the membrane transmitting the heat away, ejected molecules carry it away.”

“Yup, and that’s the key to many cooling techniques. Those cooling sprays, for instance, but a question first — which has more kinetic energy, a water droplet or the droplet’s molecules when they’re floating around separately as water vapor?”

“Lessee… the droplet has more mass, wait, the molecules total up to the same mass so that’s not the difference, so it’s droplet velocity squared versus lots of little velocity-squareds … I’ll bet on the droplet.”

“Sorry, trick question. I left out something important — the heat of vaporization. Water molecules hold pretty tight to each other, more tightly in fact than most other molecular substances. You have to give each molecule a kick to get it away from its buddies. That kick comes from other molecules’ kinetic energy, right? Oh, and one more thing — the smaller the droplet, the easier for a molecule to escape.”

“Ah, I see where this is going. The mist sprayer’s teeny droplets evaporate easy. The droplets are at air temperature, so when a molecule breaks free some neighbor’s kinetic energy becomes what you’d expect from air temperature, minus break-free energy. That lowers the average for the nearby air molecules. They slow their neighbors. Everything cools down. So that’s how sprays and refrigerators and such work?”

“That’s the basic principle.”


~ Rich Olcott

Thanks to Mitch Slevc for the question that led to this post.

The Hot Squeeze

A young man’s knock, eager yet a bit hesitant.

“C’mon in, Jeremy, the door’s open.”

“Hi, Mr Moire. How’s your Summer so far? I got an ‘A’ on that black hole paper, thanks to your help. Do you have time to answer a question now that Spring term’s over?”

“Hi, Jeremy. Pretty good, congratulations, and a little. What’s your question?”

“I don’t understand about the gas laws. You squeeze a gas, you raise its temperature, but temperature’s the average kinetic energy of the molecules which is mass times velocity squared but mass doesn’t change so how does the velocity know how big the volume is? And if you let a gas expand it cools and how does that happen?”

“A classic Jeremy question. Let’s take it a step at a time, big-picture view first. The Gas Law says pressure times volume is proportional to the amount of gas times the temperature, or P·V = n·R·T where n measures the amount of gas and R takes care of proportionality and unit conversions. Suppose a kid gets on an airplane with a balloon. The plane starts at sea level pressure but at cruising altitude they maintain cabins at 3/4 of that. Everything stays at room temperature, so the balloon expands by a third –“

Kid drawing of an airplane with a red balloon
Adapted from a drawing by Xander

“Wait … oh, pressure down by 3/4, volume up by 4/3 because temperature and n and R don’t change. OK, I’m with you. Now what?”

“Now the plane lands at some warm beach resort. We’re back at sea level but the temp has gone from 68°F back home to a basky 95°F. How big is the balloon? I’ll make it easy for you — 68°F is 20°C is 293K and 95°F is 35°C is 308K.”

“Volume goes up by 308/293. That’s a change of 15 in about 300, 5% bigger than back home.”

“Nice estimating. One more stop on the way to the molecular level. Were you in the crowd at Change-me Charlie’s dark matter debate?”

“Yeah, but I didn’t get close to the table.”

“Always a good tactic. So you heard the part about pressure being a measure of energy per unit of enclosed volume. What does that make each side of the Gas Law equation?”

“Umm, P·V is energy per volume, times volume, so it’s the energy inside the balloon. Oh! That’s equal to n·R·T but R‘s a constant and n measures the number of molecules so T = P·V/n·R makes T proportional to average kinetic energy. But I still don’t see why the molecules speed up when you squeeze on them. That just packs the same molecules into a smaller volume.”

“You’re muddling cause and effect. Let’s try to tease them apart. What forces determine the size of the balloon?”

“I guess the balance between the outside pressure pushing in, versus the inside molecules pushing out by banging against the skin. Increasing their temperature means they have more energy so they must bang harder.”

“And that increases the outward pressure and the balloon expands until things get back into balance. Fine, but think about individual molecules, and let’s pretend that we’ve got a perfect gas and a perfect balloon membrane — no leaks and no sticky collisions. A helium-filled Mylar balloon is pretty close to that. When things are in balance, molecules headed outward approach the membrane with some velocity v and bounce back inward with the same velocity v though in a different direction. Their kinetic energy before hitting the membrane is ½m·v²; after the collision the energy’s also ½m·v² so the temperature is stable.”

“But that’s at equilibrium.”

“Right, so let’s increase the outside pressure to squeeze the balloon. The membrane closes in at some speed w. Out-bound molecules approach the membrane with velocity v just as before but the membrane’s speed boosts the bounce. The ‘before’ kinetic energy is still ½m·v² but the ‘after’ value is bigger: ½m·(v+w)². The total and average kinetic energy go up with each collision. The temperature boost comes from the energy we put into the squeezing.”

“So the heating actually happens out at the edges.”

“Yup, the molecules in the middle don’t know about it until hotter molecules collide with them.”

“The last to learn, eh?.”

“Always the case.”

~~ Rich Olcott

Thanks to Mitch Slevc for the question that led to this post.

Seesaw to The Stars

I look around the playground. “Where’s the seesaw, Teena?”

“They took it away. That’s good ’cause I hated that thing!”

“Why’s that, Sweetie?”

“I never could play right on it. Almost never. Sometimes there’d be a kid my size on the other end and that worked OK, but a lot of times a big kid got on the other end and bounced me up in the air. The first time I even fell off and they laughed.”

“Well, I can understand that. I’m sure you’ve been nicer than that to the littler kids.”

“Uh-huh, except for Bratty Brian, but he liked it when I bounced him. He called it ‘going to the Moon’.”

“I can understand that, too. If things go just right you come off your seat and float like an astronaut for a moment. I bet he held onto the handles tight.”

“Yeah, I just wasn’t ready for it the first time.”

“Y’know, there’s another way that Brian’s bounces were like a rocket trip to somewhere. They went through the same phases of acceleration and deceleration.”

“Uncle Sy, you know you’re not allowed to use words like that around me without ‘splaining them.”

“Mmm, they both have to do with changing speed. Suppose you’re standing still. Your speed is zero, right? When you start moving your speed isn’t zero any more and we say you’ve accelerated. When you slow down again we say you’re decelerating. Make sense?”

“So when Bratty Brian gets on the low end of the seesaw he’s zero. When I squinch down at my end he accelerates –“

“Right, that’s like the boost phase of a rocket trip.”

“… And when he’s floating at the very top –“

“Like astronauts when they’re coasting, sort of but not really.”

“… And then they decelerate when they land. Bratty Brian did, too. I guess deceleration is like acceleration backwards. But why such fancy words?”

“No-one paid much attention to acceleration until Mr Newton did. He changed Physics forever when he said that all accelerations involve a force of some kind. That thought led him to the whole idea of gravity as a force. Ever since then, when physicists see something being accelerated they look for the force that caused it and then they look for what generated the force. That’s how we learned about electromagnetism and the forces that hold atoms together and even dark matter which is ultra-mysterious.”

“Ooo, I love mysteries! What did Mr Newton tell us about this one?”

“Nothing, directly, but his laws gave us a clue about what to look for. Tell me what forces were in play during Brian’s ‘moon flight’.”

“Let’s see. He accelerated up and then he accelerated down. I guess while he was on the seesaw seat at the beginning the up-acceleration came from an up-force from his end of the board. And the down-acceleration came from gravity’s force. But the gravity force is there all along, isn’t it?”

“Good point. What made the difference is that your initial force was greater than gravity’s so Brian went up. When your force stopped, gravity’s force was all that mattered so Brian came back down again.”

“So it’s like a tug-of-war, first I won then gravity won.”

“Exactly. Now how about the forces when you were on the merry-go-round?”

“OK. Gravity’s always there so it was pulling down on me. The merry-go-round was pushing up?”

“Absolutely. A lot of people think that’s weird, but whatever we stand on pushes up exactly as hard as gravity pulls us down. Otherwise we’d sink into the ground or fly off into space. What about other forces?”

“Oh, yeah, Mr Newton’s outward force pushed me off until … holding the handles made the inward force to keep me on!”

“Nice job! Now think about a galaxy, millions of stars orbiting around like on a merry-go-round. They feel an outward force like you did, and they feel an inward force from gravity so they all stay together instead of flying apart. But…”


“Mr Newton’s rules tell us how much gravity the stars need to stay together. The astronomers tell us that there aren’t enough stars to make that much gravity. Dark matter supplies the extra.”

~~ Rich Olcott