This is a seriously old problem that needed to be solved before we became a routinely globe-trotting species. If you have the latitude and longitude of your location and a destination, you *could* just move east/west until you match the destination’s longitude then move north/south until you match the latitude. But that’s a big waste of time, as the so-intuitively-obvious-it-barely-deserves-a-name “triangle inequality” will tell you.

As you may have heard, the shortest distance between two points is a straight line. For curved surfaces the closest you can get to a straight line is a “geodesic”, which is a straight line on small scales, but it you step back may be going all over the place. If you wrap a ribbon around a gift (or any manner of festive package) and manage to keep it flat, you’ve found a geodesic, because if the path of the ribbon changes direction, the ribbon kinks. On a sphere like Earth, where you’re stuck moving along the surface (instead of drilling straight through like some kind of compulsive, fire-proof mole), the geodesics are “great circles”. If you walk in a straight line, you’re walking on a great circle.

Doing geometry on a sphere is more difficult than geometry on a flat plane, but not by a hell of a lot. Geometry, despite what some pudunk Greek philosophers may say, is basically just playing with triangles. A nice, universal rule for any triangle is the Law of Cosines which relates the lengths of the three sides, a, b, and c, with the angle opposite one of those sides, C: .

Slap that triangle on the side of a sphere and that law becomes the Spherical Law of Cosines:

Here the meaning of a, b, and c has changed a little. Lengths on a sphere can be described by angles (drawn from the center to the surface). If you’re using radians (which you always should), those angles are . This is generally easier to use than the actual distance. For example, the angle from the north pole to the equator is 90° (obviously) while the physical distance is about 6,200 miles (different on every planet).

Lucky for us, positions on Earth are usually described in terms of angles (latitude and longitude) and not distances, so the spherical law of cosines is ready to go. Just put the corners at the north pole, where you’re at, and where you’re planning to go.

The law of cosines (either of them) relates four things: all three sides and one of the angles. If you have any three of those pieces of information you can solve for the forth. With the latitudes we have two sides and with the difference of the longitudes we got an angle.

First (because you always should) convert your angles from degrees to radians by multiplying by : , , and .

and therefore

Boom! There’s the distance. And with three sides in hand (the two given by the latitudes and the distance from, like, two seconds ago) you can find your “bearing”, the angle between north and your destination.

and so

For example, if you happen to be traveling from Nueva Guinea, Nicaragua (11.6932° N, 84.4540° W) to Nesflaten, Norway (59.6466° N, 6.7997° E), the angles involved are , , and . So the distance is

and the bearing is

If you want to walk in a straight line from downtown Nueva Guinea to uptown Nesflaten, you’d face due north, turn about 30 degrees to the right, and walk in a straight line for about two and a half months, while ignoring the Atlantic Ocean to the best of your ability.

So how do you calculate distance and bearing? A small bucket of math. Today if you want to calculate sines and cosines you break out a computer. So you may be wondering how they did this math back in the days of sailing ships (without GPS). They didn’t. Calculating was done once, then written down, then looked up many times. Once upon a time books were useful!

The people who actually did the calculating were “computers”; pitiable folk who sat in dark rooms and slowly went mad and rarely got to sail around the world.

]]>Clearly, the issue is that a spoon and a bowl aren’t osculating curves. The radius of a spoon’s curvature is typically less than an inch while a bowl’s is several inches. With different curvatures, when you drag a spoon across the bottom of an ice-cream-laden bowl, they only contact each other at a tiny point and you can only remove ice cream along a thin strip (Matching the curvatures is why you instinctively start using the spoon sideways when the ice cream is low). So if you wanna get your bowl perfectly clean, you gotta figure out how wide that strip is.

Around the point of contact there’s a gap between the spoon and bowl that gets bigger the farther away you get. But so long as the ice cream molecules are bigger than that gap, they’ll be caught. Water, at about a quarter of a nanometer across, is the smallest molecule in ice cream’s molecular menagerie.

The bottom of a circle with a radius R can be closely approximated by a parabola of the form (There’s nothing too special about parabolas; every curve that doesn’t have sharp corners can be closely approximated by circles and vice versa, it’s just that parabola math is easy). Spoons have a radius of around 1 cm. Bowls have a radius of around 7 cm. The distance from the center (the point of contact) to the edge of the strip, x, such that water molecules won’t be able to slip through the gap between the bowl and the passing spoon is given by . Solve for x and you get , so the strip should be about 5 micrometers across; on the order of a tenth of a hair’s width.

The wide ice-cream-free swaths you see in practice aren’t regions where all of the ice cream has been removed, but regions where it’s been left thin enough to see though. So, if you *really* wanted to scrape a bowl clean of every molecule of ice cream, you’d need to carefully “scan” one strip after another, about four million and change times. Just to make sure you don’t miss any (else you and your IC OCD would be forced to start over), it couldn’t hurt for those strips to overlap. Call it an even ten million passes. At, say, two passes per second (the approximate speed of a kid charging headlong into a brain freeze), it would take you almost two months to scour your bowl clean. About a week into that, you’ll get the overwhelming urge to ask for seconds. Or at least do something else.

So far, this is all a bit idealized. The Platonic perfection of parabolas doesn’t apply to actual spoons and bowls. If you zoom in close enough, ceramics are like moonscapes with plenty of room for ice cream to hide and the tip of a spoon looks more like a stainless steel mountain range. Truly, a rocky road.

As undeniably clever as the approach above is, that “10,000,000 passes with a spoon” estimate is built on unrealistic premises. Sadly, you can’t eat all the ice cream in a conventional bowl using compulsive spooning alone.

But that doesn’t mean it’s impossible eat all of the ice cream, you just have to stretch the rules a little. For example, you could eat as much as you would like, then put the bowl into a kiln. By and large, organic molecules (sugar, fat, cellulose, etc.) burn at temperatures below 500°F and turn into new compounds, and a good kiln heats things up to 1700°F. A ceramic bowl would survive the heat intact, but the chemicals that make ice cream ice cream (as opposed to ash) wouldn’t.

So if you define “eating all the ice cream in the bowl” as “there was ice cream, I ate ice cream, now it’s all gone”, a kiln is a good way to do it. But if you think destroying the evidence is somehow cheating, there is one more, perhaps not as delicious, alternative guaranteed to work.

Eat the bowl.

The empty ice cream bowl picture is from here.

The extreme close-up of a ceramic is from here.

The kiln picture is from here.

The Willy Wonka picture is from Willy Freaking Wonka.

]]>Fans of Star Trek, on the other hand, think they just read “photon torpedoes” twice.

According to the *perfectly* named Wookieepedia, proton torpedoes release clouds of high energy protons on impact. Now technically, that’s what all explosives do. By mass, atoms are almost entirely made of protons and neutrons, each of which are around 2000 times more massive than electrons, and (for most elements) show up in roughly equal numbers. So about half of the mass of practically *anything* is protons (the glaring exception is hydrogen, which is 1 proton and 1 electron).

Incidentally releasing clouds of protons, because that’s what’s in clouds of *anything*, is clearly not the idea behind proton torpedoes nor the spirit of this question. Neutron bombs (which are real things) release a heck of a lot of neutrons and not *too* much else. Proton torpedoes should be like neutron bombs, just with protons.

Once again physics, the queen of sciences and mother of buzzkills, has things to say. The thing about protons is that long before you notice them doing damage by physically impacting things, you notice their electrical charge. A proton is like a mosquito when you’re trying to sleep; the mass is not what’s important.

There are incomprehensible electrical charges buried inside of everything, we just don’t notice them because they’re normally almost perfectly balanced. One gram of protons, without accompanying electrons to bring their net charge to zero, would create an electric field strong enough strip electrons from their host atoms and ionize air more than 15 km away. It wouldn’t be too healthy for any other materials either; a field this large and intense could pull lightning bolts through granite at a range of 10 km.

So a proton torpedo with a one gram payload does a heck of a lot of damage just by existing. Luke’s aim didn’t really need to be that good.

Ironically, the big effect of detonating the torpedo and releasing the protons is that the electric field actually drops. As the last post talked about too much, you can understand a lot about electric and gravitational fields by “drawing bubbles” around them. The total field flowing out of a bubble is proportional to the amount of charge inside of that bubble. So if you’re standing far enough away, the field you feel stays about the same whether or not the torpedo has exploded. As the shell of protons expands and repels itself, the field outside of it stays about the same while the field inside of it disappears. If you’re up close, when the expanding shell of protons passes you (assuming you’re alive) you’d find yourself in a region with little or no electric field. The most destructive thing you can do with a pile of protons is not blow it up, but keep it together in as small a place as possible.

The only way to block the electric field created by a bucket of charge is to surround it with the same amount of the opposite charge. Both positive and negative charges create electric fields, but those fields counteract each other. That’s why, despite containing many kg of protons, you doesn’t destroy everything around you.

Unfortunately for BlasTech Industries (famed maker of clumsy, random blasters), counteracting electric fields is exactly what charges always do, given the chance.

So even if you intentionally store protons in you proton torpedo with the hope of weaponizing its electric field, the first thing that will happen is any nearby matter will ionize. The loosed electrons will fly toward and coat the torpedo while the stripped nuclei (which are full of protons) beat a hasty retreat. Assuming the casing is a perfect insulator, in the end you’d have a torpedo with a gram of extra protons inside, about half a milligram of electrons outside (because electrons are about 2000 times less massive than protons), and no devastating electric field.

You could keep the protons and electrons together until the torpedo hits its target, then release the protons but keep the electrons in place (I mean… somehow). Then you could transport your munitions without preemptively killing everything around them. But that would be like firing bullets tied to (very strong) rubber bands; the electrons and protons would rather fly back together than do anything else.

So “proton torpedo” may just be a code name like “tomahawk missile” (noteworthy for its lack of tomahawks). That, or George Lucas is playing a little fast and loose with physics.

]]>**Physicist**: Nature abhors a singularity. So when you see a nice glaring one (like this case, where d=0 makes the force jump to infinity) it’s worth pausing to consider how the universe finagles its way out. When you apply Newton’s Law of Universal Gravitation, , to planets and stars, d is indeed the distance between the centers of mass. But when to use that standard and why it works is a little subtle.

One of the most important ideas in physics is that the exact same laws apply to everything, everywhere, always. Nothing and nowhere gets special treatment, and that includes the center of mass.

The stuff at the center of the Earth does pull on you, but that’s nothing special; so does literally everything else. Phone books, your worst enemy, that thing you saw that one time, *everything* on Earth (and otherwise) pulls on you according to Newton’s law. Every atom pulls on every other atom.

The reason you don’t notice a pull toward each individual thing around you is that gravity is weak. The physical (as opposed to emotional) attraction between you and someone you’re walking next to is a force on the order of the weight of a mote of dust (tens of nanograms). The air moving in and out of your lungs pushes you around far, far more.

The reason you *do* notice a pull toward the Earth is that there’s a lot of Earth to do the pulling; around 100,000,000,000,000,000,000,000 times the mass of the person next to you. Sure it’s farther away (around 4,000 miles on average), but there’s a *lot* of it. Heck, it blocks half the sky (the lower half).

Finally, the reason everyone feels a pull toward the center of the Earth is that, for most intents and purposes, the Earth is a perfect sphere. No matter where you are and no matter how you look at it, you can draw a line through the middle of the Earth and the two sides will be mirror opposites of each other. As much as one half pulls you to the right, the other pulls you to the left. The collective pull of both halves is right down the middle; toward the line between them.

The same argument applies to any line that separates the two halves, and all of those lines pass through the center of mass. Technically, that’s how the center of mass is defined. So, because the mass of the Earth is spherically symmetric, gravity around here points toward the center of mass.

However! Gravity does not, in general, point toward the center of mass. Take for example the Earth-Moon system. Despite its non-contiguous-ness, it’s still a perfectly legit collection of mass. The center of mass of the Earth and Moon combined is a point just a little below the Earth’s surface. That fact is important when you consider how the two orbit each other (the Moon circles that point while the Earth kinda wobbles around it), but it is almost entirely unimportant when you consider how gravity pulls on things presently sitting on Earth’s surface.

That all said, Earth is only *almost* perfectly spherical. You may have noticed that the world is not a perfectly polished ball; there are mountains, oceans, gigatons of missing single socks, and that non-uniformity penetrates well below the surface. Those variations in the distribution and density of mass means that Earth’s gravity changes (very, very slightly) from place to place, both in magnitude and direction.

By flying a pair of satellites in formation over the Earth and measuring the distance between them we can detect them speeding up and slowing down with respect to each other as one, then the other, passes over differently-gravity’d regions.

Finally, to more directly address the original question, since gravity is a collective pull toward each individual piece of mass and not necessarily a pull toward the *center* of mass, Newton’s universal law of gravitation has to be applied with a little finesse. works fine (with d the distance to the center) as long as you’re *outside* of a sphere of mass, it stops working as soon as you’re inside because the mass above you counteracts the pull of the mass below you. In the case of a bullet shot through a rubber ball, this is why the gravitational force never jumps to infinity; the equation that says it should ceases to apply. The universe always has an obnoxious way to get out of having to deal with singularities.

Because gravity is an “inverse square force” (the “” in “” ) it has the remarkable property that the total gravitational field pointing into any closed “bubble” is proportional to the amount of mass contained within that bubble. Inverse square laws also describe the way light gets dimmer with distance: so if you build a glass bubble around a light source, no matter how big or what shape the bubble is, the same total amount of light flows out of it.

The beauty of this is that it allows us to do what should be an arduous calculation (adding up the contribution from every tiny chunk of matter) in a manner that appeals to physicists: stupid easy and deceptively impressive. One of the immediate conclusions of this is that, *as long as you’re outside of the sphere of mass in question*, there’s no difference between the gravity of a sphere of mass and the same amount of mass concentrated at a point. So if the Sun collapsed into a black hole, everything in the solar system would continue to orbit it as though nothing had happened, because the gravity would only have changed below where the surface of the Sun used to be.

The second cute conclusion of this is that if you’re inside of a sphere of mass, only the layers below you count toward the gravity you feel. So the closer you are to the center, the less mass is below you, and the less gravity there is. If you were in an elevator that passed through the Earth, you’d know you were near the center because there’d be no gravity. With the same amount of mass in every direction, every atom in your body would be pulled in every direction equally, and so not at all. A sort of gravitational tug of war stalemate.

Newton’s Universal Law of Gravitation applies to everything (hence the “universal”). But instead of applying to the centers of mass of pairs of big objects, it applies to every possible pair of pieces of matter. In the all-to-common-in-space case of spheres, we can pretend that entire planets and stars are points of mass and apply Newton’s laws to those points, but only as long as we don’t need perfection (which is usually). As soon as things start knocking into each other, or the subtle effects of their not-sphere-ness become important, you’re back to carefully tallying up the contribution from every chunk of mass.

The pedestal with nothing on it used to have a guy stabbing a bird. The picture is from here.

The GRACE stuff is from NASA.

]]>When you describe something from your own perspective you say things like “it’s ten feet in front of me” or “it’s to my left” or “it passed me by at 30 mph”. You personally define a set of directions (forward, left, etc.) and distances (however far away something is) relative to yourself. Far more subtly, your perspective also includes time. The “future direction” is just as personal and subjective as all of the other direction and distance stuff. The subjectiveness of time was one of the great insights of Einstein’s relativity.

A short, more pedantic answer is that the observer determines the coordinate system. That’s just the mathematical way of saying that when you talk about the location and time that things happen, you just describe them in terms of your location and what your watch says.

We’re so accustomed to sharing a coordinate system, like (so called) universal time or latitude and longitude, that it’s easy to fall under the assumption that there really is such a thing as “correct coordinates”. In reality, such “universal coordinates” are useful only because they’re something a lot of people have bothered to agree on. And to be fair, it is a lot more useful to tell someone where you are in terms of street corners or latitude and longitude than it is to use your own personal coordinate system, which would amount to saying “I am where I am”. Philosophically interesting, but not useful.

So while you could describe an observer’s trajectory through a city using some kind of standard coordinate system, that’s *not* the observer’s perspective. That’s a description of their location using someone/thing else’s perspective. Form their own perspective, an observer doesn’t observe their position changing so much as they observe the city moving around them.

As you read this, you would be well within your rights to say “the screen is two feet in front of me” (or however far it is) or more succinctly “the location of the screen is x=2” (where the x direction is forward, the y direction is left, and the z direction is up). From anyone else’s perspective, the location of the screen is different. For the person sitting next to you on the subway (if you’re reading this on a subway), the location of the screen might be “x=2 and y=3” or even more succinctly “(2,3,0)”.

If you’ve ever taken an intro physics course, you’ve been subjected to the parable of the guy kicking a ball off a cliff. The problem is something like “there’s this guy at the top of a 20 m tall cliff who, in a sudden pique of ludophobia, kicks a ball off of the cliff so that it is initially traveling sideways at 3 m/s. Where does it land?”. The answer, you will be thrilled to learn, is about 6 m from the base of the cliff.

When you’re setting up this problem you’re faced with an ancient conundrum that has been haunting physicists since long before the age of cartography: what coordinates should I use? The most correct answer is: whatever lets you be as lazy as possible.

If you put the center of your coordinates at the base of the cliff, with the axes aligned with the ground and the cliff (red axes in picture above), then one axis gives you height off the ground while the other tells you distance from the cliff. Very convenient. In this case the base of the cliff is at (0,0), the ball is kicked at (0,20), and lands in the water at (6,0). You *could* use the kicker’s point of view (yellow axes), in which case you would describe the base of the cliff as 20 m below them, and the ball lands 20 m down and 6 m out, (6,-20).

The beauty of physics is that it’s capable of imitating the universe’s supreme deference. You can write physical laws without ever specifying coordinates, allowing us to define the stage (make up coordinates) and let the universe play out however it likes. So if you really, *really* wanted to, you could use completely arbitrary coordinates (blue axes). But don’t. The level of the water is no longer just “z=0” or “z=-20”, it’s an equation. Gravity no longer points in the “negative z direction”, but off to the side. You can still do the problem and by applying the exact same physical laws, but you have to do a lot more work and that’s categorically unacceptable for a physicist.

In normal space, direction is part of the coordinate system (forward, left/right). We’re already used to that notion when we talk to each other (as in “You’ve got something on the left side of your face. No *my* left, *your* right. Well it’s all over the place now.”). And you already know how to change your coordinate directions. If you want your “left direction” to become your “forward direction” you just rotate 90° to the left. If you want your “up” to become your “forward”, just rotate backward 90° (lie down). Easy.

In relativity, the future is just another direction. Obviously time is a *little* special. You can measure distances in any spacial direction using a ruler or string, but to measure “distance” in the time direction you need a clock. So to define an observer in relativity, you define a coordinate system with them at the center and a time defined by what their clock says.

It’s hard to picture the “time direction”. Firstly because it requires thinking in four-dimensional terms and secondly because clocks and rulers are super different. That’s why physicists futz about with math and coordinates all the time; they allow us to extend our intuition from what we can picture to what we can’t.

It turns out that you can do something similar to changing direction with time. Exchanging one direction in space for another is called “rotation” (no big deal). Exchanging one direction in space for the time direction is called a “Lorentz Boost” or, if you don’t speak physicist, it’s called “start moving in that direction”. Someone facing a different direction relative to you will have a different perspective on what the “forward direction” is and someone *moving* relative to you will have a different perspective on what the “future direction” is.

So when you hear things like “when something moves very fast it experiences less time” it’s important to parse out whose time you’re talking about and even what moving means. This is actually a statement about how things behave in *your* perspective. As the observer, time and space are defined relative to *your* position and clock. When someone is moving, they’re moving relative to you and according to your coordinate system. When someone is moving through time slower, it’s according to your clock. The real headaches kick in when you realize that from the perspective of the other person, you’re the one who’s moving and experiencing time slower. This feels like a contradiction, but keep in mind that different observers all have their own clocks and coordinates. There’s a lot more detail on that here.

Here on Earth we’re all moving at about the same speed. The fastest someone else is likely to be moving relative to you is at most 2000 mph (if you happen to be both on opposite sides of the Earth and on the equator). Relativistic effects, including disagreeing about time, are only worth talking about when your relative velocities are an appreciable fraction of light’s modest 670,000,000 mph. The most accurate clocks in the world are capable of detecting disagreements induced by walking slowly (relative velocities of less than 1 meter per second). But we humans don’t care about differences of a few parts in a quintillion, so while the differences in our clocks are detectable (when we work *really* hard to detect them), they aren’t worth worrying about.

It’ll be at the

You can read about the show and get tickets here.

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