Is this some form of conservation of angular momentum?

**Physicist**: Actually, it’s some form of conservation of angular momentum.

Assuming you still have at least one foot on the ground, falling over is just a rotation that stops either comically or sadly or both. You start upright and 90° later you’re prone. Everything that moves is subject to the conservation of momentum: if you push something one way, then you’ll be pushed the other way. Similarly, everything that rotates is subject to the conservation of angular momentum: if you rotate something one way, then you’ll be rotated the other way.

If you start to fall in some direction, then you want your body to rotate in the opposite direction. Rotating your arms in the direction of the fall causes the rest of you to rotate back to upright.

Someone with gigantic hands and stick-thin-but-long arms would practically never fall over. The more mass you have far from the pivot point (shoulders) the greater the “moment of inertia“. Something with a lot of mass is hard to move or stop (it has a lot of “inertia“). Something with a high moment of inertia is hard to turn and to stop turning. In fact, that’s why flywheels are designed the way they are. Most of their mass is far from their pivot so that they have a high moment of inertia without being ridiculously heavy.

Until we can find the coveted giant-hands-long-arms gene, tightrope walkers will be forced to continue using poles to keep their balance. They use their pole to keep balanced in exactly the same way the rest of us ground-dwellers use our arms. If you can believe it, the search for this gene isn’t a priority for most geneticists.

The same trick we (humans with arms) use to keep our balance is the same trick that space craft use to rotate in space. Here on the ground we can push off of things. If you want to face a new direction, use your feet to push on the ground and voilà: a new vista lays before you (or another wall if you’re inside).

However, space is notable for its remarkable dearth of stuff. There’s nothing in space to push on, so spacecraft engineers (notable for their remarkable plenitude of cleverness) literally provide stuff for their spacecraft to push: flywheels. By turning a tiny flywheel clockwise a lot, the rest of the spacecraft turns counter-clockwise a little.

]]>**Physicist**: Matter in deep space tends to take the form of gas. Liquids basically don’t happen, and solids are pretty rare. The cosmic microwave background ensures that everything is *at least* as hot as about 2.7K, but in general the gases we see out there are quite a bit hotter. Fortunately, there’s so little of it that you’d barely notice.

For example, in the core of the Orion Nebula the temperature of the gas is around 10,000K, but even if you were there your biggest problem wouldn’t be getting burned, it would be freezing to death. Also suffocation. Really, if you’re going to travel in space at all, bring a spacesuit.

Temperature is a measure of the average random speed of a material (technically, it’s the “variance” of the velocities). That “random” bit is important. For example, the air you’re (hopefully) breathing isn’t sitting still, it’s moving every-which-way at around half a km per second. Air bounces around so much that it doesn’t get very far (about 1 ten millionth of a meter before it gets bounced), but it’s all still moving pretty fast.

By comparison, Halley’s comet is moving at a few dozen km/s (it varies between about 1 km/s and 50 km/s). However, it’s frozen because that movement isn’t random. Its atoms are all moving really fast, but they’re all moving together.

Here comes the point. In deep space you mostly find individual atoms cruising along at high speeds. In order to define a temperature you look at lots of atoms passing through a region and see how random those trajectories are. Typically: they’re very random and very fast, so the temperature of those gases is thousands of degrees Celsius (or Kelvin or Fahrenheit for that matter).

It’s not unusual for things to be randomly traveling through space really fast: space is full of stuff taking forever to go from nowhere to nowhere through nothing at break-neck speeds. Generally, unless there’s a good reason to do otherwise, the individual atoms in space are traveling in every-which-way. A “good reason” is often running into something, or being caught up in an accretion disk. The majority of molecules in deep space are traveling in a straight line, very fast, without (strongly) interacting with anything else for years at a time.

A natural question to ask is: Isn’t space cold? If this interstellar gas is so damnably hot, then why doesn’t it cool off? The answers are: “yes it is” and “you have to think about why things cool off”. Light is created by *accelerating* charges. Traveling in a straight line involves no acceleration. If things bounce off of each other a lot, then they change direction a lot and *that* involves (for lack of a better word) a lot of accelerating.

This has nothing to do with either the spirit or letter of the question, but one of the most terribly cool things ever is that entire stars obey the same rules; they travel along at high speeds without interacting with anything else for huge periods of time. While atoms scatter by bouncing off of each other, stars interact with each other through their gravity (it’s very, very rare for stars to actually run into each other). If you stand back far enough, you find that the stars in a galaxy act a lot like a gas. You can even describe the “temperature” of a galaxy and talk about the movement of its stars in terms of thermodynamics. For example, “hot galaxies” are those with lots of randomly moving stars, and these galaxies literally evaporate (eject stars into inter-galactic space) and “cool” as a result. As they “cool” the randomness of their remaining stars’ movements decrease and the galaxy itself tends to contract.

]]>Science started in the 1750’s when a cadre of ultra-wealthy nobles decided to use their extreme means to build complex and bewildering devices for the express purpose of, as Mikhail V. Lomonosov put it (in Russian), “… the obfuscation of the truth from all, the befuddlement of the masses, the erosion of spiritual pursuits, and to [waste a lot of] time.”.

In his letters to fellow conspirator Cantor, Newton boasted about their contributions to science and mathematics saying “What we do today, let it not be mistaken, is the most elaborate and vexing gaff ever perpetrated. This truly is a godly joke against which all other humour can scarcely be compared.”.

Within the conspiracy there are supporters and detractors. Einstein, crushed by guilt, finally recanted in 1960 saying (in German) “I made it all up. I thought it would be funny, but then things got out of hand.” At the other extreme are examples like Gallileo, who left his middle finger on display with a plaque that read (in Italian) “May all the Earth sit and spin like a plate upon my bird”. This plaque was later removed, ostensibly for being offensive, but in reality for accidentally revealing a truth about the Earth.

The scientific community didn’t become truly organized until the early 20th century in order to squelch public knowledge of ghosts and telepathy. Today psychic scientists like James Randi use their secret powers to “prove” that other psychics don’t exist by messing up their vibes. During a meeting of the NSF inside their secret volcano lair, NSF director Dr. Córdova was accidentally recorded speaking without her human face mask on: “Tricking people into injecting their children with autism and hiding all the health benefits of coal is easy. That’s Tuesday morning. The hardest part of my job is keeping all the free energy devices off the market.”

The scientific conspiracy was perhaps best summed up by Carl Sagan from the after-life “Why did I do Cosmos? Are you serious? Why did man pretend to go to the Moon? Why do we hide global cooling or make up germs? Why do we systematically spread bizarre and fantastic lies about the nature of all of existence, generation after generation? Because it’s hilarious.”

]]>The universe does a lot of stuff (for example, whatever you did today), but literally everything that ever happens increases entropy. In some sense, the increase of entropy is equivalent to the statement “whatever the most *overwhelmingly* likely thing is, that’s the thing that will happen”. For example, if you pop a balloon there’s a *chance* that all of the air inside of it will stay where it is, but it is overwhelmingly more likely that it will spread out and mix with the other air in the room. Similarly (but a little harder to picture), energy also spreads out. In particular, heat energy always flows from the hotter to the cooler until everything is at the same temperature (hence the name: “thermodynamics”).

If you get in front of that flow you can get some work done.

“Usable energy” is energy that hasn’t spread out yet. For example, the Sun has lots of heat energy in one (relativity small) place. Ironically, if you were in the middle of the Sun, that energy wouldn’t be accessible because there’s nowhere colder for it to flow (nearby).

The spreading out of energy can be described using entropy. When energy is completely and evenly spread out and the temperatures are the same everywhere, then the system is in a “maximal entropy state” and there is no remaining useable energy. This situation is a little like building a water wheel in the middle of the ocean: there’s plenty of water (energy), but it’s not “falling” from a higher level to a lower level so we can’t use it.

The increase of entropy is a “statistical law” rather than a physical law. You’ll never see an electron suddenly vanish and you’ll never see something moving faster than light because those events would violate a physical law. On the other hand, you’ll never see a broken glass suddenly reassemble, not because it’s impossible, but because it’s *super* unlikely. A spontaneously unbreaking glass isn’t *physically* impossible, it’s *statistically* impossible.

However, when you look at really, really small systems you find that entropy will sometimes increase. This is made more explicit in the “fluctuation theorem“, which says that the probability of a system suddenly having a drop in entropy decreases exponentially with size of the drop.

For example, if you take a fistful of coins that were in a random arrangement of heads and tails and toss them on a table, there’s a chance that they’ll all land on heads. That’s a decrease in the entropy of their faces, and there is absolutely no reason for that not to happen, other than being unlikely. But if you do the same thing with two fistfuls of coins it’s not twice as unlikely, it’s “squared as unlikely” (that should be a phrase). 10 coins all landing on heads has a probability of about 1/1,000, and the probability of 20 coins all landing on heads is about 1/1,000,000 = (1/1,000)^{2}. The fluctuation theorem is a lot more subtle, but that’s the basic sorta-idea.

The “heat death of the universe” is what you get when you starting talking about the repercussions from ever-increasing entropy and never stop asking “and then what?”. Eventually every form of *useable* energy gets exhausted; every kind of energy ends up more-or-less evenly distributed and without an imbalance there’s no reason for it to flow anywhere or do work. “Heat death” doesn’t necessarily mean that there’s no heat, just no concentrations of heat.

But even in this nightmare of homogeneity we can expect occasional, local decreases in entropy. Just as there’s a *chance* that a broken glass will unbreak, there’s a *chance* that a pile of ash will unburn, and there’s a *chance* that a young (fully-fueled) star will accidentally form from an fantastically unlikely collection of scraps. There’s even a chance of a fulling functioning brain spontaneously forming. But just to be clear, these are all really unlikely. Really, *really* unlikely. As in “in an infinite universe over an infinite amount of time… maybe“. We do see entropy reverse, but only in tiny quantities (like fistfuls of coins or the arrangements of a few individual molecules). Something like the air on one side of a room (that’s in thermal equilibrium) suddenly getting 1° warmer while the other gets 1° colder would literally be the least likely thing that’s ever happened. The universe suddenly “rebooting” after the heat death is… less likely than that. Multivac interventions notwithstanding.

Those events that look like decreases in entropy have always been demonstrated to be either a matter of not taking everything into account or just being wrong.

Long story short: yes, after the heat death there should still be occasional spontaneous reversals of entropy, but they’ll happen exactly as often as you might expect. If you break a glass, don’t hold your breath. Get a new glass.

]]>Worse, you would think that momentum would go up hand in hand with kinetic energy, when the formulas above instead show the latter going up much faster due to the exponent. This also doesn’t make sense.

I’m sure you can do some math to show why it has to be this way, but can you explain in non-math terms why kinetic energy and momentum behave this way?

**Physicist**: This is pretty unintuitive. In fact, historically this was a whole thing. Buckets of profoundly smart folk argued and debated about whether velocity (momentum) or velocity squared (energy) was the conserved quantity. Turns out it’s both. The difficulty is first that energy can change forms and second that up until the 20th century lab equipment was *terrible* (and often home-made).

Force is mass times acceleration: F=ma. If you apply a force over a *time* you get momentum and if you apply force over a *distance* you get energy. Acceleration times time is velocity, so it should more-or-less make sense that force times time is momentum: . What’s a lot less obvious is energy.

A decent way to think about force and kinetic energy is to consider a falling weight. Gravity applies a constant force and thus a constant acceleration. If you tie a string to that weight you could power, say, a clock. Every meter it’s lowered it provides the same amount of energy, so lowering it 2 meters provides twice the energy as lowering it 1 meter.

Now imagine the weight free-falling that distance (instead of being slowly lowered). After the first meter it’ll already be moving, so it’ll fall through the second meter faster and in less time. The velocity gained is acceleration times time, so since it spends less time falling through that second meter, the falling weight spends less time accelerating and gains less speed.

But it still has to gain the same amount of energy every meter it falls. Otherwise weight-powered clocks would act *really* weird (a chain twice as long would yield only √2 as much energy). That means that at higher speeds you gain the same amount of energy from a smaller increase in speed. Or (equivalently) once you’re moving faster, the same increase in speed produces a greater increase in energy. This sometimes seems to produce paradoxes, but doesn’t.

With a little work and some calculus (see the answer gravy below) you can make this a lot more rigorous and you’ll find that the relationship between energy and velocity is exactly . In fact, figuring out this sort of thing is a big part of what calculus is for.

If it bothers you that energy doesn’t scale proportional to velocity, keep in mind that we’ve got that covered: momentum. Ultimately, both momentum and energy are just names for numbers that can be calculated and for which the total never changes. That which we call momentum by any other name would be as conserved.

**Answer Gravy**: Energy or work is force times distance: E=FD. When all the variables are constant finding the work done is just a multiplication away. However, when the variables aren’t constant finding the work done requires integration. The question of this post is one of the big reasons behind why calculus was originally invented. If you want to learn intro physics then please, for your own sake, learn intro calculus *first*. It is so much easier to talk about position, velocity, and acceleration (intro physics) when you can say “acceleration is the derivative of velocity and velocity is the derivative of position”. If you start physics with just a *little* calculus background, then you and your physics professor will high-five at least twice daily. Guaranteed.

Instead of a single distance with a constant force, we chop up the distance into lots of tiny pieces dx long and add them up. So a better, more universally applicable way of writing “E=FD” is , where the force is written “F(x)” to underscore that it may be different at different locations, x.

What we’ll calculate is the energy gained by an object that starts at rest, is pushed by a force F(x) over a distance D, and moves from position x=0 at time t=0 to position x=D at time t=T.

When we say the object started “at rest” we mean “v(0)=0″. Whatever v(T) is, it’s the velocity of the object when we’re done. So, the energy gained by an object that starts at rest and is pushed up to some speed v is .

Huzzah for calculus!

]]>So, you call that new number “j” (not to be confused with “j” from engineering, which is actually just “i” and presumably stands for “jamaginary number”). On the face of it, there’s nothing wrong with that; if we can make up i and work with it (to great effect), then making up j shouldn’t be terribly different. In the same way that we can write complex numbers as A+Bi, we should be able to write these new numbers as A+Bi+Cj; “trinions” as it were. However, it turns out that introducing a “j” requires us to also introduce a “k” (that also does the same thing as i and j).

Here’s why. You start by saying “i^{2} = j^{2} = -1″ and then asking “ij = ?”. You begin to get a sinking feeling when you square it: (ij)^{2 }= i^{2}j^{2} = (-1)(-1) = 1. This implies that ij = 1 or -1. But ij = 1 means that j = -i and ij = -1 means that j = i. There are more rigorous (confusing/complicated) ways to do this, but they ultimately boil down to “dude, we need another number”. That number is k (for “kamaginary” maybe).

So we’ve got i^{2} = j^{2} = k^{2} = -1 and ij = k. Fine. But there’s a big problem: quaternions can’t be commutative (mathematicians would call this big problem an “interesting property”, because they’re so chipper). “Commutative” means that order doesn’t matter, but for quaternions it must. Here comes a contradiction:

Firstly: (ij)^{2} = k^{2} = -1. This is basically a definition. It’s “True”.

Secondly (with commutativity): (ij)^{2} = (ij)(ij) = ijij = i^{2}j^{2} = (-1)(-1) = 1. Savvy readers will note that 1 ≠ -1. This can be fixed by declaring that ij = -ji.

Thirdly (declaring that ij = -ji): (ij)^{2} = (ij)(ij) = i(ji)j = i(-ij)j = -i^{2}j^{2} = -(-1)(-1) = -1. Fixed!

So far, this whole thing has been about why quaternions have the weird properties they do: there needs to be an i, j, *and* k, and you have to give up commutativity. Complex numbers are written “A+Bi” where i^{2} = -1. Quaternions are written “A+Bi+Cj+Dk” where i^{2} = j^{2} = k^{2} = -1, ij = k, jk = i, ki = j, and reversing any of these last three flips the sign.

One of the most profoundly cool things about quaternions is that they have their own form of Euler’s equation. When , . This can be derived the same way the regular Euler equation is derived, but using the fact that .

At this point it’s entirely natural (for a mathematical masochist) to ask “alright, but what if there were *yet another* square root of -1?”. Well it turns out that the next jump is harder and requires *seven* things that square to -1. Concerned at the prospect of running out of letters, clever mathematicians usually label these e_{1}, e_{2}, e_{3}, e_{4}, e_{5}, e_{6}, e_{7}. where (e_{1})^{2} = (e_{2})^{2} = … = (e_{7})^{2} = -1. An octonion number is written “A + Be_{1} + Ce_{2} + De_{3} + Ee_{4} + Fe_{5} + Ge_{6} + He_{7}“, where each of these (capital) letters is a real number. When you make the jump to octonions you not only lose commutativity you lose associativity, which makes everything terrible. With octonions you can’t say that (ab)c = a(bc), which is a big loss.

Some terribly insightful old soul might now be driven to inquire “alright, but what if there were *still more* square roots of -1?”. Sure. Enter the Cayley Dickson construction to create a “ladder” of as many of these number systems as your heart may ever desire, doubling in complexity every time.

Here’s the idea: you’ve start with a number system, then you take pairs of those numbers and slap a couple of rules on them. Complex numbers are just a pair of real numbers with some algebra glued on. For example, and . You may as well write this and . In addition to addition and multiplication, complex numbers also have an operation called “complex conjugation” (denoted with a bar or asterisk) which flips the sign of the imaginary part of a complex number. For example, or equivalently . The same operation exists for quaternions. For example, .

The Cayley Dickson construction defines numbers “higher up the ladder” as pairs of numbers from “lower down the ladder”. So a complex number, Z, is a pair of real numbers, A and B, which we can write Z=A+Bi={A,B}. A quaternion number, Z, is a pair of complex numbers, A+Bi and C+Di, which we can write Z=A+Bi+Cj+Dk=A+Bi+Cj+Dij=(A+Bi)+(C+Di)j={A+Bi,C+Di}. You’ll never guess how you can write an octonion.

Addition is handled like this , multiplication is handled like this , and conjugation is handled like this . For the jump from real to complex numbers those bars (conjugates) don’t do anything, but they’re important for each of the higher number systems. With this weird looking formalism in hand you can go from real numbers to complex numbers to quaternions to octonions to sedenions and so on and on and on (if you *really* want to).

It turns out that these higher number systems are useful. Complex numbers are ridiculously useful. Quaternions have a lot of interesting and *fairly* intuitive uses, like modeling rotations in 3 dimensions (which coincidentally is where we live) in part because they don’t have “special angles” that mess them up (e.g., the north pole is difficult to work with because it doesn’t have a definable longitude, but quaternions don’t have “north pole type problems”). While octonions are useful, they’re not useful in any easy to describe ways (when was the last time you *really* needed 8 dimensions for a problem?). Turns out they’re useful in string theory and presumably the higher number systems are useful as well. The harder mathematicians try to make mathematics that’s “pure” and free of the burden of being useful, the better they end up making our physics and computers.