explain xkcd - User contributions [en]

3090: Sail Physics

2025-05-17T14:21:53Z

162.158.137.155:

{{comic
| number = 3090
| date = May 16, 2025
| title = Sail Physics
| image = sail_physics_2x.png
| imagesize = 699x263px
| noexpand = true
| titletext = Turning in other directions can be accomplished by using a magnetized centerboard and ocean currents, since a current flowing through a magnetic field induces a Laplace force.
}}

==Explanation==
{{incomplete|This page was created by a wind-blown electron. Don't remove this notice too soon.}}

This comic shows what initially appears to be an explanation of how boats sail upwind. However, it quickly devolves into misinformation; instead of even mentioning {{w|tacking (sailing)|tacking}}, it dives into a totally wrong explanation involving electrons and magnetism.

This humour works at another level — almost all interaction of physical things with each other at macro scale (humans and boat sized objects) are electromagnetic in nature (this is considering, that at this scale, among the 4 fundamental interactions, only electromagnetic interactions are of reasonable strength). So one unaware of sailing mechanics may start to explain the situation with electromagnetism, and could come to this line of thinking, but it is wrong. If we are to consider this, we find that either no force is appearing in the direction shown, or very little.

{{w|Triboelectric effect|Triboelectric effect}} is acquisition of static charge by friction between two objects, which in turn depends on effective interaction surface area. But the charge acquired would be very small, and not be comparative to gravity by orders of magnitudes (gravity is a weaker force, but involves a whole planet's-worth of mass normally dominating the static-effects of all but the lightest particles/hairs/scraps of tissue, or significantly buoyant toy balloons). There is more charge generated due to the running of the boat through water, in a realistic situation. Following on from the initial assumption, however, the comic assumes motion in a downstream direction, effectively due to normal forces on the sail. Further, the comic applies this motion as of that of a charged object in Earth's magnetic field. If we are to consider this magnetic field to be roughly parallel to Earth's surface, which would be true for situations close to the equator (as Earth's magnetic field is effectively that of dipole bar magnet, which is roughly aligned with our planetary rotational axis). The net Lorentz force on the boat would be F = qvB, and since both velocity of both and magnetic field lines are in the same plane (as viewed in the above sketches, in a top-down manner), then the force would be perpendicular to the plane. That is, along the direction of gravity for the boat, which will not cause any motion in plane. If we consider magnetic field lines to be perpendicular to Earth's surface (close to poles), even then the magnitude of the force is very small. If we consider a net charge of 1 coulomb across the boat (which is an extremely high amount of charge, at which point, the charge would try to accumulate on sharp objects, and due to this high charge density, either charge will leave the boat through coronal discharge, or the sharp objects would start to break from the boat (this is due to high self electrostatic energy, which is roughly proportional to square of charge density divided by radius of curvature of object, if the bonding between object is not strong enough, then this electrostatic energy can overcome this bonding energy), assuming that this charge does not cause the ship to break or the charge to leak out to water body, as the charge would like to spread to minimise the electrostatic energy), and if we consider field strength of order 0.0001 Tesla (close to typical values on surface), and take wind speeds of 10000 m/s (also very high) then the net force on ship would be 1 N, which for a boat of mass 10 kg (less for a typical boat, but is appropriate for a makeshift raft) cause a net acceleration of 0.1 m/s^2. Which is not tiny, but due to friction across the boat, would be cancelled out (by 1 or 2 orders of magnitude). This is considering the best possible case. For more reasonable charge values, this force would be about a trillion times weaker, and hence this will not cause any practical motion.{{citation needed}}

If the described effect were actually significant, it wouldn't always make boats turn upwind. A positively charged boat would turn counter-clockwise on the northern hemisphere, where the magnetic field points down, and clockwise on the southern hemisphere, where the magnetic field points up. Thus it would turn upwind or downwind depending on which side of the boat faces the wind. A negatively charged boat would turn in the opposite direction. To beat to windward it would be necessary to switch between two sails made of different materials – one sail to accumulate a positive charge, and another to accumulate a negative charge.

==Transcript==
{{incomplete transcript|Don't remove this notice too soon.}}

[Panel 1]

How sailboats use physics to sail upwind:

[A schematic boat with a sail is shown (top-down view). Winds shown with directional arrows, pointing in the direction of the sail (going towards 4:30 on a clock face).]

[Panel 2]

1. Wind passing over the sail strips away electrons via the triboelectric effect.

[Schematic similar to panel 1, but with charged ions shown across both sides of the sail, representing the triboelectric effect.]

[Panel 3]

2. The positively charged boat is blown downwind; its movement in Earth's magnetic field produces a Lorentz force.

[The same schematic, except a force vector is shown in the direction of the wind, and a perpendicular force vector (along 1:30 on a clock face) is shown with a dashed arrow.]

[Panel 4]

3. The Lorentz force acts perpendicular to the direction of motion, redirecting the boat upwind.

[A net force vector is shown perpendicular to the downstream vector.]

{{comic discussion}}<noinclude>

Talk:2966: Exam Numbers

2024-08-01T14:04:14Z

162.158.137.155:

pre-algebra: 4, calculus: pi^2 / 4 (about 2.467), physics: cosmological constant: depends on how you measure it [[Special:Contributions/162.158.167.48|162.158.167.48]] 18:11, 31 July 2024 (UTC)

Game theory: -5x10⁶ (maybe helpful, maybe not... just be thankful I didn't include an ''i'' factor in there somewhere...) [[Special:Contributions/172.70.162.185|172.70.162.185]] 18:20, 31 July 2024 (UTC)
:Interesting; I went with ∞+10. So, between our answers, that makes the average...
:[[User:ProphetZarquon|ProphetZarquon]] ([[User talk:ProphetZarquon|talk]]) 05:21, 1 August 2024 (UTC)

Could somebody reformat all the math here in whatever LaTeX plugin this wiki uses? --[[Special:Contributions/162.158.222.102|162.158.222.102]] 18:35, 31 July 2024 (UTC)
:Probably not, because the MathML here is broken. But, also, nothing I see requires anything particularly complicated, it can all stay in fairly straightforward (standardly formatted) text. [[Special:Contributions/141.101.98.224|141.101.98.224]] 18:44, 31 July 2024 (UTC)
I had to look up "TREE(3)." Seriousness aside, I think the largest number would be the astrological sign 1 that has its end_points_ as galaxy clusters. [[Special:Contributions/172.68.245.184|172.68.245.184]] 19:26, 31 July 2024 (UTC)
:Which astrological sign? Search engines aren't helping. [[User:Onestay|Onestay]] ([[User talk:Onestay|talk]]) 20:41, 31 July 2024 (UTC)
::The nonexistent one I just made up that looks like a "1." 😃 [[Special:Contributions/172.71.222.6|172.71.222.6]] 21:06, 31 July 2024 (UTC)
:'OAK'? 'ELM'? 'ASH?' 'BOX'? 'YEW'? [[Special:Contributions/141.101.98.165|141.101.98.165]] 08:52, 1 August 2024 (UTC)

If infinity _is_ a number, it might be a possible solution to the game theory question. The average of any set of numbers that includes infinity is infinity, and infinity + 10 is still infinity. I probably wouldn't try that in most classes, but a game theory professor might approve "gaming" the system, as it were.
:If I would prefer no-one (else) to win, I might submit -∞ as my answer. [[Special:Contributions/172.70.90.74|172.70.90.74]] 20:13, 31 July 2024 (UTC)
::If I really wanted to mess with them, I would submit i. [[Special:Contributions/172.70.160.248|172.70.160.248]] 08:54, 1 August 2024 (UTC)

I did a bit of a deep dive into wikipedia and the googology wiki and the answer to the last question depends on a few things (along with assuming ZFC). If transfinite ordinals count as numbers, then those at the end of {{w|List of large cardinal properties}} take the cake (if i'm reading it right). Otherwise, something based off [https://googology.fandom.com/wiki/Rayo%27s_number Rayo's number] is the best googologists have come up with so far. [[Special:Contributions/172.69.246.149|172.69.246.149]] 20:18, 31 July 2024 (UTC)Bumpf

:How about "On, in the context of MK set theory"? MK is a standard way to extend ZFC by allowing classes as mathematical objects, so On (the class of all set-size ordinals) is a class-sized "ordinal". But MK doesn't allow proper classes to be contained in any object, so "On+1" doesn't exist except as a definable hyperclass. Thus, On is the biggest "number" in a model of MK set theory.

Isn’t the joke in the pre-algebra that it would require algebra in order ro calculate? [[Special:Contributions/172.68.70.135|172.68.70.135]] 20:36, 31 July 2024 (UTC)
:Yes. I agree that it would be worth adding wording along the lines that “the joke here is that you need algebra to solve the equation”. [[User:Dúthomhas|Dúthomhas]] ([[User talk:Dúthomhas|talk]]) 20:56, 31 July 2024 (UTC)
: I interpreted the 'pre-' bit as being more like 'proto-' - i.e. it's not fully proper algebra, but it's the kind of work you would do in preparation for tackling proper algebra.[[Special:Contributions/172.68.186.156|172.68.186.156]] 08:58, 1 August 2024 (UTC)

You know, formatting math on this wiki would be a lot easier if the Math extension were correctly installed, but evidently it's not: <math>\int_0^\pi x \sin^2 x \;dx</math> [[User:Zmatt|Zmatt]] ([[User talk:Zmatt|talk]]) 22:22, 31 July 2024 (UTC)

Is that integral really correct? I asked Wolfram Alpha and it gave me
: integral x sin^2(x) dx = 1/8 (2 x (x - sin(2 x)) - cos(2 x)) + constant
which does not seem to be the same as
: −2x sin(2x)+cos(2x)−2x)/28 + C.
But maybe there's something with half-angle formulas that makes them the same? … but I don't think so, they don't evaluate the same for x=0. [[User:JohnHawkinson|JohnHawkinson]] ([[User talk:JohnHawkinson|talk]]) 02:56, 1 August 2024 (UTC)

:Yup, looks like it was supposed to be
:: -(2x sin(2x)+cos(2x)-2x^2)/8
:but they messed up the places of the negation and square.
:Though the important part here isn't what it is at any f(x), but what it is for any f(x)-f(y). In this particular case, f(pi)-f(0). [[Special:Contributions/162.158.41.121|162.158.41.121]] 04:49, 1 August 2024 (UTC)

As to '''biggest numbers:''' I thought most people would say the answers revolved around "nine-stuffing." For a kindergartener, stuff in as many bare <code>9</code>s as possible. For a postgrad, mix in exponentiation and write your numbers even smaller than a kindergartener can. <code>9^9^9^9...</code> or perhaps <code>99^99^99...</code> or…I'm not sure what's optimal. Of course, I'm not math postdoc ;) Or maybe some integrals or big-∏ notation. [[User:JohnHawkinson|JohnHawkinson]] ([[User talk:JohnHawkinson|talk]]) 12:41, 1 August 2024 (UTC)

== What is a number? ==

Infinity is _not_ a number. [[User:Dúthomhas|Dúthomhas]] ([[User talk:Dúthomhas|talk]]) 19:39, 31 July 2024 (UTC)

Infinity is absolutely not a number, and is the one answer I would mark as unambiguously wrong for the last one. Just say TREE(G_64) or something. [[Special:Contributions/162.158.154.31|162.158.154.31]] 20:15, 31 July 2024 (UTC)
:This is correct. No one in post-grad math would write “infinity” and expect that answer to work. Infinity is NOT a number except for seven-year-olds. Yet the explanation above continues to posit it as a possible correct answer. [[User:Dúthomhas|Dúthomhas]] ([[User talk:Dúthomhas|talk]]) 20:49, 31 July 2024 (UTC)
::I qualify as a "post-grad math", and yet, I think infinity would have been a perfectly valid answer. Let me explain. The term "number" without further context is a bit vague, because there are several possible generalizations of natural numbers (something that presumably everyone agrees to call a "number"), and they are not compatible, ie. there is not a single generalization that generalizes them all. So we have to choose which generalization makes sense in the current context. Since the question is about thinking how big a number is, I naturally thought that the adequate generalization would be one that focuses on the order on natural numbers, ie. ordinals. In that case, my answer to this question would be "the class of numbers I can think of is not bounded, therefore there is no such thing such as a 'biggest number I can think of'". But if I had to write down a big number, I would write ε_{ε_{ε_{...}}} up until I filled the page, because that's the most efficient way I know to write a big, *big* infinity. Which is a number. (and I'm not seven, just to be clear) [[User:Jthulhu|Jthulhu]] ([[User talk:Jthulhu|talk]]) 08:35, 1 August 2024 (UTC)
:In IEEE floating point math, Infinity is ''not'' Not A Number. The latter is an indication of error (in a context where errors can't be signalled immediately) and an entirely separate concept to infinity. But both are not Normal Numbers. Or even Denormalized Numbers. Floating point math is a whole lot trickier than it appears to be at first glance, and only extremely tangentially related to mathematical reals. --[[Special:Contributions/172.68.205.54|172.68.205.54]] 00:48, 1 August 2024 (UTC)
::I would have written this, but I saw that your comment already explained the two points I would have made, so, well, well done! [[User:Jthulhu|Jthulhu]] ([[User talk:Jthulhu|talk]]) 08:35, 1 August 2024 (UTC)
:If I write a song titled "Infinity" that was part of an opera, then it would be a [https://en.wikipedia.org/wiki/Musical_number number]. [[Special:Contributions/162.158.175.141|162.158.175.141]] 13:26, 1 August 2024 (UTC)
A number, by definition, is a construct used to classify and/or compare values. How rigorous this needs be for one limits the extent to which they accept things as being a number. Even things like "apple" could be interpreted as (dimensioned) numbers, with a possible value being "1 fruit"; In that regard, one may consider things like apple=orange<grapes.

Just "infinity" is nearly useless in this regard, as it's "no end thing". Usually interpreted (when necessary) as the countable infinite cardinal x=aleph_null, this prevents most useful comparisons, including dimensional analysis since x^n=x for all counting (aka. finite positive integer) n. Spacetime may or may not be boundless, but we can't tell how many edges may or may not loop. Is it infinity? Yes. Is it infinite? God only knows. Can you *count to it*? God can. Does that make it a number? Depends. Is "infinity plus one" a sane concept? No, it can't be finite, ordinal, and/or real in a way addition is defined; It's without end, and if you could add to it, that would indicate an end.

In contrast, classification has its roots in trade, and barter, and tipping. How much of a thing is enough, but not too much. Somebody may accept between 1/2 and 2/3 of a pie you're splitting, because less wouldn't be fair and more may give them a stomach ache; Is 3<=6x<=4 a number? It's similar in uselessness to "infinity", but whether something is less or more can at least still be established within its range. In the limit, Surreal numbers are the principal example of classification, taking the arithmetic mean of the maximum and minimum of their lower and upper bounds, or the predecessor or successor, or zero. For example, y={y|1} is the biggest number less than one, with z<=y<1 for all z<1. It's less than one, but not any "smaller" than one, with an immeasurably infinitesimal difference 0<1-y.

Choice of axioms is very important for all this, since its full extent can render everything except finite non-negative integers "not a number" (by Presburger Arithmetic), or allow everything up to and including unique antichain cardinalities (by Martin's Maximum).

The sixth power of the smallest ordinal with the cardinality of the continuum in the constructed universe (w_1^6 where beth_n=C(w_n)) is the biggest number I can personally conceptualize, although I can consistently work with w_2 in this system as well. Does the fact that this is infinite make it any less useful as a number than 2.5? No. It says I can think accurately about all the standard ways of comparing things in up to 6 infinitely divisible dimensions. Just because one cannot necessarily picture something others can't doesn't mean it doesn't exist. If a one-eyed person can only see a 2 spatial + 1 temporal dimensional image, that doesn't mean depth doesn't exist, it just means it's "hidden" from that perspective. 3+1+2 has two "hidden" dimensions compared to normal 3+1 spacetime, and beth_1 is infinitely divisible unlike the quantum (at most beth_0) nature of our known universe, but I can still work with 3+1+1, and 3+1+2 in the same way people can think about a (possibly looping) universe where everything can be bigger or smaller, and spatial geometry itself may be some degree of spherical, and people have been working with fractions since antiquity, so why should I limit myself to what other people can grasp?

In summary: "number" is too vague for claiming most things "aren't" to be reasonable. Infinite values (that aren't just "infinity", that's vague enough by itself to be almost as unreasonable) are just one one example of a valid answer most people seem to be up in arms about. [[Special:Contributions/162.158.41.181|162.158.41.181]] 01:06, 1 August 2024 (UTC)

:All right, all right. I yield. That’s some... _impressive_ reasoning. If we are going to redefine words to meaninglessness then there is no hope of engaging in useful discussion. I’m sure Randall will at least get a good laugh out of the idea that post-grad math students would submit “infinity” as the largest number they could think of. I still think it a disservice to readers to posit infinity as a _valid_ answer, though. [[User:Dúthomhas|Dúthomhas]] ([[User talk:Dúthomhas|talk]]) 05:05, 1 August 2024 (UTC)

Y'all, the answer is clearly 1. Sincerely, someone who has studied probability.[[Special:Contributions/162.158.137.155|162.158.137.155]] 14:04, 1 August 2024 (UTC)