9 ms·

Very Wrong Math

192 points by breadbox 2y ago

BalinKing 2y ago
Related Wikipedia article: https://en.wikipedia.org/wiki/String_girdling_Earth#Implications https://en.wikipedia.org/wiki/String_girdling_Earth#Implicat.... The takeaway is that the extra length of the arc is likely much smaller than one would intuitively expect. The problem is usually framed like so: If you wrapped a rope around the earth, how much more rope would you need to add so that it would be 1 meter above the ground at all points? The answer is only 2π meters!
- shortrounddev2 2y ago
  (2pi * (n + 1)) - (2pi * n) -> 2pi * (n + 1 - n) -> 2pi * 1 -> 2pi If I remember my algebra correctly. Someone else check my work I'm a dropout
  - freeopinion 2y ago
    For convenience, we set τ=2pi. :-) x = τ(r+1) - τr = τ(r+1-r) = τ(1) = τ
    - shortrounddev2 2y ago
      How do you pronounce that symbol?
      - Tistron 2y ago
        Tau. cf vihart: https://www.youtube.com/watch?v=jG7vhMMXagQ&t=0s https://www.youtube.com/watch?v=jG7vhMMXagQ&t=0s
      - freeopinion 2y ago
        https://doc.rust-lang.org/stable/std/f64/consts/constant.TAU.html https://doc.rust-lang.org/stable/std/f64/consts/constant.TAU... https://docs.python.org/3/library/math.html#math.tau https://docs.python.org/3/library/math.html#math.tau https://www.google.com/search?q=tau*1 https://www.google.com/search?q=tau*1 https://docs.oracle.com/en/java/javase/21/docs/api/java.base/java/lang/Math.html#TAU https://docs.oracle.com/en/java/javase/21/docs/api/java.base... https://learn.microsoft.com/en-us/dotnet/api/system.math.tau?view=net-9.0 https://learn.microsoft.com/en-us/dotnet/api/system.math.tau...
      - layer8 2y ago
        https://tauday.com/tau-manifesto https://tauday.com/tau-manifesto https://en.wikipedia.org/wiki/Tau https://en.wikipedia.org/wiki/Tau
- kurthr 2y ago
  The only issue I see with this is that as a classic physics trope, we've approximated the earth as a sphere. If, instead we approximate it as a fractal... then the distance is infinite, or at least highly dependent on the thickness of the rope! The error in the original is assuming that the radius is proportional to the height above the earth (Earthradius=0?).
  - Dylan16807 2y ago
    > infinite, or at least highly dependent on the thickness of the rope The latter. But that's only if it's not somewhat taut. Some tension brings it closer to a circle and makes the actual thickness pretty unimportant. But I like the idea overall. It means that lifting up the string makes it smoother and it actually needs less length. How's that for being unintuitive?
    - kurthr 2y ago
      Exactly, if you're only 1cm off the surface you follow every nook and cranny. If you're 10km off the surface only Everest is a blip.
  - seanhunter 2y ago
    We actually model the earth as a very large spherical cow. This is approximately the same for most purposes but ends up being more convenient. P.S. Not a physicist, but my child is studying maths and physics at Uni at present, so I have it on good authority that this is still going on. They told me in their first week one of their classes had a worked example where the lecturer used the phrase "Assume the penguin's beak is a cone".
    - davrosthedalek 2y ago
      A spherical cow /in vacuum/
    - kergonath 2y ago
      > I have it on good authority that this is still going on Do you mean making simplifying assumptions to make a problem tractable? Of course it’s still going on. It has to be, otherwise you just cannot do anything. > Assume the penguin's beak is a cone It is impossible to consider the true shape of a penguin’s beak for several reasons: - you’d need to go all the way down to the electron clouds of the atoms of the beak, at which point the very concept of shape is shaky - every penguin has a different beak so even if you describe perfectly one of them, it does not necessarily make your calculation more realistic in general. There is a spectrum of approximations one can make, but a cone is a sensible shape at a first order. It’s also simple enough that students can actually do it without years of experience and very advanced tools. What do you think they should do instead?
- gsf_emergency 2y ago
  This could be why dimensional analysis is one of the few things from physics class that can't be drilled enough.. Without forcefully dumping the geometric "intuition", this would still feel counterintuitive to me!
- nayuki 2y ago
  And the text about the airplane problem was added on 2024-11-26: https://en.wikipedia.org/w/index.php?title=String_girdling_Earth&diff=prev&oldid=1259596804 https://en.wikipedia.org/w/index.php?title=String_girdling_E...
- travisjungroth 2y ago
  > The takeaway is that the extra length of the arc is likely much smaller than one would intuitively expect. Maybe it’s because I’m a pilot and we never account for altitude when measuring distance, my intuition puts the difference at “effectively zero”. I also have it internalized that the earth’s atmosphere is very thin.
syntex 2y ago
just 2piR and then extra h change the result very little fraction. How is that counter-intuitive :)
nejsjsjsbsb 2y ago
Are we talking spacetime?
kubb 2y ago
Curious, does the air being thinner affect flight time?
- Waterluvian 2y ago
  I think in a way that’s why planes fly so much further up than you’d think they’d need to. They want more consistent and minimal atmospheric conditions. Less air means less energy means less turbulence, I think? If you’re talking about friction… oooh that’s an interesting one. Intuitively yes. But is it also negligible?
- seanhunter 2y ago
  Presumably this would affect drag significantly. Here are the equations of motion of an aircraft https://eaglepubs.erau.edu/introductiontoaerospaceflightvehicles/chapter/airplane-equations-of-motion/ https://eaglepubs.erau.edu/introductiontoaerospaceflightvehi... ...and indeed it does. Here is a discussion https://aviation.stackexchange.com/questions/24641/what-is-the-relation-between-an-airplanes-altitude-and-the-drag-it-is-experienc https://aviation.stackexchange.com/questions/24641/what-is-t...
- jhanschoo 2y ago
  There's no straightforward answer because there are many factors that affect flight time and fuel economy, including the aerodynamics of the plane and the engine technology. I hazard a guess that for commercial airplanes these are chosen primarily for reasons of fuel economy per seat and then that determines the model's designated cruising altitude. For a particular model, flying above the model's cruising altitude should lead to lower fuel efficiency.
- aja12 2y ago
  From what I've learned reading AdmiralCloudberg's plane crashes analysis [1]: altitude heavily matters in fuel consumption. Jet planes use a lot less fuel at a higher altitude, up to the point that a plane on the verge of running out of fuel at a medium altitude might manage to squeeze in 50 or 100 more miles of flight by climbing 5000 feet, even accounting for the increased fuel consumption during climb. I guess that correlates with speed as well. Turbofan engines, on the other hand, are more fuel efficient than jet engines at lower altitudes, hence they remain common for interstate transit. The difference seems to be directly caused by the effect of air "thickness" on the engines. [1] https://admiralcloudberg.medium.com/ https://admiralcloudberg.medium.com/
seanhunter 2y ago
I have seen a very similar (incorrect) argument used to justify the idea of a flat earth. A builder on youtube made the argument (with a similar out of scale drawing of the earth) that if he drops a plumb bob and makes a right angle so he has a straight horizontal line and then goes across that line for a bit and drops another plumb bob, the two lines he has dropped are parallel, "proving" that the surface of the earth must be parallel to the horizontal line and therefore flat and not curved. If the earth's surface was actually curved he argued then the two lines he has dropped should tilt slightly inward towards each other. Which of course they do. The earth is just much much much bigger than in the diagram so the effect is within the margin of error for the measurement he was taking. As a meta point, our intuition often fails us hilariously when we are dealing with stuff that is out of the scale we have commonly seen in our lives. We joke about LLMs hallucinating but I'm not convinced we are so superior when we are outside our personal "training data".
- munch117 2y ago
  Ah, but would they actually be parallel on a flat earth? Say the earth is disc-shaped. Then the center of gravity is only directly beneath you if you're standing at the exact center. You get ever-so-slightly not parallel lines, just like on a round earth. The fun part of a disc-shaped earth comes as you move towards the sides, and gravity, still pointing towards the center, makes you stand at an increasingly acute angle to the surface. The ground beneath you will then appear like one big endless mountainside, with an increasingly steep slope the further away from the center that you get.
  - jimmaswell 2y ago
    I'm considering what flat-surfaced shape you could construct with equal gravitational pull at all points. Maybe something where the center is thin as a point, the edges have a lot of depth, and they curve towards the center either convex or concave. Might run some calculus to figure it out.
    - t_mann 2y ago
      That way you should be able design a disc-shaped earth with constant strength of the gravitational force on the whole surface. But it would still have a center of mass (likely lying outside the shape you're describing, in the void beneath the center point), and the direction of the force should still be pointing towards that center, no? So the problem the GP has described, that you're starting to tilt as you move towards the edge, should remain in principle.
teo_zero 2y ago
And of course pi = 22/7! ;)
- necovek 2y ago
  That's actually a lot closer to the actual value of pi than the implied difference in the article vs the 0.15% actual difference in path length. As in, the illustration would be less wrong if it only used 22/7 for Pi and correctly portrayed dimensions of Earth and flight heights.
bruce511 2y ago
Even if the math of the arc length was correct (and you don't need to be a math professor to figure out it isn't) there's another logic misstep. Implied in the caption is that the speed is the same at all heights (given that an increase in distance is implied as an increase in time.) This is again obvious nonsense - speed is a function of thrust versus drag, and it's safe to say that both of those are affected by air density. It becomes even less true once one gets to space. There height is a function of speed which means that to "catch up" something in front of you, you need to slow down.
- mastermedo 2y ago
  > It becomes even less true once one gets to space. There height is a function of speed which means that to "catch up" something in front of you, you need to slow down. Can you expand on this? My brain is not connecting the dots.
  - db48x 2y ago
    He is talking about orbital mechanics, rather than free space. When you are in an orbit, the shape of the orbit is determined by your speed. At every distance from the center of the object you are orbiting (such as the Earth), there is a speed that makes your orbit a circle. If you are going at any other speed then your orbit will be an ellipse instead. Too fast and your orbit rises higher above the Earth. Too slow and it dips back down closer to it. If you try to “catch up” with an object ahead of you in your orbit by speeding up you will only turn your orbit into an ellipse that gets further away from the Earth, and thus further away from the object you were trying to catch. Instead of catching it you’ll go up and over it. As Niven wrote, “forward is up, up is back, back is down, and down is forward”. It’s rather counterintuitive at first. Playing KSP can help you get a feel for it, especially once you start docking multiple craft together.
    - davrosthedalek 2y ago
      Just to point out here what's different between "space" and "not space": "Space" assumes no "height control",i.e. ways to exert force "down or up" along the earth-object direction. That's obviously not true for a plane. If you can exert force in that direction, you can change speed and keep the shape of the trajectory around earth constant.
chrismorgan 2y ago
I think the funny thing about this article is this numeric error (though not so egregious as the one that caused the article!): > The mean radius of the earth is actually 3,459 miles or over 18 million feet. That’s off by 500 miles; the correct figure is 3,959 miles. That makes it almost 21 million feet, and yields a ratio of about 1.0013378, even smaller than the quoted 1.0015.
prmph 2y ago
One question I've always had with this: How does the rotation of the earth affect an airplane's flight time, if any? And how does this change with altitude?
- kergonath 2y ago
  It does not really, at least not directly. What matters is relative velocity compared to the starting and final locations, and relative to the air around the aircraft. It just happens that there are very powerful atmospheric currents that go west to east (those are due to the earth’s rotation, among others phenomena). So, when flying towards the east, catching these currents can significantly reduce flying time. When flying towards the west, we want to avoid them by flying below or elsewhere.
  - prmph 2y ago
    Thanks for this explanation; quite interesting. But it still seems to me that there might be a gravitational/inertial effects at play as well. At a (hypothetical) infinite altitude, it can no longer be said the the plane is moving perfectly in lock-step with the gravity/rotational acceleration of the earth. This implies the inertia of the plane relative to the rotation of the earth still has an effect at lower altitudes. The effect might be tiny, but would be interesting to learn more about it nonetheless.
    - BlueTemplar 2y ago
      What gravity/rotational acceleration ? Something like this does exist under general relativity : https://en.m.wikipedia.org/wiki/Frame-dragging https://en.m.wikipedia.org/wiki/Frame-dragging However, > This does not happen in Newtonian mechanics for which the gravitational field of a body depends only on its mass, not on its rotation.
      - prmph 2y ago
        Yes frame-dragging seems to be the name of the concept I was thinking of. Cool.
752963e64 2y ago
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n4r9 2y ago
Don't know if anyone mentioned this yet, but presumably the flight path does not follow a normal vector to gain height, but generally something more diagonal in the direction of travel.
pessimizer 2y ago
Does "Remember the high you go the further it'll have to travel" really need to be debunked? Did the "design and construction firm" spell "drill" with one "l"?
simplicio 2y ago
Seems intuitively obvious. On a flat Earth the two distances would be the same, and while the Earth isn't flat, its close enough to approximate a flat surface for most purposes, so you'd expect the differences in the two arcs to be ~0
lynguist 2y ago
I would do it like this: Approximate Earth as a flat line. (The 5000ft path is close enough that it is also represented by the flat line. This is the 5000ft path.) Then make the 33000ft path which is a slightly looser line on top of this line. This new path is not 4 times longer. Just a little bit raised, because 33000 ft is “nothing” compared to Earth. (To become 4x longer we would go deep into outer space and back.)
wittjeff 2y ago
"the high you go" reinforces my initial assumption that this is self-filtering clickbait.
JimmyWilliams1 2y ago
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mppm 2y ago
xkcd.com/386
svilen_dobrev 2y ago
Charles Petzold.. His c++ book stood on the shelf behind me. 30y ago. heh :)
1970-01-01 2y ago
All models are wrong, but some are useful. ⇒ Some models are wrong and useless.
fargle 2y ago
i've seen that exact image posted semi-regularly on various reddit and facebook groups. (it's one of 500 things i hate about those sites. but for some communities that's where the information lives, that's where marketplace lives, etc., so i'm stuck with it) these kind of things are intentionally wrong "puzzles" that are designed to get hundreds of people mad and post rebuttals to "drive engagement" or whatever. the pictures of a wheel with sledgehammers and chains and jacks with lugnuts plainly still in place and a post "how can i get this off it's stuck and i've tried everything". sigh... it's just another form of trolling. sure enough, notice the sibling comments here. how many nice people took the time to patiently explain the fallacy(ies) for the 1000th time. then the pedants who correct the grammar/math/etc. in the 98% correct explanations. then the "true believers"/trolls who don't get it and argue back. and so on. https://xkcd.com/386/ https://xkcd.com/386/
juresotosek 2y ago
haha crazy
csours 2y ago
Reminds me of this classic: https://www.politifact.com/factchecks/2020/mar/06/msnbc/bad-math-msnbc-bloombergs-ad-spending-wasnt-enough/ https://www.politifact.com/factchecks/2020/mar/06/msnbc/bad-... “Bloomberg spent $500 million on ads. The U.S. population is 327 million. He could have given each American $1 million and still have money left over.” https://www.youtube.com/watch?v=6egeUxIEQnM https://www.youtube.com/watch?v=6egeUxIEQnM
jodrellblank 2y ago
Earth in the picture is scaled to roughly 300,000 feet per pixel; Earth's surface and both flying altitudes would be the same pixel if drawn to scale. (~42M feet diameter shown in ~134 pixels).
quantified 2y ago
Holy crap is one drop of stupid consuming a lot of mental energy. This is after the drop of stupid that was Terence Howard's "1 x 1 = 2" physics rant fell on everyone's head. Individual drops quickening into rain would drown us all, apparently. Do serious people like Charles Petzold (here) and in other venues address this stupid out of fear that the stupid spreads, or because they just can't stand someone being wrong somewhere like a cognitive itch that must be scratched? If one troll flooded the zone with 30 of these over a month, mayhem would ensue. Absent knowing the true origin for this diagram, we don't even know if was stupid or malicious.
tenthirtyam 2y ago
xkcd what-if has a great picture of what a typical orbital flight is really like: https://what-if.xkcd.com/58/ https://what-if.xkcd.com/58/ tl;dr: Our atmosphere is surprisingly thin.

192 points by breadbox 2y ago

Related Wikipedia article: https://en.wikipedia.org/wiki/String_girdling_Earth#Implications https://en.wikipedia.org/wiki/String_girdling_Earth#Implicat.... The takeaway is that the extra length of the arc is likely much smaller than one would intuitively expect. The problem is usually framed like so: If you wrapped a rope around the earth, how much more rope would you need to add so that it would be 1 meter above the ground at all points? The answer is only 2π meters!

shortrounddev2 2y ago
(2pi * (n + 1)) - (2pi * n) -> 2pi * (n + 1 - n) -> 2pi * 1 -> 2pi If I remember my algebra correctly. Someone else check my work I'm a dropout
- freeopinion 2y ago
  For convenience, we set τ=2pi. :-) x = τ(r+1) - τr = τ(r+1-r) = τ(1) = τ
  - shortrounddev2 2y ago
    How do you pronounce that symbol?
    - Tistron 2y ago
      Tau. cf vihart: https://www.youtube.com/watch?v=jG7vhMMXagQ&t=0s https://www.youtube.com/watch?v=jG7vhMMXagQ&t=0s
    - freeopinion 2y ago
      https://doc.rust-lang.org/stable/std/f64/consts/constant.TAU.html https://doc.rust-lang.org/stable/std/f64/consts/constant.TAU... https://docs.python.org/3/library/math.html#math.tau https://docs.python.org/3/library/math.html#math.tau https://www.google.com/search?q=tau*1 https://www.google.com/search?q=tau*1 https://docs.oracle.com/en/java/javase/21/docs/api/java.base/java/lang/Math.html#TAU https://docs.oracle.com/en/java/javase/21/docs/api/java.base... https://learn.microsoft.com/en-us/dotnet/api/system.math.tau?view=net-9.0 https://learn.microsoft.com/en-us/dotnet/api/system.math.tau...
    - layer8 2y ago
      https://tauday.com/tau-manifesto https://tauday.com/tau-manifesto https://en.wikipedia.org/wiki/Tau https://en.wikipedia.org/wiki/Tau
kurthr 2y ago
The only issue I see with this is that as a classic physics trope, we've approximated the earth as a sphere. If, instead we approximate it as a fractal... then the distance is infinite, or at least highly dependent on the thickness of the rope! The error in the original is assuming that the radius is proportional to the height above the earth (Earthradius=0?).
- Dylan16807 2y ago
  > infinite, or at least highly dependent on the thickness of the rope The latter. But that's only if it's not somewhat taut. Some tension brings it closer to a circle and makes the actual thickness pretty unimportant. But I like the idea overall. It means that lifting up the string makes it smoother and it actually needs less length. How's that for being unintuitive?
  - kurthr 2y ago
    Exactly, if you're only 1cm off the surface you follow every nook and cranny. If you're 10km off the surface only Everest is a blip.
- seanhunter 2y ago
  We actually model the earth as a very large spherical cow. This is approximately the same for most purposes but ends up being more convenient. P.S. Not a physicist, but my child is studying maths and physics at Uni at present, so I have it on good authority that this is still going on. They told me in their first week one of their classes had a worked example where the lecturer used the phrase "Assume the penguin's beak is a cone".
  - davrosthedalek 2y ago
    A spherical cow /in vacuum/
  - kergonath 2y ago
    > I have it on good authority that this is still going on Do you mean making simplifying assumptions to make a problem tractable? Of course it’s still going on. It has to be, otherwise you just cannot do anything. > Assume the penguin's beak is a cone It is impossible to consider the true shape of a penguin’s beak for several reasons: - you’d need to go all the way down to the electron clouds of the atoms of the beak, at which point the very concept of shape is shaky - every penguin has a different beak so even if you describe perfectly one of them, it does not necessarily make your calculation more realistic in general. There is a spectrum of approximations one can make, but a cone is a sensible shape at a first order. It’s also simple enough that students can actually do it without years of experience and very advanced tools. What do you think they should do instead?
gsf_emergency 2y ago
This could be why dimensional analysis is one of the few things from physics class that can't be drilled enough.. Without forcefully dumping the geometric "intuition", this would still feel counterintuitive to me!
nayuki 2y ago
And the text about the airplane problem was added on 2024-11-26: https://en.wikipedia.org/w/index.php?title=String_girdling_Earth&diff=prev&oldid=1259596804 https://en.wikipedia.org/w/index.php?title=String_girdling_E...
travisjungroth 2y ago
> The takeaway is that the extra length of the arc is likely much smaller than one would intuitively expect. Maybe it’s because I’m a pilot and we never account for altitude when measuring distance, my intuition puts the difference at “effectively zero”. I also have it internalized that the earth’s atmosphere is very thin.

syntex 2y ago

just 2piR and then extra h change the result very little fraction. How is that counter-intuitive :)

nejsjsjsbsb 2y ago

Are we talking spacetime?

kubb 2y ago

Curious, does the air being thinner affect flight time?

Waterluvian 2y ago
I think in a way that’s why planes fly so much further up than you’d think they’d need to. They want more consistent and minimal atmospheric conditions. Less air means less energy means less turbulence, I think? If you’re talking about friction… oooh that’s an interesting one. Intuitively yes. But is it also negligible?
seanhunter 2y ago
Presumably this would affect drag significantly. Here are the equations of motion of an aircraft https://eaglepubs.erau.edu/introductiontoaerospaceflightvehicles/chapter/airplane-equations-of-motion/ https://eaglepubs.erau.edu/introductiontoaerospaceflightvehi... ...and indeed it does. Here is a discussion https://aviation.stackexchange.com/questions/24641/what-is-the-relation-between-an-airplanes-altitude-and-the-drag-it-is-experienc https://aviation.stackexchange.com/questions/24641/what-is-t...
jhanschoo 2y ago
There's no straightforward answer because there are many factors that affect flight time and fuel economy, including the aerodynamics of the plane and the engine technology. I hazard a guess that for commercial airplanes these are chosen primarily for reasons of fuel economy per seat and then that determines the model's designated cruising altitude. For a particular model, flying above the model's cruising altitude should lead to lower fuel efficiency.
aja12 2y ago
From what I've learned reading AdmiralCloudberg's plane crashes analysis [1]: altitude heavily matters in fuel consumption. Jet planes use a lot less fuel at a higher altitude, up to the point that a plane on the verge of running out of fuel at a medium altitude might manage to squeeze in 50 or 100 more miles of flight by climbing 5000 feet, even accounting for the increased fuel consumption during climb. I guess that correlates with speed as well. Turbofan engines, on the other hand, are more fuel efficient than jet engines at lower altitudes, hence they remain common for interstate transit. The difference seems to be directly caused by the effect of air "thickness" on the engines. [1] https://admiralcloudberg.medium.com/ https://admiralcloudberg.medium.com/

seanhunter 2y ago

I have seen a very similar (incorrect) argument used to justify the idea of a flat earth. A builder on youtube made the argument (with a similar out of scale drawing of the earth) that if he drops a plumb bob and makes a right angle so he has a straight horizontal line and then goes across that line for a bit and drops another plumb bob, the two lines he has dropped are parallel, "proving" that the surface of the earth must be parallel to the horizontal line and therefore flat and not curved. If the earth's surface was actually curved he argued then the two lines he has dropped should tilt slightly inward towards each other. Which of course they do. The earth is just much much much bigger than in the diagram so the effect is within the margin of error for the measurement he was taking. As a meta point, our intuition often fails us hilariously when we are dealing with stuff that is out of the scale we have commonly seen in our lives. We joke about LLMs hallucinating but I'm not convinced we are so superior when we are outside our personal "training data".

munch117 2y ago
Ah, but would they actually be parallel on a flat earth? Say the earth is disc-shaped. Then the center of gravity is only directly beneath you if you're standing at the exact center. You get ever-so-slightly not parallel lines, just like on a round earth. The fun part of a disc-shaped earth comes as you move towards the sides, and gravity, still pointing towards the center, makes you stand at an increasingly acute angle to the surface. The ground beneath you will then appear like one big endless mountainside, with an increasingly steep slope the further away from the center that you get.
- jimmaswell 2y ago
  I'm considering what flat-surfaced shape you could construct with equal gravitational pull at all points. Maybe something where the center is thin as a point, the edges have a lot of depth, and they curve towards the center either convex or concave. Might run some calculus to figure it out.
  - t_mann 2y ago
    That way you should be able design a disc-shaped earth with constant strength of the gravitational force on the whole surface. But it would still have a center of mass (likely lying outside the shape you're describing, in the void beneath the center point), and the direction of the force should still be pointing towards that center, no? So the problem the GP has described, that you're starting to tilt as you move towards the edge, should remain in principle.

teo_zero 2y ago

And of course pi = 22/7! ;)

necovek 2y ago
That's actually a lot closer to the actual value of pi than the implied difference in the article vs the 0.15% actual difference in path length. As in, the illustration would be less wrong if it only used 22/7 for Pi and correctly portrayed dimensions of Earth and flight heights.

bruce511 2y ago

Even if the math of the arc length was correct (and you don't need to be a math professor to figure out it isn't) there's another logic misstep. Implied in the caption is that the speed is the same at all heights (given that an increase in distance is implied as an increase in time.) This is again obvious nonsense - speed is a function of thrust versus drag, and it's safe to say that both of those are affected by air density. It becomes even less true once one gets to space. There height is a function of speed which means that to "catch up" something in front of you, you need to slow down.

mastermedo 2y ago
> It becomes even less true once one gets to space. There height is a function of speed which means that to "catch up" something in front of you, you need to slow down. Can you expand on this? My brain is not connecting the dots.
- db48x 2y ago
  He is talking about orbital mechanics, rather than free space. When you are in an orbit, the shape of the orbit is determined by your speed. At every distance from the center of the object you are orbiting (such as the Earth), there is a speed that makes your orbit a circle. If you are going at any other speed then your orbit will be an ellipse instead. Too fast and your orbit rises higher above the Earth. Too slow and it dips back down closer to it. If you try to “catch up” with an object ahead of you in your orbit by speeding up you will only turn your orbit into an ellipse that gets further away from the Earth, and thus further away from the object you were trying to catch. Instead of catching it you’ll go up and over it. As Niven wrote, “forward is up, up is back, back is down, and down is forward”. It’s rather counterintuitive at first. Playing KSP can help you get a feel for it, especially once you start docking multiple craft together.
  - davrosthedalek 2y ago
    Just to point out here what's different between "space" and "not space": "Space" assumes no "height control",i.e. ways to exert force "down or up" along the earth-object direction. That's obviously not true for a plane. If you can exert force in that direction, you can change speed and keep the shape of the trajectory around earth constant.

chrismorgan 2y ago

I think the funny thing about this article is this numeric error (though not so egregious as the one that caused the article!): > The mean radius of the earth is actually 3,459 miles or over 18 million feet. That’s off by 500 miles; the correct figure is 3,959 miles. That makes it almost 21 million feet, and yields a ratio of about 1.0013378, even smaller than the quoted 1.0015.

prmph 2y ago

One question I've always had with this: How does the rotation of the earth affect an airplane's flight time, if any? And how does this change with altitude?

kergonath 2y ago
It does not really, at least not directly. What matters is relative velocity compared to the starting and final locations, and relative to the air around the aircraft. It just happens that there are very powerful atmospheric currents that go west to east (those are due to the earth’s rotation, among others phenomena). So, when flying towards the east, catching these currents can significantly reduce flying time. When flying towards the west, we want to avoid them by flying below or elsewhere.
- prmph 2y ago
  Thanks for this explanation; quite interesting. But it still seems to me that there might be a gravitational/inertial effects at play as well. At a (hypothetical) infinite altitude, it can no longer be said the the plane is moving perfectly in lock-step with the gravity/rotational acceleration of the earth. This implies the inertia of the plane relative to the rotation of the earth still has an effect at lower altitudes. The effect might be tiny, but would be interesting to learn more about it nonetheless.
  - BlueTemplar 2y ago
    What gravity/rotational acceleration ? Something like this does exist under general relativity : https://en.m.wikipedia.org/wiki/Frame-dragging https://en.m.wikipedia.org/wiki/Frame-dragging However, > This does not happen in Newtonian mechanics for which the gravitational field of a body depends only on its mass, not on its rotation.
    - prmph 2y ago
      Yes frame-dragging seems to be the name of the concept I was thinking of. Cool.

752963e64 2y ago

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n4r9 2y ago

Don't know if anyone mentioned this yet, but presumably the flight path does not follow a normal vector to gain height, but generally something more diagonal in the direction of travel.

pessimizer 2y ago

Does "Remember the high you go the further it'll have to travel" really need to be debunked? Did the "design and construction firm" spell "drill" with one "l"?

simplicio 2y ago

Seems intuitively obvious. On a flat Earth the two distances would be the same, and while the Earth isn't flat, its close enough to approximate a flat surface for most purposes, so you'd expect the differences in the two arcs to be ~0

lynguist 2y ago

I would do it like this: Approximate Earth as a flat line. (The 5000ft path is close enough that it is also represented by the flat line. This is the 5000ft path.) Then make the 33000ft path which is a slightly looser line on top of this line. This new path is not 4 times longer. Just a little bit raised, because 33000 ft is “nothing” compared to Earth. (To become 4x longer we would go deep into outer space and back.)

wittjeff 2y ago

"the high you go" reinforces my initial assumption that this is self-filtering clickbait.

JimmyWilliams1 2y ago

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mppm 2y ago

xkcd.com/386

svilen_dobrev 2y ago

Charles Petzold.. His c++ book stood on the shelf behind me. 30y ago. heh :)

1970-01-01 2y ago

All models are wrong, but some are useful. ⇒ Some models are wrong and useless.

fargle 2y ago

i've seen that exact image posted semi-regularly on various reddit and facebook groups. (it's one of 500 things i hate about those sites. but for some communities that's where the information lives, that's where marketplace lives, etc., so i'm stuck with it) these kind of things are intentionally wrong "puzzles" that are designed to get hundreds of people mad and post rebuttals to "drive engagement" or whatever. the pictures of a wheel with sledgehammers and chains and jacks with lugnuts plainly still in place and a post "how can i get this off it's stuck and i've tried everything". sigh... it's just another form of trolling. sure enough, notice the sibling comments here. how many nice people took the time to patiently explain the fallacy(ies) for the 1000th time. then the pedants who correct the grammar/math/etc. in the 98% correct explanations. then the "true believers"/trolls who don't get it and argue back. and so on. https://xkcd.com/386/ https://xkcd.com/386/

juresotosek 2y ago

haha crazy

csours 2y ago

Reminds me of this classic: https://www.politifact.com/factchecks/2020/mar/06/msnbc/bad-math-msnbc-bloombergs-ad-spending-wasnt-enough/ https://www.politifact.com/factchecks/2020/mar/06/msnbc/bad-... “Bloomberg spent $500 million on ads. The U.S. population is 327 million. He could have given each American $1 million and still have money left over.” https://www.youtube.com/watch?v=6egeUxIEQnM https://www.youtube.com/watch?v=6egeUxIEQnM

jodrellblank 2y ago

Earth in the picture is scaled to roughly 300,000 feet per pixel; Earth's surface and both flying altitudes would be the same pixel if drawn to scale. (~42M feet diameter shown in ~134 pixels).

quantified 2y ago

Holy crap is one drop of stupid consuming a lot of mental energy. This is after the drop of stupid that was Terence Howard's "1 x 1 = 2" physics rant fell on everyone's head. Individual drops quickening into rain would drown us all, apparently. Do serious people like Charles Petzold (here) and in other venues address this stupid out of fear that the stupid spreads, or because they just can't stand someone being wrong somewhere like a cognitive itch that must be scratched? If one troll flooded the zone with 30 of these over a month, mayhem would ensue. Absent knowing the true origin for this diagram, we don't even know if was stupid or malicious.

tenthirtyam 2y ago

xkcd what-if has a great picture of what a typical orbital flight is really like: https://what-if.xkcd.com/58/ https://what-if.xkcd.com/58/ tl;dr: Our atmosphere is surprisingly thin.