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Very Wrong Math

charlespetzold.com

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Re: Very Wrong Math

#11

Post by seanhunter »

kurthr wrote: post #4
BalinKing wrote: post #2

Related Wikipedia article: 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!

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?).

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".

Re: Very Wrong Math

#12

Post by seanhunter »

kubb wrote: post #9

Curious, does the air being thinner affect flight time?

Presumably this would affect drag significantly. Here are the equations of motion of an aircraft https://eaglepubs.erau.edu/introductiontoaerospaceflightvehi...

...and indeed it does. Here is a discussion https://aviation.stackexchange.com/questions/24641/what-is-t...

Re: Very Wrong Math

#13

Post by jhanschoo »

kubb wrote: post #9

Curious, does the air being thinner affect flight time?

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.

Re: Very Wrong Math

#14

Post by seanhunter »

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".

Re: Very Wrong Math

#15

Post by aja12 »

kubb wrote: post #9

Curious, does the air being thinner affect flight time?

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/

Re: Very Wrong Math

#16

Post by shortrounddev2 »

freeopinion wrote: post #7
shortrounddev2 wrote: post #3

Earlier quoted context omitted.

(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

For convenience, we set τ=2pi. :-) x = τ(r+1) - τr = τ(r+1-r) = τ(1) = τ

How do you pronounce that symbol?

Re: Very Wrong Math

#19

Post by bruce511 »

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.

Re: Very Wrong Math

#20

Post by gsf_emergency »

BalinKing wrote: post #2

Related Wikipedia article: 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!

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!

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