The Drag Equation (Empire State Building v Eiffel Tower) - Numberphile

Tom Crawford discusses drag - then we drop the Empire State Building and Eiffel Tower into the ocean... Check out Brilliant (get 20% off their premium service): brilliant.org/numberphile (sponsor)
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[@numberphile]

Check out Brilliant (get 20% off their premium service): brilliant.org/numberphile (sponsor) Watch raw slow motions from filming of this video: kzbin.info/www/bejne/qX21XmWpidplZsk

[@knowit5248]

Please help me My subscribers are stoped increasing 😭😭😭😭

[@geoffreycraigwilliams4104]

Dr use

[@johncavanaugh3960]

Your view numbers are out of wack. KZbin...

[@kellerkind6169]

So this made me think: What about fine sand as a "fluid" (sinking into sand, maybe quicksand even)? Also this really made me tink about drag and air density in terms of rockets being launched, how its really important to keep their profile as low as possible (2d projected area) and about how much air density is changing whilst the rocket is working its way through the athmosphere

[@whiterottenrabbit]

What about the Foodskey channel? Care to finally tell the 20k+ subscribers there that you abandoned Foodskey in favour of more profitable channels like Numberphile or Computerphile?

[@theCodyReeder]

So this video randomly started playing in my pocket. It was just after finishing a meal while I was heading to rinse off my plate. I thought the sound in the beginning was coming from my stomach and I got really concerned for a moment that I had eaten something very bad. 😆

[@fonkbadonk2957]

You're one of the very few people whom I'd believe ANY day that they think they might have ingested something horrible and being 100% reasonable about that fear.

[@pigpiggypigbigpig681]

OK. Well... I’m glad you turned out to be fine lol!

[@Triantalex]

??

[@sbmathsyt5306]

Shout out to Brady for the amazing perfectly timed graphics, they really add to Tom's explanation in this video.

[@pmcpartlan]

Thanks!

[@sbmathsyt5306]

@@pmcpartlan Fair play if this is you! Great work!!

[@pmcpartlan]

@@sbmathsyt5306 Sorry, being a bit cheeky. :) But genuinely love comments like this, glad to hear the animations add to the clarity for you, thank you

[@TomRocksMaths]

Pete's the man.

[@sbmathsyt5306]

@@pmcpartlan Fair play not cheeky at all. I think it really takes the videos to the next level.

[@ashantraveler]

The problem is that the equation is missing the upward thrust (buoyancy force) on each object. In the air, it won't make much of a difference. But in water, it sure does.

[@micayahritchie7158]

Yes water is much more dense than air

[@charanvantijn541]

Yes, the empire state building probably has a very low specific mass. It might even float! Now that we know the mass, does anybody know the volume inside the empire state building?

[@randms2fake]

According wikipedia Empire State Building has 200500 m2 of commercial and office space. Assuming that space is at least 2 meters tall, it gives minimum volume that's about 400000 m3. So ESB would replace at least 400 million kg of water. It would float.

[@martindf6831]

Although the empire building is probably quite empty It would fill with water when dropping and it definitively wont float. However they should have taken in account Arquímedes force

[@andrewsnow7386]

I think it's totally fair to assume the Empire state building floods internally for their thought experiment, but they should have said they were ignoring buoyancy. Since the wrecking ball and Eiffel tower are both made almost entirely from the same material (steel/iron), considering buoyancy for these wouldn't change the relative speed for these two. But, the Empire state building has a lot of lighter materials, glass, stone, and brick. And if you consider the interior, I'll bet there is a lot of wood paneling, plastic carpets, etc. It would be very difficult to come up with an average density, but it's got to be substantially less than that of steel. Thus, if the flooded buoyancy was considered for the Empire state building, I bet it would have the slowest terminal velocity.

[@EternalLoveAnkh]

Best two moments in this video: "Lead is a metal" and "It's not that hard to imagine a 2D projection of a bowling ball". Haha! RJ

[@Triantalex]

false.

[@hunterhandley4602]

they really "imma compare a bowling ball and an absolute unit of a human"

[@ReynaSingh]

This man is equally dedicated to math and Pokemon 😅

[@aronklein8835]

I noticed his tattoos too

[@y2536524]

and SpongeBob

[@happygimp0]

more like physics than math

[@ethanbartiromo2888]

Currently so am I

[@kevin9681]

I think you meant Meth!!?

[@ANTONIOMARTINEZ-zz4sp]

I love Tom Crowford's videos. The enthusiasm and energy he puts on his explanations are infectious. Looking forward to seeing the next one!

[@TomRocksMaths]

[@syberrus]

It would be great if it was a science. But it is a fake science and results are wrong

[@starpetalarts6668]

We really need to redefine fluid so that cats aren't fluids. Let's say: "fluids forms into containers but cannot be a cat."

[@WGSen]

cats aren't fluids?

[@DrunkTortilla]

A fluid was taught to me as something that deforms continually when shear force is applied. I think this is the more textbook definition compared to how its defined in the video.

[@douglaspantz]

@@DrunkTortilla so, a cat?

[@pickle_256]

@@douglaspantz i believe so

[@outside8312]

Cats are scientifically considered a semi fluid

[@SJrad]

“Objects of different mass fall a the sa-“ Drag: “I’m gonna stop you right there”

[@stevecummins324]

Also inverse square nature gravitational attraction.

[@baumundallesandere]

... fall at the same speed in a vacuum.

[@hybmnzz2658]

@@stevecummins324 more negligible than drag

[@gran4404]

Every physics teacher: but we'll just ignore that for now

[@Triantalex]

false.

[@battleroyale1760]

No human was harmed while calculating the terminal velocity.

[@fatmn]

While? No. After? Maybe.

[@geoffstrickler]

Well, it might have made a few humans’ brain hurt.

[@battleroyale1760]

@@geoffstrickler haha😁

[@lotoa3383]

Tom: we have to take into account the holiness Me: are we throwing the Vatican too?

We should throw the pope out of a plane to test.

[@Dilholio]

Well we are taking mass into account

[@mortisCZ]

@ His ornate garments would work like a wing.

@@mortisCZ So not very hol(e)y? I knew it, he's a scammer.

[@arik9112]

I love it when ya bring in tom loved it! thanks!

[@TomRocksMaths]

[@theimmux3034]

Now I wanna see the Empire State Building make a descent into the Mariana Trench.

[@thekingoffailure9967]

I love it when Tom's on! He's so enthusiastic and its inspiring to see a young person every once in a while

[@TomRocksMaths]

[@N.I.R.A.T.I.A.S.]

Happy new year, Brady. Thanks for the awesome content!

[@jide7765]

13:04 as a skydiver, I can say that Vt = 50m/s (average) not 37m/s (flat on belly position)

[@mishal0404]

So wt do u think would change in the values he gave

[@aleksapetrovic7088]

@@mishal0404 he had big ranges on drag coefficient and the average area of the human, I'd say 37m/s is close to 50m/s given the rough estimations

[@jide7765]

@@mishal0404 Surface and drag coefficient.

[@mgancarzjr]

Knees are bent in a lazy W position.

[@sayochikun3288]

other comments are right. the way he put the "nice numbers" is not an averge human description.

[@razielhamalakh9813]

23:18 "There's no air." There is though. It's just very thin.

[@TheAlps36]

Even the entire universe has a density (I think it's about 1x10^-27kg/m^3) which is virtually negligible but still not nothing

[@vblaas246]

What's it composed of then? And what pressure? At average ground level (gravitationally).

[@RebelKeithy]

@@vblaas246 There is a wikipedia article on the moon's atmosphere. It's slightly more dense than the interplanetary medium. It says pressure is around 0.3 nPa and varies throughout the day. The entire atmosphere of the moon is around 10 metric tonnes.

[@martintuma9974]

It sounded like "there is no eh...".

[@TheAlps36]

@@RebelKeithy that's still surprisingly heavy!

[@PhilBoswell]

Since the Empire State Building is partially hollow, would the buoyancy have any appreciable effect?

[@markc7884]

Yep! The forces caused by escaping air etc would also cause it to rotate and spin, making this a whole lot more complicated. But with objects so large and even taking gasses into account which cause buoyancy, it's easier to run a simulation than to calculate it for objects specifically.

[@rolfs2165]

@@markc7884 Or you could just decide to treat it as one solid object and ignore gasses, open windows, and whatever else.

[@TheNameOfJesus]

The Empire State Building would probably float, and never sink, unless you broke the windows. But then breaking the windows would make the building's cross section partly hollow, like the Eiffel Tower.

[@alexanderzellmer951]

yes. let`s do this! Empire Stat Building: i found the mass of 365,000 t confirmed on wikipedia. I estimated the ground area from the block size at 50m x 110 m. Assuming it is approximately a tall pyramid with an effective height of about 350 m. Floor by height by 1/3 ...that gives ~ 640000 m³. so densitiy is 0.56 t/m³. in other words, it floats like a log of wood. wrecking ball: extrapolated from 0.67 m² cut area, it`s 0.46 m radius. that makes it 0.4 m³ or ~0.4 t(water) to subtstract off the 2,25 t weight. 1.85t/2.25t = correction factor for buoyancy ~ 0.82. sqrt(0.82) ~ 0.91. So corrected vmax is 11m/s x 0.91 = 9.91 m/s Eiffel tower: Assuming it`s bare iron, no encased air or else - density is 7.9 t/m³. so reduction of buoyancy is 1/7.9 ~ 13% of its mass. sqrt(0,87) = 0.93 x original vmax = 11.2 m/s Eiffel Tower wins. Second wrecking ball. Empire State Building fails(?) to sink.

[@thatchapthere]

It's important to clarify that as long as an object is more dense than water there's no such thing as "bouyancy"

[@Glass-vf8il]

Adding bouncy to the latter example might’ve been an interesting take too. Would if I’d have a significant effect

[@u1b2]

was going to make the same comment. buoyancy is a big factor especially in dense fluids like water.

[@cerwe8861]

In the Water i would also consider Buoyancy, in the Air it should be really small but i think it would habe a bit of an effekt in the Water.

[@NateCrownwell]

I love videos with Tom he is epic!

[@TomRocksMaths]

[@zman97211]

Brady+Tom go sky diving, next on Numberphile! I'd love that ...

[@TomRocksMaths]

I'm in if Brady is?

[@rwbishop]

@@TomRocksMaths It's a lot of fun... jumping is about all I did throughout the 70's & early 80's. Do a tandem or two first as you will definitely be in a state 'sensory overload'. (You might even get 'hooked' & decide to continue jumping... it happens to the best of them!)

[@archivist17]

Tom Crawford videos are the best in a very high quality feed!

[@nacoran]

Does this mean I'd fall faster because I have dimples?

[@IceMetalPunk]

Small correction: the Moon *does* have an atmosphere, it's just so incredibly thin/sparse that for most practical purposes, it can be ignored. But it's not zero, so that terminal velocity calculation will be finite, though extremely large.

[@greggregoryst7126]

Would an object even have a chance to accelerate enough to get to the terminal velocity? If only some kind of a huge-huge parachute?

[@SanyaJuutilainen]

​@@greggregoryst7126 It's very unlikely, the parachute would have to be astronomically huge. There are ~100 molecules per cubic cm where on Earth's it's roughly 100 billion billion per one, meaning difference of 16 orders, even square rooted that's too much. The total mass of lunar "air" is given as 25 thousand kilograms - for the whole Moon.

[@greggregoryst7126]

​@@SanyaJuutilainen So then can we calculate terminal velocity, like the OP assured?

[@SanyaJuutilainen]

@@greggregoryst7126 We can. It will be very large, but we can :)

[@marcolammers]

I'm pretty sure the wrecking ball vs eiffel tower vs empire state building is wrong. And I suppose it's because Tom Crawford probably estimated the weight of the Empire State Building using a fundamental misunderstanding about what buildings are. First, very short why it cannot be right : A wrecking ball is a solid object with a fundamentally compact geometry. It combines high density and low drag. The Empire State building is anything but a solid object, and its geometry has by definition more drag than a sphere. Therefore, the Empire State Building can only be slower than a wrecking ball Secondly : The Empire State building not being a solid object is a tremendous understatement. Much of construction is the art of encapsulating as much volume as possible with as little material as possible. The material density of a building is extremely low, almost all of the volume of a building is void. If sealed off properly, the empire State Building would easily float.

[@driesdebie3021]

I really like this guy!

[@TomRocksMaths]

[@Pheebs77]

I was lost after 5 minutes but you are all excellent, engaging teachers. Schoolkids who love maths would adore you 😍😍

[@seanoneill7085]

This brought back memories of the fluid dynamics classes I took in college. I earned a Bachelor of Science in Mechanical Engineering (about 33 yrs ago) . This was really very interesting and I will revisit my text book. I forgot how interesting it is. Thank you.

[@sanjogkarki]

I could watch this video again and again. Wow informative.

[@TomRocksMaths]

[@syberrus]

And wrong

[@Mattiaeragiapreso]

22:22 no, Galileo Galilei did in fact think about drag because he said that *if it was not for the air*, the object would have hitten the ground at the same time regardless of their masses. And that is exactly true.

[@andriypredmyrskyy7791]

One thing I learned doing aircraft modelling, the term C_D is doing a lot of heavy lifting. The drag coefficient can be a function of... anything. The "2D projected area" A is basically just a placeholder too, to make the units work out. It's typical for modelling to include some reference area for A, and to make C_D a function of whatever is relevant, relative angle (alpha, angle of attack, and/or beta, sideslip angle), sometimes it can be a function of even seemingly irrelevant stuff like propeller speed, outside air temperature, velocity (if you're characterizing different flow regimes). I thought this equation was so cool learning it the first time, and then it became significantly less cool when I found out that basically C_D is where all the modelling happens. We're just hiding the complex physics in C_D. The last thing is that if you disagree with parts of the formulation I've mentioned or has been seen in this video, usually those discrepancies are also hidden in the modelling done for C_D.

[@lukewalker4497]

Hey guys, great work i love it. You missed something essential in the water case: BOYANCY. So the "net gravity" pulling down is actually m * g MINUS (rho of the fluid * Volume of the Body * g). For air of 1000 times lower density than the falling body its no issue but for in-water, the sink rate reduces much by boyancy, so sink rates be even less than you calculate. To re-illustrate: imagine the "falling body" is actually a lighter-than-air Balloon filled with Helium. It still has some (small) "pull down" force m*g so will "sink" in your math. work out. Never the less it will actually * rise up* towards terminal velocity ZERO, at a high-enough point where boyancy and gravity balance out.... there you have it. cheers!

[@jakeroosenbloom]

Loving the animation!

[@MrJasoon13]

Great episode ! Thank you to the both of you.

[@TomRocksMaths]

[@danielschack2810]

Wow, a new numberphile video!

[@Potti314]

Awesome - thanks for the great entertainment :-) That was very interesting and enjoyable to watch.

[@TomRocksMaths]

[@jursamaj]

Actually, the Moon *does* have an atmosphere, it's just 3e-15 atm. Even inter-galactic space has a tiny atmosphere (less than 1 atom per m^3).

[@sterlingburnsides9712]

How did I just KNOW Crawford was gonna be the guest for this topic lol

[@kasuha]

Gravity does not change in case of throwing things into water, but I'm afraid we should not neglect buoyancy in such case. Concrete weight about 2400 kg per m3 in the air but when you put it in the water, it's about a ton less. Iron loses about the same, but comes in with about three times the density so in ratio it doesn't lose that much.

[@ishanjoshi7551]

Shouldn't you consider buoyant force while dropping stuff into water... Like wouldnt it change the force balance and in turn change the terminal velocity values we obtained?

[@killerbee.13]

Yes, it does make a big difference in fact-the Empire State Building would actually float, and the Eiffel Tower would beat the wrecking ball. Assuming, that is, that the windows on the Empire State Building are invincible and airtight. Realistically of course they'd break and the building would fill with water, and that would change its coefficient of drag so I can't estimate its terminal velocity. However, if you were to replace the air with water *without* breaking the windows, it would sink, so I ran the numbers: Adjusted for buoyancy, the wrecking ball (concrete, d=2.5) has a terminal velocity of 8.9 m/s, the Eiffel Tower (steel, d=7.8) has 10.2 m/s, and the ESB (steel, concrete, and water; d=2.8) has 10.5 m/s Also, while I was at it, I decided to work out the terminal velocity for the air-filled ESB (airtight invincible windows, d=0.375) (which floats, but the drag equation works for upwards movement too), which is -17.3 m/s. So, were it to be pulled all the way to the bottom and released from there, it would float up to the surface faster than any of them sink to the bottom, by a large margin.

[@hareecionelson5875]

23:30 Damn it Brady, you've broken maths again

[@dqrksun]

XD

[@karthikramakrishna4092]

Brady's excitement about the drag race is epic.

[@Omnifarious0]

One thing you didn't take into account for skydiving is that the terminal velocity decreases as the person drops because the air density increases. For the ocean, this isn't so much the case, because water isn't very compressible.

[@aryefasten3477]

To visualize the 2D shape of an object, you can think of the shadow it would cast. Hence the ball casting a circle.

[@kurumi394]

Aristotles: heavy things fall faster Me in grade school: fair enough Me in junior high/high school: dumbass Me in uni: fair enough

[@brycering5989]

Density ;)

[@kurumi394]

@@brycering5989 Yeah, that just about sums it up.

[@skyforger8102]

As a skydiver this video is great, but it also shows the discrepancies between the math and the real world. A skydiver falls at about 120mph or 54m/s when falling belly to earth like described in the video.

[@TheMazyProduction]

13:22 nice

[@hectooooor]

I agree

[@srwapo]

nice.

[@jackeea_]

No-one: Engineers: yo what's the drag coefficient of the eiffel tower

[@karlesmcquade2863]

I really enjoy Tom's clarity, enthusiasm, and dedication to his discipline! And more generally, as an American, I really appreciate hearing and getting to know the English equivalents for certain words in our shared language. But I gotta say, "candy floss" is a really gross synonym for "cotton candy."

[@TomRocksMaths]

[@marchanselthomas]

I was thinking of watching Netflix and ended up here and enjoyed this video way too much than I should have.

[@saniulsk436]

During my college days i found it difficult to understand but the way you explain the thing makes it lucid to understand.

[@TomRocksMaths]

:)

[@diegosanchez894]

That bowling ball terminal velocity is *nice*

[@jimbruttan69420]

2:10 - when the hip kid in the street turns out to be a mathematics professor

[@jmunt]

the moment I saw the thumbnail I new it was gonna be Tom

[@WTFAnimatonsHD]

23:44 c : let me introduce myself

[@ronisaiba9623]

Love the artworks.

[@ejn1011]

"Mass: That matters." *I see what you did there!

[@seytanuakbar3022]

Maximum velocity for head-first dive is 50-60 m/s. It depend on type of clothes.

[@jide7765]

as a skydiver, I can say that Vt = 50-60 m/s (180-210 km/h) for the "flat on belly" position. Head down, it's about 80-95 m/s (300-350 km/h) These are average numbers.

[@seytanuakbar3022]

@@jide7765 Skydivers use closed suit. Everyday clothes creates much more drag. There is much deference how wind affects me when I run in summer and winter outfits.

[@jide7765]

@@seytanuakbar3022 I know, I'm doing free fly with coarse fabric clothes (more drag) and RW with slick clothes (less drag). And head down with coarse fabric clothes is not 50-60 m/s. That velocity is for belly-flat position with everyday clothes.

[@nicks210684]

@@jide7765 yeah his issue was using an area of 1m2 which is far bigger than most people.

[@JohnDlugosz]

You neglected to consider that the person will open his parachute. His projected area and drag coefficient will go up by orders of magnitude at a certain altitude.

[@rulocastaguzman6310]

Great video, you should do one on the Lift Force now!

[@stephenbeck7222]

If you’re part of a school group then you can get a pretty cheap live demo of the first half of the video at an “iFly” indoor skydiving center, including hand calculating (approximating) your own terminal velocity and checking it personally by flying in the wind tunnel. Pretty fun.

[@whatitmeans]

About the experiment in the moon: if you pick a small feather (light item), and you left it over the cover of big thick book (heavy item), and then, you take the book on you hands horizontally to the Earth surface with the feather on top, and you left fall the book, since there is no air touching the feather, both objects will fall to the ground at the same speed, been both items always in contact with each other (i.e., you don't need to be in the moon to figure out things fall at same speeds if there is no friction)

[@davidpearman4792]

The slowest a skydiver can fall is about 110mph (roughly 50m/s) without extremely baggy jumpsuits. Doing less than 100mph would be very difficult. There might be a bit of error coming from projected area USPA B-51191

[@jide7765]

Exactly! As a skydiver, I can say that Vt = 50-60 m/s (180-210 km/h) for the "flat on belly" position. Head down, it's about 80-95 m/s (300-350 km/h) These are average numbers.

[@rwbishop]

​@@jide7765 Indeed! And Vt can be varied easily/dramatically by mere changes of body position.

[@EmilyMGin]

Loving the content on fluids!!! This type of stuff gets really complicated and boring in university so I'm astonished how entertaining this video is!

[@axelnils]

This would be high school physics though, right?

[@TomRocksMaths]

Glad you enjoyed it Emily :)

[@EmilyMGin]

@@axelnils Personally for me in Canada, I didn't get into the nitty gritty of drag until I took Fluid Mechanics in Uni! You're right that they introduce the concept of drag in high school though!

[@aestheticvibez.]

The intro was interesting

[@baoboumusic]

Oh it's Tom! Instant like. I've never been so excited about fluids - well, except IPAs maybe.

[@TomRocksMaths]

[@doubledarefan]

You overlooked 2 things: 1: The density of air decreases with altitude. When a falling object nears Vt, it then will only slow down. 2: Saltwater is denser than fresh water. Instead of the ocean, maybe use Lake Baikal for the hypothetical tower dunks.

[@ragnkja]

17:41 “The average density of the Pacific Ocean is 1036 kg/m^3” He also said that the difference in air density was small enough to ignore.

[@kedarjoshi1706]

The ball moves at a nice velocity!

[@OriginalPiMan]

I recommend giving indoor skydiving a try if you ever get the opportunity. All the falling sensation with none of the splat danger. My family all loved it, but I didn't. But I also don't regret trying it.

[@TomRocksMaths]

I have done this and can confirm it was great fun :)

[@SimonDoer]

Seeing how Tom used alot of approximations in this video (e.g. the projective area of the Eiffel tower), it would be interesting to see a video on fermi approximation.

[@TomRocksMaths]

So, who had the correct answer for the drag race?

[@aviralsood8141]

For some reason I guessed Eiffel Tower to be the fastest. Also wouldn't the buoyant force make the apparent mass of the less dense concrete ball lower than the two steel buildings and so make it fall slower?

[@samp-w7439]

13:19 Nice.

[@BluesJayPrince]

This is my favorite guy (I’m an aerospace engineer so I’m biased)

[@TomRocksMaths]

[@andrewhouchin5812]

As someone who's bungee jumped and skydived (attached to an instructor), I found it to be scarier to jump for bungee jumping. I think it was because I was all alone bungee jumping and I could see the ground much more directly. Skydiving just felt surreal.

[@General12th]

I bet buoyancy would have a big effect on things falling through water, since the density of water is in the same order of magnitude as the density of concrete and steel (while air is only a percent of a percent). In fact, since concrete is less dense than glass and steel, I bet taking buoyancy into account would make the wrecking ball _lose_ the race!

[@nytmare3448]

Nice Terminal velocity for the bowling ball

[@TheRealDoctorBonkus]

Aaaww yeah I love Tom Crawford!

[@Salien1999]

I knew exactly who the speaker would be before I even clicked. Can't find a guy more enthusiastic about fluid dynamics. 😂

[@TomRocksMaths]

[@KM683250]

I can't wait to share this with my physics class! But now I want a video that shows how the drag coefficient is calculated.

[@neruneri]

I love that this man exists. Science isn't just for posh old men, it's for everyone, and Tom is a great representation of the younger generation!

[@TomRocksMaths]

[@CompanionCube]

except he can‘t do basic fact checking, 133kmh is completely wrong for a 100kg human falling vertically

[@RFC-3514]

11:37 - If there's one object whose projection is *not* hard to visualise, it's a sphere. It looks like a circle from _any_ direction.

[@trevcam6892]

I think that the Empire State Building would have a higher terminal velocity because the windows would probably break and the building would fill up with water thereby expelling the air and reducing its natural buoyancy. Love this channel. Keeps my aged brain agile (maybe). I'd be interested in seeing how the drag equations can be used to explain how increasing speed of a land based vehicle results in a non linear increase in fuel consumption.

[@Ojisan642]

I realized at the end, the fact that your velocity keeps increasing is how an orbit works. But the velocity’s vector has an effect in that case. The acceleration isn’t in a straight line, it’s in an ellipse.

[@peterkelley6344]

Numberphile ought to ask Adam Savage if it would be acceptable to refer to Buster (the Mythbusters dummy) in examples like this. Tom Crawford always comes across like an Assistant or an intern in a math department. I think he deserves an appearance above that. He knows his math.

[@jamesnorlin1273]

Did you account for the buoyancy? That’s a lot of displaced volume for the Empire State building.

[@josephcote6120]

Would it be pedantic to ask if buoyancy is usually ignored? If the Empire State building was completely hollow and sealed, the lift for a big displacement would be pretty huge. It might even float.

[@Eylrid]

One way to think about the mass/area part of the equation: Think of the average column of material above a section of the bottom surface. That column is how much weight that section has to hold up. The width of the Empire State Building is so big that the column of mass over a given area is a lot more than either the bowling ball or the Eiffel Tower, or at least that was my guess. Running the numbers the bowling ball is 3358 kg/m^2, the Eiffel Tower is 12000 kg/m^2, and the Empire State building is 13035 kg/m^2. I'm honestly surprised that the Eiffel Tower and Empire State Building have such similar mass/area.

[@DoktorApe]

I saw that demonstration with the hammer and the feather on the moon on live television when I was a little kid. ;)

[@fugmopoly]

i love numberphile😌

[@ambassadorkees]

Without air, rho is zero, but once you travel fast enough, you get far from any body of mass, and you loose gravity. That's why you won't ever get even close to the speed of light.

[@garrettducat5769]

I love this video, but he's wrong! The empire state building would float. He didn't include the buoyant force in any of the equations. This is negligible with dense objects in air, but its actually really important when the fluid is water. The empire state building has a volume of ~2,250,000 m^3, and at 365,000,000 kg that's only a density of 162 kg/m^3. It would need to fill with water before it started to sink. After that it would sink at the rate he described. Also, the concrete wrecking ball would have appreciable buoyant force too since concrete is far less dense than steel. It would move through water at ~81% of the rate he describes, or 9 m/s. The Eiffel tower would not hold air and would win, but it would only move at ~93% the rate, or 11.2 m/s.

[@deinauge7894]

... however the empire state building would not sink with this speed when filled with water. same reason as with the other two objects. i suspect it is made of concrete, so same percentage less velocity as you calculated for the ball

[@eagl-3]

This Man is awesome.

[@TomRocksMaths]

[@hyperboloidofonesheet1036]

On the moon there is an atmosphere, but it is only about 100 molecules per cubic centimeter. At that density, I think the Reynold's number is so small that you're sitting well in laminar flow conditions, so the drag equation doesn't apply. And to echo everyone else: buoyancy matters -- the empire state building would float.

[@riccardo9297]

YES! Another video on fluid mechanics with Tom!

[@TomRocksMaths]

[@mrfmashups]

15:32 Ah yes my favourite building in the world, The Wrecking Ball

[@RagaarAshnod]

Can you do a video on drag coefficient (Cd)?

[@Drachenbauer]

A kite con only fly because of the drag. If it´s pulled through a fluid, the drag pushes it up. On the moon an astronaut can pull it as hard as they want, it will not take of from the ground and the sail becomes not even deformed into the typical wing-profile-curves between the sticks of the frame.

[@cortek123]

Followed the mathematics, but got lost when he said, "candy floss". Haha