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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
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?
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.
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?
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.
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
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.
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. 😆
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.
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.
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.
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.
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.
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.
@@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.
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.
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?
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.
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.
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 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.
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?
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.
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.
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.
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!
@23:31, well actually, even deep space isn't a perfect vacuum so there is stil drag on the moon, in insterstellar and in intergalactic space. Interplanetary space has a density of a few particles per cm3 so it's so small that it will not matter for human height drop but technically, the density is not zero on the moon ;) Although I didn't do the math but I suspect that any object would reach relativistic speeds before this drag becomes of any relevance (except if you're using a sail or something shaped like one and light enough) so this drag equation will not apply but one would need another one to do the calculations.
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.
@@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.
@@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.
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.
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.
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.
@@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!)
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.
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.
Since they are all steel and concrete I don't imagine there will be much difference. All the masses will be replaced by apparent masses which will still be in approximately the same ratio. Will have to run the calculations to be sure though.
According to what I saw in wikipedia, the density of steel (about 7800 - 8000 kg/m3) is much higher than the density of concrete (between 2000 and 2600, depending on the type of concrete). Therefore, a sphere of about 2300kg medium-heavy concrete would have a volume of 1 m3, and thus a buoyancy of 1000kg in water, resulting in an "apparent weight" of 1300kg, which leads to a reduction of the terminal speed by the factor SQRT(1300/2300) = 0.75 ... a reduction of about 25 percent, whereas the terminal speed of a pure steel object (like the Eiffel tower) would be slowed down only by about 6.5 percent ( = 1-sqrt(7/8) )
@@tobyk.4911 Yeah you're right, I double checked. Also I think that the higher viscosity of water than air might reduce some of the hole-induced air drag of the Eiffel Tower, but I don't know much about fluids. Does a turbulent flow equivalent of viscosity exist, and what effect does it have on drag?
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
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.
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)
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."
@@DavidRichfield agree, there should be a dependency, but from the video above I can hear that drag coefficient only depend on the object, but not the media
@@ovsds he already mentioned that the drag coefficient depends on the density of the medium, and around 02:00 he mentions that it depends on the Reynolds number, which is ρ.v.D/μ where μ is the viscosity.
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.
From those figures, shouldn’t the Empire State float? Also, how does the far lower Re in water after the drag coefficients? I’m guessing the quoted values are for similar velocities in *air*.
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.
What about object density? I mean, what if you fully seal the building and take out all the air, hipotetically up to a density lower than water, it will never sink. How is that included on the equations? Another example, what if that ball has lower density than air? It will not fall, it will raise (gain altitude), like helium globes, etc. What i mean is how relative 'negative' density is considered! In other words, how can be raise velocity be calculated (negative fall velocity).
The ratio of a falling object’s velocity to its terminal velocity varies with time according to a hyperbolic tangent function. Specifically V = tanh(tg/v*) Where v* terminal velocity and V = v/v*
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.
@@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!
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.
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!
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.
Something doesn't add up I'm afraid. Skydivers in the classic belly to earth position (like the one in the video) drop at around 120 mph or around 55m/s, which is a significantly different figure from what you calculate in the video (I won't spoil it for anyone who hasn't seen it). Obviously things like height and weight make a difference, with tall light people slower than short heavy people.
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.
Hey Tom. Imagine a car driving on a freeway (thus, turbulent flow) without other vehicles in front. Will the Drag Coefficient be the same if the car drives at 30 km/h, 90 km/h or 120 km/h? I assume not, but I don't know how to demonstrate it. Will it change by a lot, or will the change be irrelevant? Does the amount by which it changes starts to be irrelevant after a certain speed is reached?
Why would the drag coefficient change? The shape is the same. Wouldn't only the total drag change essentially in proportion to velocity? I'm not a physics surgeon, so quite possible that's all wrong.
Well, try to invert the Drag formula and isolate the C_d. You will see that it depends on F_d/ v². So, what happens when v and F_d change? I've always wondered that, but found no definite answer.
@@andreastedile But isn't Fd generally going to change roughly in proportion to v²? I don't doubt that Cd changes to some degree due to aerodynamic effects but my guess would be not significantly unless it is like an airplane, with changing aerodynamic characteristics depending on velocity, but some of that is also related to the cross-sectional area changing. I dunno. Seems more complicated the more I think about it. You have me curious now.
@@andreastedile The V in the equation stands for Terminal velocity only. 30 km/h is not the terminal velocity of a car being dropped through the air and accelerating under gravity, it is an arbitrary speed that you have decided to accelerate the car up to using the engine. EDIT: the top speed of an F1 car depends on air resistance, and if the acceleration of the engine is constant, you might get a nice estimate for the top speed by using the engine acceleration in place of g
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.
Wouldn't the drag being uneven across the structure make them come down more like a dart if they fell long enough? Would that be enough to affect the outcome?
I need to say that "A" isn't per definition the frontal area: in aerospace it's also typically the "top" surface area (ie for wings in an aircraft). Important thing is that definition of A is always the same, and it combines with Cd to define the shape.
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Please help me
My subscribers are stoped increasing 😭😭😭😭
Dr use
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
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?
Is it nescessairy to fill even thís little piece of public space with an ad? Please, where in is nót an ad? Happy 2021 i have to add.
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.
Yes water is much more dense than air
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?
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.
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
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.
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. 😆
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.
OK. Well... I’m glad you turned out to be fine lol!
??
Shout out to Brady for the amazing perfectly timed graphics, they really add to Tom's explanation in this video.
Thanks!
@@pmcpartlan Fair play if this is you! Great work!!
@@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
Pete's the man.
@@pmcpartlan Fair play not cheeky at all. I think it really takes the videos to the next level.
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
false.
I love Tom Crowford's videos. The enthusiasm and energy he puts on his explanations are infectious. Looking forward to seeing the next one!
It would be great if it was a science. But it is a fake science and results are wrong
We really need to redefine fluid so that cats aren't fluids. Let's say: "fluids forms into containers but cannot be a cat."
cats aren't fluids?
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.
@@DrunkTortilla so, a cat?
@@douglaspantz i believe so
Cats are scientifically considered a semi fluid
This man is equally dedicated to math and Pokemon 😅
I noticed his tattoos too
and SpongeBob
more like physics than math
Currently so am I
I think you meant Meth!!?
Since the Empire State Building is partially hollow, would the buoyancy have any appreciable effect?
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.
@@markc7899 Or you could just decide to treat it as one solid object and ignore gasses, open windows, and whatever else.
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.
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.
It's important to clarify that as long as an object is more dense than water there's no such thing as "bouyancy"
they really "imma compare a bowling ball and an absolute unit of a human"
“Objects of different mass fall a the sa-“
Drag: “I’m gonna stop you right there”
Also inverse square nature gravitational attraction.
... fall at the same speed in a vacuum.
Every physics teacher: but we'll just ignore that for now
false.
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.
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.
Well we are taking mass into account
@ His ornate garments would work like a wing.
@@mortisCZ So not very hol(e)y? I knew it, he's a scammer.
13:04 as a skydiver, I can say that Vt = 50m/s (average) not 37m/s
(flat on belly position)
So wt do u think would change in the values he gave
@@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
@@mishal0404
Surface and drag coefficient.
Knees are bent in a lazy W position.
other comments are right. the way he put the "nice numbers" is not an averge human description.
I love it when Tom's on! He's so enthusiastic and its inspiring to see a young person every once in a while
No human was harmed while calculating the terminal velocity.
While? No. After? Maybe.
Well, it might have made a few humans’ brain hurt.
@@geoffstrickler haha😁
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.
23:18 "There's no air." There is though. It's just very thin.
Even the entire universe has a density (I think it's about 1x10^-27kg/m^3) which is virtually negligible but still not nothing
What's it composed of then? And what pressure? At average ground level (gravitationally).
@@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.
It sounded like "there is no eh...".
@@RebelKeithy that's still surprisingly heavy!
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.
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?
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.
Happy new year, Brady. Thanks for the awesome content!
I love it when ya bring in tom
loved it!
thanks!
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.
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.
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?
@@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.
@@SanyaJuutilainen So then can we calculate terminal velocity, like the OP assured?
@@greggregoryst7126 We can. It will be very large, but we can :)
So, who had the correct answer for the drag race?
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?
2:55 Does the fact that the cube turned diagonally presents a larger projected area, cancel out the gains from the lower Cd?
Adding bouncy to the latter example might’ve been an interesting take too. Would if I’d have a significant effect
was going to make the same comment. buoyancy is a big factor especially in dense fluids like water.
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.
Now I wanna see the Empire State Building make a descent into the Mariana Trench.
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.
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.
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!
@23:31, well actually, even deep space isn't a perfect vacuum so there is stil drag on the moon, in insterstellar and in intergalactic space. Interplanetary space has a density of a few particles per cm3 so it's so small that it will not matter for human height drop but technically, the density is not zero on the moon ;) Although I didn't do the math but I suspect that any object would reach relativistic speeds before this drag becomes of any relevance (except if you're using a sail or something shaped like one and light enough) so this drag equation will not apply but one would need another one to do the calculations.
During my college days i found it difficult to understand but the way you explain the thing makes it lucid to understand.
:)
Maximum velocity for head-first dive is 50-60 m/s. It depend on type of clothes.
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.
@@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.
@@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.
@@jide7765 yeah his issue was using an area of 1m2 which is far bigger than most people.
Tom Crawford videos are the best in a very high quality feed!
I was lost after 5 minutes but you are all excellent, engaging teachers. Schoolkids who love maths would adore you 😍😍
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.
I love videos with Tom he is epic!
The frontal area of the Eiffel Tower 750m2? Wouldn't the multiple steel beams one behind the other add to frontal surface?
Another problem: the density of the Empire State is 343 kg/m3. It would float.
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.
I could watch this video again and again. Wow informative.
And wrong
No-one:
Engineers: yo what's the drag coefficient of the eiffel tower
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.
Brady+Tom go sky diving, next on Numberphile! I'd love that ...
I'm in if Brady is?
@@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!)
Aristotles: heavy things fall faster
Me in grade school: fair enough
Me in junior high/high school: dumbass
Me in uni: fair enough
Density ;)
@@brycering5989 Yeah, that just about sums it up.
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.
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.
How big of an effect would buoyancy have on the falling through water case?
Since they are all steel and concrete I don't imagine there will be much difference. All the masses will be replaced by apparent masses which will still be in approximately the same ratio. Will have to run the calculations to be sure though.
According to what I saw in wikipedia, the density of steel (about 7800 - 8000 kg/m3) is much higher than the density of concrete (between 2000 and 2600, depending on the type of concrete). Therefore, a sphere of about 2300kg medium-heavy concrete would have a volume of 1 m3, and thus a buoyancy of 1000kg in water, resulting in an "apparent weight" of 1300kg, which leads to a reduction of the terminal speed by the factor SQRT(1300/2300) = 0.75 ... a reduction of about 25 percent, whereas the terminal speed of a pure steel object (like the Eiffel tower) would be slowed down only by about 6.5 percent ( = 1-sqrt(7/8) )
@@tobyk.4911 Yeah you're right, I double checked. Also I think that the higher viscosity of water than air might reduce some of the hole-induced air drag of the Eiffel Tower, but I don't know much about fluids. Does a turbulent flow equivalent of viscosity exist, and what effect does it have on drag?
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
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.
@@jide7765 Indeed! And Vt can be varied easily/dramatically by mere changes of body position.
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.
Does this mean I'd fall faster because I have dimples?
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)
This is my favorite guy (I’m an aerospace engineer so I’m biased)
I was thinking of watching Netflix and ended up here and enjoyed this video way too much than I should have.
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."
To visualize the 2D shape of an object, you can think of the shadow it would cast. Hence the ball casting a circle.
4:33 . . . We have been driving boats the wrong way through water since the beginning of time. The "pointy end" should be the stern!
2:10 - when the hip kid in the street turns out to be a mathematics professor
but Empire State Building has air stuck inside, how boiance would interfere there.
just open the windows
Great episode ! Thank you to the both of you.
But what's about viscosity, did I miss any assumptions? Pretty much sure density is not the only medium parameter that affects.
The drag coefficient depends on the Reynolds number, which depends on the viscosity.
@@DavidRichfield agree, there should be a dependency, but from the video above I can hear that drag coefficient only depend on the object, but not the media
@@ovsds he already mentioned that the drag coefficient depends on the density of the medium, and around 02:00 he mentions that it depends on the Reynolds number, which is ρ.v.D/μ where μ is the viscosity.
Awesome - thanks for the great entertainment :-) That was very interesting and enjoyable to watch.
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).
the moment I saw the thumbnail I new it was gonna be Tom
If I dropped multiple objects of consistent weight from a height of 100 meters what shape would fall the slowest?
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.
I have done this and can confirm it was great fun :)
I really like this guy!
From those figures, shouldn’t the Empire State float? Also, how does the far lower Re in water after the drag coefficients? I’m guessing the quoted values are for similar velocities in *air*.
I'm guessing the building is flooded from the inside.
Oh it's Tom! Instant like. I've never been so excited about fluids - well, except IPAs maybe.
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.
What about object density? I mean, what if you fully seal the building and take out all the air, hipotetically up to a density lower than water, it will never sink. How is that included on the equations?
Another example, what if that ball has lower density than air? It will not fall, it will raise (gain altitude), like helium globes, etc.
What i mean is how relative 'negative' density is considered! In other words, how can be raise velocity be calculated (negative fall velocity).
How did I just KNOW Crawford was gonna be the guest for this topic lol
The ratio of a falling object’s velocity to its terminal velocity varies with time according to a hyperbolic tangent function.
Specifically V = tanh(tg/v*)
Where v* terminal velocity and V = v/v*
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.
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!
This would be high school physics though, right?
Glad you enjoyed it Emily :)
@@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!
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.
Who else knew it was gonna be Tom Crawford after reading the title?
13:22 nice
I agree
nice.
Brady's excitement about the drag race is epic.
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!
15:32 Ah yes my favourite building in the world, The Wrecking Ball
That bowling ball terminal velocity is *nice*
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.
Something doesn't add up I'm afraid.
Skydivers in the classic belly to earth position (like the one in the video) drop at around 120 mph or around 55m/s, which is a significantly different figure from what you calculate in the video (I won't spoil it for anyone who hasn't seen it). Obviously things like height and weight make a difference, with tall light people slower than short heavy people.
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.
YES! Another video on fluid mechanics with Tom!
Hey Tom. Imagine a car driving on a freeway (thus, turbulent flow) without other vehicles in front. Will the Drag Coefficient be the same if the car drives at 30 km/h, 90 km/h or 120 km/h? I assume not, but I don't know how to demonstrate it. Will it change by a lot, or will the change be irrelevant? Does the amount by which it changes starts to be irrelevant after a certain speed is reached?
Why would the drag coefficient change? The shape is the same. Wouldn't only the total drag change essentially in proportion to velocity? I'm not a physics surgeon, so quite possible that's all wrong.
Well, try to invert the Drag formula and isolate the C_d. You will see that it depends on F_d/ v². So, what happens when v and F_d change? I've always wondered that, but found no definite answer.
Read wikipedia about the Drag coefficient, it raises a similar question
@@andreastedile But isn't Fd generally going to change roughly in proportion to v²? I don't doubt that Cd changes to some degree due to aerodynamic effects but my guess would be not significantly unless it is like an airplane, with changing aerodynamic characteristics depending on velocity, but some of that is also related to the cross-sectional area changing. I dunno. Seems more complicated the more I think about it. You have me curious now.
@@andreastedile The V in the equation stands for Terminal velocity only. 30 km/h is not the terminal velocity of a car being dropped through the air and accelerating under gravity, it is an arbitrary speed that you have decided to accelerate the car up to using the engine.
EDIT: the top speed of an F1 car depends on air resistance, and if the acceleration of the engine is constant, you might get a nice estimate for the top speed by using the engine acceleration in place of g
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.
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.
23:50 Regardless of mass, hmm. So, a photon in space and a gravitational field would accelerate past the speed of light? What am I missing?
So this is how SpaceX work our Max Q?
How do they do it since air density/drag constantly changes as they rise?
This Man is awesome.
Loving the animation!
Wouldn't the drag being uneven across the structure make them come down more like a dart if they fell long enough?
Would that be enough to affect the outcome?
I need to say that "A" isn't per definition the frontal area: in aerospace it's also typically the "top" surface area (ie for wings in an aircraft). Important thing is that definition of A is always the same, and it combines with Cd to define the shape.
Wow, a new numberphile video!
Nice Terminal velocity for the bowling ball
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!
except he can‘t do basic fact checking, 133kmh is completely wrong for a 100kg human falling vertically
Followed the mathematics, but got lost when he said, "candy floss". Haha
Nobody expected that the Holiness of the Eiffel tower matters this much :)