Help with terminal velocity of a floating city?

JenniferLee · Apr 13, 2016

Hello,

So I have a city in my sci-fi novel. It's shaped like a metal washer, with a total diameter about 4.8 km and the inner "hole" diameter about 1.5 km, and it's floating about 13 km above ground with the aid of handwaving and antigravity.

I want to drop it. And I'd like some help figuring out how long it would take for it to hit the ground.

I know the terminal velocity formula is v = the square root of ((2*m*g)/(ρ*A*C)) but I can't seem to find the drag coefficient of a washer (or flattened bagel or... I don't even know the name of the shape), and mathematics has never been my friend, so I was wondering if someone could give me a hand?

I don't need an exact time, but a general idea would be nice. I don't know if it would take a minute, several minutes...

The weight of the city is approximately 700000000 kg. Assume earth gravity and atmosphere (I don't know where exactly earth air starts to thin, but they've done extensive terraforming and atmosphere alterations, so it shouldn't be an issue here). Also assume that the city wouldn't flip over or wobble too much. The person dropping it wants to scare the people on it, not send them flying off the city.

Thank you so much for any help. I've been trying to figure this out for weeks.

Aerial · Apr 13, 2016

Just to address the "where the air thins" comment: At 13km (~8 miles) up you're at the very top of the altitude range commercial jets cruise at (a little above, really). There's very little oxygen at that altitude and cabin depressurization leads to unconsciousness in 30 seconds or less. Commercial aircraft cabins are pressurized to approximately 8,000 ft altitude, 6,000 ft if they're going for more passenger comfort. So whatever your floating city's technology, it's going to have to have some kind of pressure vessel to keep enough pressurized atmosphere in the city for people to breathe.

You may have more luck finding a drag coefficient if your city is torus-shaped or (half) spherical, which is probably required due to the need to have a pressurized atmosphere around it.

morngnstar · Apr 13, 2016

There's not that much variation in drag coefficient. If you assume C = 1 you'll still get an order-of-magnitude approximation.

cmhbob · Apr 13, 2016

I bet Stark still has the numbers from when Novi Grad fell.

"Dropping" something that big will cause people to go flying, though. Even in Age of Ultron, when the city falls, you see some people float for a second or three. If he's going to drop something, but not all the way, that sudden stop is going to cause some structural issues with buildings and such

King Neptune · Apr 13, 2016

Another concern is the angle at which it will fall. If it manages to remain perfectly level, then you'll get one answer, but if it is at any other angle it will tend to have significant horizontal travel. There's only one way to know for sure, but I don't think anyone has a spare floating city to use in the experiment.

morngnstar · Apr 13, 2016

Possibly it will tilt until it's sideways, but not stop there and flip, and tumble over and over.

Another issue is it might never reach terminal velocity, or not for most of its fall.

It all depends on the time scale for each of the motions. Will it reach the ground or terminal velocity first? Will it tip over or reach the ground first? It's hard to have an intuition at these scales.

blacbird · Apr 13, 2016

With a gravitational attraction like that of Earth, something solid, with the density typical of construction material (concrete, steel, stone) weighing 700 million kilograms isn't going to encounter any significant air resistance.

Just don't be under it when you time it.

caw

frimble3 · Apr 13, 2016

Am I missing something, or unless there's really specific (and really fast) antigravity, isn't the sudden stop at the end of a 13 km fall going to kill pretty much everybody aboard the city? ie people falling off buildings, cars driving off cliffs, skydivers with 'chute failure.

morngnstar · Apr 13, 2016

I did a back-of-the envelope and found about 8 min. using rho = 1.2 and C = 1. That's the density for Earth at sea level. Using rho = 0.36 for 11km altitude gives 4.5 min. The actual time would be a weighted average, which is quite complex because the density varies by a complex formula, therefore the velocity varies by an even more complex formula, as well as the complication of spending different amounts of time at each elevation. But in any case the result would be somewhere within that range, assuming that drag coefficient. It could be slightly less or more with a different coefficient, but it's minutes, regardless.

Free-fall time would be 51 seconds, whereas at freefall it would exceed terminal velocity in 3 to 6 seconds, so it will reach close to terminal velocity and drag does matter. Drag always matters if you go fast enough.

morngnstar · Apr 13, 2016

frimble3 said:
Am I missing something, or unless there's really specific (and really fast) antigravity, isn't the sudden stop at the end of a 13 km fall going to kill pretty much everybody aboard the city? ie people falling off buildings, cars driving off cliffs, skydivers with 'chute failure.

So don't stop it suddenly.

morngnstar · Apr 13, 2016

I checked your mass figure and this works out to a density per unit area of 4.3 grams per square centimeter. That means if your city were made of, say, titanium (density 4.5 grams per cubic centimeter), it could only be one centimeter thick. Even if I grant that you might have some advanced material that could be lighter and / or stronger, I would think the humans might want to pile up some stuff on top of the city that would add up to more than 4 grams on every square centimeter. Are you sure your mass figure is reasonable?

King Neptune · Apr 13, 2016

The actual mass per unit of downward facing area would be very significant. If the mass is small, then the ciry would drift slowly like a leaf after it got to the thick atmosphere. It would be the difference between dropping a book and dropping a single sheet of paper.

WeaselFire · Apr 14, 2016

JenniferLee said:
... they've done extensive terraforming and atmosphere alterations ...

Then you could likely change the speed at which your city drops to whatever you need for your story. The other posts are correct, given your parameters, a 5-8 minute drop would end with a splat. And, your mass of a city that size is woefully light. Plus, it is very unlikely to fall in perfectly horizontal stance, tipping and sliding would be expected.

But then, a floating city like this would not normally be expected either, so feel free to tweak to your needs.

Jeff

JenniferLee · Apr 16, 2016

Okay, so first off, apologies for posting the question and never coming back; I joined here and immediately forgot my password for a while.
So, thank you all for the input, and an 8 minute fall time is a helpful estimate.
Yes, I realize the oxygen levels and the issue of whether it will fall straight and all the problems on the city when this happens. These are all made more or less moot by my novel's worldbuilding and technology. It's a space opera, soft sci-fi, bordering on fantasy kind of thing. "Artificial gravity" is a term my characters throw around a lot, and in the next draft I will nail down exactly how it all works, but in a nutshell yes, a very thin plate is capable of supporting the city.
And as for the damage of stopping fast, the goal is of course to stop it very slowly. Though it would be interesting if none of my characters realize that...

Many thanks.

cmhbob · Apr 16, 2016

Something else to consider.

You suggested your city weighs about 700,000,000 kg. That works out to 771,617 US tons.

For perspective, after 9/11, the Fresh Kills landfill received 1.6 million US tons of debris from the World Trade Centers, or about twice the weight of your city (1,451,495,584 kg). Not sure what kind of wrench that throws into your equations. I know you said soft sci-fi, but I just thought I'd give you some perspective for size.

King Neptune · Apr 16, 2016

One image that came to mind when I thought about this city was that of aerostats in Orion Shall Rise by Poul Anderson. I can't remember the details, by it was lighter than air, but that depended on its altitude.

Help with terminal velocity of a floating city?

JenniferLee

Aerial

There is no spoon.

morngnstar

cmhbob

Did...did I do that?

King Neptune

morngnstar

blacbird

frimble3

Heckuva good sport

morngnstar

morngnstar

morngnstar

King Neptune

WeaselFire

Benefactor Member

JenniferLee

cmhbob

Did...did I do that?

King Neptune