“The Kinematics AOU Meta You Didn’t Know You Needed,” by quiescentire
Curator’s Note: One of my favorite kinds of meta, although I can’t claim to understand it well, is what I’m playfully calling STEM meta – meta that takes on the science and math of media worlds. In this wonderful piece, quiescentire interrogates the actual destructive force of Ultron’s threat to Earth in Marvel’s Age of Ultron.
I’ve seen AOU twice, soon to be thrice, and thoroughly enjoyed this utterly bananas movie. There are enough plot holes to drive a barge through, which I’m happy to ignore in the name of watching this clusterfuck unfold onscreen while laughing maniacally to myself, but I have one very large problem:
I can remain silent no more on the subject of wildly inaccurate falling-object damage projections.
I listened to Ultron monologue about his dastardly plan of dropping a rock onto the Earth in order to wipe out the human race like the Biblical flood and thought to myself: really? Because the reason celestial impact events are so destructive is that celestial objects (comets, meteors, asteroids, and other space junk) are going really, really fast when they hit the Earth.
For context, let’s talk about asteroids. Your average stony asteroid impacts the Earth at a speed of about 17 km/s — you read that right, 17 kilometres per SECOND [1]. Large asteroid impacts are so destructive because that’s a fuckload of kinetic energy — even a middling-sized asteroid, with a diameter of 7 metres, enters the Earth’s atmosphere with as much kinetic energy as an atomic bomb [2].
Ultron, however, is “just” dropping a chunk of Sokovia from 18,000 feet. How fast will that piece of rock be travelling when it hits the ground?
I needed to make several assumptions to complete this calculation.
- How big is this rock?
- Allow the chunk of Sokovia (“TCOS”) lifted by Ultron to be a cut-in-half-sphere with a diameter of 1 km. Yeah, yeah, it could be bigger or smaller or differently shaped, I know.
- The volume of TCOS is thus 261,799,388 cubic metres.
- For argument’s sake, let’s say TCOS is made entirely of granite, which has a density of about 2.75 grams per cubic centimetre or 2750 kilograms per cubic metre.
- The mass of TCOS is thus 7.19948 x 10^11 kilograms.
- How fast is this rock going to be travelling when it impacts the Earth?
- The first time I watched AOU, I thought that Ultron was just letting TCOS free-fall from above 18,000 feet or so. It took two more viewings to figure out what the hell was going on in that whole sequence, but let’s start out by discussing this basic scenario. More complex scenarios will be discussed in Part II, don’t worry.
- That means our starting velocity is 0 metres per second and we have 5486.4 metres to speed up.
- Let’s assume there’s no atmosphere because fuck calculating a drag coefficient over varying atmospheric densities during 6 kilometres of freefall, I’ll give the advantage to Ultron here if it means I don’t have to look this up.
- Use your kinematic equations. Final velocity = square root of (initial velocity^2 + 2*9.807 m/s^2 * 5486.4 m) = 328 m/s
This is fast, but NOWHERE NEAR the velocity of a for-real asteroid (remember: 17 km/s is = 17,000 m/s).
So, how much kinetic energy (KE) is TCOS carrying when it makes impact?
- Well, more kinematic equations: KE = 1/2 * mass * velocity^2
- TCOS thus has 0.5 x 7.19948 x 10^11 kg x (328 m/s)^2 = 3.87355 x 10^16 Joules of kinetic energy by the time it hits the ground.
- Converting this amount to the equivalent in TNT [3], this is 9.25 megatons of TNT.
Nearly 10 megatons of TNT? That sounds like a lot! It’s certainly more than atomic bombs in WWII. However, hydrogen bombs yield the equivalent of 25 – 100 megatons of TNT, and the total global nuclear arsenal contains the destructive capacity of about 7,000 megatons. A stony asteroid of “just” 100 metres in diameter has an impact energy of 38 megatons … and that size of asteroid impacts the Earth about every 5,000 years with minimal global effects [4]. Anatomically modern humans have been around for about 200,000 years [D], so we’ve seen 40 impacts of four times the size of Ultron’s with no appreciable consequences.
I’m not claiming that Ultron isn’t causing a significant amount of destruction, I’m merely pointing out that dropping a chunk of Sokovia onto the Earth via freefall from the lower atmosphere generates nowhere near enough kinetic energy needed to simulate the extinction-level impact event that Ultron claims he’s interested in.
For example, a stony asteroid of the same size as that chunk of Sokovia (1 km diameter) would release 46,300 megatons of energy upon impact and leave a crater 13.6 km wide [4]. And even that kind of impact even happens relatively frequently, around every 500,000 years or so — nothing like true mass extinction events, which happen on a frequency of many tens of millions of years [5]. The meteorite impact that happened at the end of the Cretaceous period, the one associated with the extinction of the dinosaurs, left a crater 180 km in diameter. Ultron’s big rock is kind of pitiful in comparison.
As a scientist, here’s my advice to Ultron. If you want to create an extinction-level impact event, you need to generate a force on the scale of teratons, not megatons. If you wanted to match the impact that created the Chicxulub crater that wiped out the dinosaurs then we’re talking 100 teratons (= 1 x 10^8 megatons) but IMHO even 1 teraton would be sufficient to kick humanity in the teeth. This means you need to increase your energy release by a factor of 100,000. You can do this by changing several of your variables:
- Use a larger rock. If your TCOS was 46 kilometres in diameter, you’d reach 1 teraton of kinetic energy release upon impact. This is a little unwieldy, sure, but the math works out.
- Drop the rock from a higher altitude. If you drop the original TCOS from 500,000 kilometres away, you’ll get close to 1 teraton of energy upon impact. For reference, the altitude of the International Space Station is 330 – 410 km, so this could be somewhat tricky. Plus I imagine the acceleration due to gravity would change over distance as the object approaches Earth, and I already stated I’m not doing more complicated calculations to take those sorts of things into account so fuck you very much.
- Abandon this ridiculous plan and go for the nuclear launch codes like you originally planned, or any number of other possible plans like really aggressive geoengineering [6] or perhaps a highly infections plague of nanobots. *call me
A FINAL NOTE: the moon is only 340,000 kilometres away, AND it’s way bigger than a chunk of Sokovia. Ultron, I appeal to your interest in supporting the mad sciences and urge you to knock the moon out of orbit and make it smack into the Earth. This will most definitely generate over a teraton of energy upon impact, AND has the additional bonus of an attack on the human psyche for the approximately ten seconds we’d all spend wondering what the hell was happening to the moon. Plus, the moon frankly has had it coming what with making the ocean slosh around weirdly (a.k.a. “tides”) and lighting up my room at night when I’m trying to sleep.
In conclusion: I can accept a lot of things, but the idea that dropping a big rock from from the lower atmosphere [7] onto the Earth’s surface could have even a small chance of wiping out humanity is just a failure at math and basic kinematics. Ultron, I really thought you’d be better than that.
[1] http://en.wikipedia.org/wiki/Impact_event
[2] http://en.wikipedia.org/wiki/Little_Boy
[3] http://en.wikipedia.org/wiki/TNT_equivalent
[4] http://impact.ese.ic.ac.uk/ImpactEffects/
[5] http://en.wikipedia.org/wiki/Extinction_event#Patterns_in_frequency
[6] http://en.wikipedia.org/wiki/Stratospheric_sulfate_aerosols_%28geoengineering%29
[7] Yes, 20,000 feet is still well within the troposphere: http://www.srh.noaa.gov/srh/jetstream/atmos/layers.htm
The most frequently-asked questions about my original post were:
- Why did you pick 18,000 feet for the maximum altitude?
- I thought that chunk of Sokovia (TCOS) had a vibranium core?
- Didn’t Ultron plan to make the engines inside TCOS reverse thrust at some point, so it wouldn’t just free-fall, it would actually accelerate towards the Earth like a downward-pointing rocket?
And of course these are all totally on point!
So, first things first: it took me until my third viewing, when I was really focused on picking out these exact details, to really figure out what the hell was going on in that final action sequence. I’ve done my best to use as accurate numbers as possible, but as with all modelling exercises, “garbage in – garbage out”.
- I picked 18,000 feet because Maria Hill says “18,000 feet and climbing” at one point in the final action sequence (I actually had used 20,000 feet after my first viewing, but I’ve tightened that up now). It’s unclear what altitude TCOS actually reaches in the movie, but the fact that not only superheroes but also civilians are still running around performing such activities as “fighting robots” and “breathing” rather than laying unmoving on the ground suggests they are not really very high up. Anything above 6000 m (19,685 feet) is considered an extreme altitude for humans [A], and ascending rapidly from low altitudes is a surefire way to make yourself very sick and/or die.
So it appeared as if Ultron did not get TCOS up very high. I don’t think he evil-monologued about exactly how high up he had originally planned to send his TCOS before dropping it; however, even if Ultron dropped the rock from 60,000 feet (18,288 metres) – the highest certified altitude of a Concorde jet, and over twice as high as the 8000m mountaineering “death zone” that is fatal to humans – you still only end up with 28 megatons of kinetic energy upon impact. It’s just not anywhere near enough for a mass extinction event … not even if he drops it from the goddamn moon.
- I was under the impression that vibranium was super-light and super-strong, like magic titanium. If it’s not a dense (heavy) material, then it’s not going to help much. And I also thought that the world’s vibranium reserves were quite limited – in the first Cap movie, Howard Stark says he used up all of the vibranium in the world to make Steve’s shield back in 1940-whatever, so presumably there’s not much of the stuff kicking around. Wakanda has a monopoly over its production, and it sounded like Ultron wanted to take it from Klaue to build himself a new super-strong body, not to jam into Sokovia all willy-nilly.
But let’s assume that Ultron thinks to add any dense, heavy material – iron or lead or something – to his big rock in case the vibranium doesn’t work out. Even then, given the scale of the object’s mass we’re talking about here, adding even a million metric tonnes of extra stuff (which I imagine is far more vibranium than exists in the world) doesn’t actually matter that much. Another million tons of “stuff” adds only 0.139% more mass than the original 1 km diameter chunk of Sokovia, because that chunk of Sokovia is already so enormously big to begin with. And because the relationship between kinetic energy and mass is linear (rather than a squared function, as with velocity) it would increase kinetic energy by a similarly paltry amount. Remember, we would have needed to use a chunk of Sokovia that was 46 km in diameter to approach 1 teraton of kinetic energy upon impact from 18,000 feet.
- YES and it took me until the third go-round to figure this out! This point, more than any other, gave me pause. Just how much additional acceleration are we talking about here? What’s the best-case scenario? How can we make Ultron’s dumb plan actually work?
A few thoughts:
– Ultron didn’t intend to drop the rock from a mere 18,000 feet, so let’s give him the benefit of the doubt and put the thing way, way up at 410 km altitude, which I’m choosing because that’s the most distant orbit of the International Space Station. I think we can all agree that that’s pretty high.
– Ultron’s vibranium engine appears to provide thrust without using up any fuel (???), so we’ll ignore the changes in mass that would result from burning most of your fuel payload in order to reach your maximum altitude, prior to “reversing thrust” and sending that thing downwards.
– Let’s still assume there’s no atmosphere, because I’ll be damned if I’m going to calculate the drag coefficient on an object moving at accelerating speed through a changing atmospheric density, my poor laptop would explode.
Through the magic of algebra, we can calculate how much additional acceleration would be necessary to create the kind of impact cause a mass extinction. Running with our 1 teraton threshold for mass extinction, a 1 km diameter half-sphere of granite would require an additional acceleration of 14,300 m/s^2 … on top of Earth’s gravity’s contribution of 9.807 m/s^2. Thanks, gravity. It’s like you’re not even trying.
At this velocity profile, TCOS’s descent would last 7.57 seconds. Covering 410 km in less than 8 seconds … now you’re talking real meteor speeds! The original freefall from 18,000 feet would take 33.45 seconds.
Let’s convert this to equivalents in g-force because what the fuck a metre per second squared actually feels like is just a string of question marks in my head.
For comparison [B]:
– When you stand on the Earth at sea level, you’re experiencing 1 g. Congratulations, how wonderful.
– The space shuttle maximum during launch and re-entry is 3 g, or 29 m/s^2
– Apollo 16 reached 7.19 g upon re-entry, or 70.6 m/s^2
– The fatal threshold for humans is ~100 g, or 982 m/s^2 — this is the equivalent of crashing your car into a wall at 100 km/hr, which is a rather rapid deceleration
For Ultron’s plan to work, we need to sustain a constant 14,300 m/s^2 acceleration, which is approximately 1458 g-units. That’s a lot — a lot, a lot, a lot. Some things do reach those very large accelerations but they are all either very small objects, and/or that acceleration lasts for only a fraction of a second: the maximum acceleration of a piston firing on a formula one car engine, for example, or the strike of a jellyfish stinger. Nothing like moving a 1 km diameter chunk of granite through hundreds of kilometres of atmosphere, sustained for several seconds.
So Ultron’s magic vibranium engine would have to produce an insanely large acceleration — over 1400 times more than that of Earth’s gravity — and also be shooting in from orbit in order for this plan to work. Is it a particularly plausible doomsday device? Ehh … I remain unconvinced.
[A] http://en.wikipedia.org/wiki/Altitude_sickness#Definitions
[B] http://en.wikipedia.org/wiki/Orders_of_magnitude_(acceleration)
“The Kinematics AOU Meta You Didn’t Know You Needed,”©quiescentire, Part I originally posted on 4 May 2015
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Thank you for this.
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