I don’t think it’s a spoiler to say that the machines are trying to take over in the Netflix show The Mitchells vs. the Machines. I mean, there’s obviously some type of conflict. But in case you haven’t seen it yet, here is your official warning: I’m going to use the movie to do some fun estimation problems about the robot apocalypse.
Maybe I’ve already said too much.
You’re still here? Great.
Let’s get right to the important stuff: The machines have decided that they would be better off without all those pesky humans, so they are gathering up all the people and putting them into seven giant 128-story rockets. Their plan is to use small flying pods to pick up each of the approximately 8 billion humans on Earth and bring them to one of these rockets. That leads us to the Mitchells. The Mitchells are just your normal everyday family with everyday problems—except they are pretty much the last free humans on Earth, and they have to save everyone. Other than that, they are just like the rest of us.
This gives us some questions to answer.
How Long Would It Take to Gather All the Humans?
I’m not a robot trying to take over the world. But if I was, I might decide to put one of these giant rockets on each continent. I mean, there are seven rockets and seven continents—right?
But the number of people in Antarctica is basically just a rounding error compared to the populations of India or China. So let’s use population density instead.
Here is where I am going to put the rockets. I haven’t optimized the locations; these are just rough guesses:
Of course this is a 2D projection of the surface of the Earth (and not the actual surface of the Earth). This means that some of the distances and sizes are distorted. Still, I think I have roughly minimized the distance a human must travel to get to a rocket. You could create an algorithm using the population density to find the seven best locations—but I will leave that up to real robots.
You can’t launch your rocket until all the humans are on board. That means that you have to wait for the people who are the farthest away to travel to the launch site. Looking at the rocket location in North America, I’m going to approximate that people from as far as 3,000 kilometers away will be traveling to the site. (People in Alaska are further away than that, but they might be able to get to the sites in Asia.) With this distance, I just need the travel speed to calculate the time.
In case you haven’t yet seen the movie (and you should), the machines use some type of hexagon-shaped force field pods to carry the humans. (Don’t worry about those humans; they are fine. Each pod has Wi-Fi, so everything’s good.) Just as a guess, the height of a pod looks to be around 1.5 meters tall. But how fast do the pods move?
I found a quick scene that shows the motion of a pod from the side view. If I assume it’s 1.5 meters tall, I can use video analysis to mark the location of the pod in each frame. From that position data and the time data (from the frame rate), I get a speed of 14.5 meters per second. Let’s just go with 15 m/s (about 33 mph).
With that, I can calculate the time it takes for the last person to board a rocket bound for nowhere (according to the machines).
That’s just 55 hours. Not too bad. This is a rough estimation, but it couldn’t be too wrong. Let’s say the person who lives the farthest away has to travel 4,000 kilometers instead, and at a slower speed of just 10 m/s. That would still take only 111 hours, still less than a week of travel time. At least this delay gives the Mitchells time to figure out a plan to save the world.
How Big Would These Rockets Be?
I’m not sure what year this movie is supposed to take place, but it seems like it’s in modern times. So let’s say there are still 8 billion humans on the planet who are evenly distributed among the seven rocket sites. That means that each rocket would carry 1.14 billion people—each in their own hexagonal pod.
Based on what we see in the movie, there are two design constraints for this rocket. First, they have to be 128 “stories” tall. (I mean, they said that right in the movie.) The second is that they are sort of in the shape of a V. Each one consists of a vertical wall along with another wall at a 33 degree angle. Here is a sketch:
Since we know the height is 128 stories, and one story in a typical building equals 3.3 meters, then I can find the number of hex pods that will stack up in the inverted V. But that leaves a question: How many layers of hex pods would you need for all 1.14 billion pods?
OK, let’s get to this. I will start with the side area of the V. I’m going to assume it’s a right triangle and the vertical side is 128 x 3.3 = 422.4 meters tall, with a vertex of 33 degrees. Now I just need to find the total side area of the rocket V and divide by the area of each hex pod to find the number that fit in one vertical layer. After that, I can find the number of layers needed for all those people.
Since you might want to use your own estimated values, I put it all in Python. Here are my results. (Yes, you can click the pencil and change the values to make yourself happy.)
But here you see a problem: That 128-story rocket would have to have a width of 38 kilometers. That’s 23.6 miles! The robots sure build weird-looking fat rockets. Oh well, I guess they know best. They are the machines, after all.
How Much Energy Would It Take to Launch Them?
Now you’ve got this giant rocket full of people and hex pods. You want to send it into space, never to return. How much energy would that take?
The robots obviously aren’t going to use rockets with chemical fuel—that’s so barbaric only a human would do it. But either way, I can calculate the energy needed. There are two important parts of this calculation: What is the mass of each V-rocket, and how fast does it need to go?
First, let’s take the mass. I’ll use a rough estimate that most of the mass is due to all those humans. (It’s possible that the pods are just some type of force field with no mass, and that the structure of the rocket itself is super efficient and has negligible mass compared to the people.) So, if there are 1.14 billion people per rocket and their average mass is 75 kilograms, that’s a total mass of 8.55 x 1010 kilograms.
If the robots want these humans to never come back, they would need to launch the rocket with a velocity equal to the escape velocity. Here is my more detailed explanation of escape velocity—but essentially it’s the minimum speed needed to reach an infinite distance from the Earth. Using the mass and radius of the Earth, this is a velocity of 1.118 x 104 m/s.
Now for the energy. Since the V-rocket is moving, it has kinetic energy. We define kinetic energy as:
Then I can calculate the kinetic energy per rocket to be 5.3 x 1018 joules.
But what does that even mean? Well, if you pick up a textbook from the floor and put it on a table, that’s about 10 joules of energy. So 5.3 x 1018 joules is an enormous amount. Just to put this into perspective, in 2015 humans collectively used about 6 x 1020 joules (based on this data). So, one rocket would require almost 1 percent of all the energy used by all of humanity in an entire year. But I guess if you really want to get rid of us, that would be worth it. Right?
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