I often miss some cool stuff the first time I watch a movie. This is probably a good thing—it shows that I'm focused on the story and not the small details. In this case, the movie is 2016's Captain America: Civil War and the scene involves the density of a character named Vision.
OK, I am going to give a SPOILER ALERT—but if you haven't seen this movie yet, I have a feeling you won't be upset about spoilers. Anyway, this scene doesn't reveal any huge plot points.
So here's the deal. Vision is trying to keep Wanda (Scarlet Witch) safe in the Avengers' headquarters. Hawkeye comes to help her leave, but Vision catches them. Although Vision could easily defeat Hawkeye, the same cannot be said for the powers of Scarlet Witch. Scarlet Witch has some ability to control matter—and in this case it appears that she can activate Vision's powers. One of Vision's primary powers is his ability to change his density.
So with a bit of magic, Scarlet Witch increases Vision's density up to the point were he becomes too massive to move. He grows so massive that he breaks through the floor. With Vision out of the way, Wanda and Hawkeye are free to leave and finish the rest of the movie.
Now for the fun part. What was the density and mass of Vision when he crashed through the floor? How about a quick review of density? Take a look at these five objects.
These blocks are all different, but there is something similar about them. If you took the three blocks on the left, they all have the same mass (about 45 grams). The three blocks on the right all have the same volume (I'm disappointed that they are almost exactly 1 cubic inch—they should have some value in cm3). But wait! What if you take the mass of each block and divide by its volume? This is how we define density. The density is a property that doesn't depend on the size of the object, just its material. So the two white objects (on the ends) have different volumes and different masses, but the same density. The same is true for the two black objects.
To estimate the mass and density of Vision, I need some particular event that gives a hint about his mass since you can't "see" the mass of an object. Yes, you guessed it: I can use moment that Vision breaks through the floor to estimate his mass.
Here is what I'm going to do. I'm going to assume the floor is made of concrete and that the gravitational force on Vision (due to his large mass) is enough to exceed the compressive strength of concrete to initiate the break.
What is "compressive strength"? This is the pressure a material can withstand before breaking. Yes, it's the pressure and not the force (remember that pressure is the force divided by the contact area). This is why you can more easily break a material with a sharp pointy object than you can with a big flat object. The pointy object has a smaller area and therefore you get a bigger pressure for the same amount of force.
But what about the compressive strength of concrete? It's perhaps between 20 and 40 mega Pascals (MPa) where a Pascal is the same as one Newton per square meter. This means that if the floor breaks, I know the pressure from the force between Vision and the floor. If I estimate his contact area, I can then calculate the force and next his masses.
Really, the only thing left to estimate is the contact area. I could perhaps do a more detailed analysis, but I think it's fine to just get a rough value. What about a contact area that is a rectangle with a length of 1 meter and a width of 0.5 meters? That would put the area at 0.5 m2. I'm going with that.
Oh, one more thing. If I want to calculate the density of Vision, I also need his volume. He looks like a normal human—at least in terms of size. Humans have a density close to 1000 kg/m^3 (the density of water). If a human has a mass of 75 kg, the volume would be around 0.075 m3. I'm going with that value.
Let's crunch the numbers. I'm including the calculations in this python script so that you can put your own values in (if you don't like mine). Just click the "pencil" to edit and "play" to run it if you change any of the values.
Just to be clear, that is massive. The density is extreme (it's not neutron-star-level density though). Actually, it's sort of difficult to visualize a mass that large. How about this? What would be the size of a spherical asteroid of that same size? If the asteroid is made of normal stuff, it might have a density of 3,000 kg/m^3. With the same mass as Vision, a spherical asteroid would have a diameter of around 10 meters (30 feet). That's one big old rock.
You know (or you should know) that I can't just stop there. There are many questions left unanswered. I would normally just assign these as homework, but let me answer two of these questions for you.
Would there be a noticeable gravitational force between Vision and Hawkeye due to the large mass?
There is a gravitational interaction between all objects with mass. Normally on the surface of the Earth we only deal with the gravitational force between an object and the other. Interactions between two objects (like people) are usually so small that you would never be able to measure them. In this case, however, one of those people has a giant mass.
The magnitude of the gravitational force depends on both the masses of the objects and the distance between them. If you assume the objects are point masses (not true but an OK approximation), then the following equation calculates the force.
The G is just the universal gravitational constant with a value of 6.67 x 10-11 Nm2/kg2. If I assume a distance of 1.5 meters between Hawkeye and Vision, the gravitational force between them would be 0.0034 Newtons. That is a pretty tiny force. In fact, if you put a paperclip on top of Hawkeye's head, the weight of this paperclip would be more than twice the gravitational pull from Vision. I don't think Hawkeye would notice it.
Assuming Scarlet Witch increases Vision's density at a constant rate, how long will it take for him to have a mass equivalent to the Earth?
If you watch a clip of the scene, it seems clear that Scarlet Witch starts influencing Vision's mass when his head gem turns from yellow to red. Vision drops to his knees 13.9 seconds later. The floor also starts to crack at this point. Finally, after 20.4 seconds, Vision crashes through the floor.
Assuming a constant rate for the increase of mass (and thus density), the mass increases at 100,000 kilograms per second. If this mass increase rate stays constant, it would take 5 x 1019 seconds to get up to the mass of the Earth (6 x 1024 kg). Hint: that time is super, super, super long. It's not going to happen. But it was still fun to calculate.
Here are a few more homework questions for you:
- How long (assuming a constant mass increase rate) until Vision's mass reaches the point where Hawkeye gets pulled to Vision?
- If you consider the relationship between mass and energy (E = mc2), how much energy would it take to increase Vision's mass? What about the power? How does this compare to the power output of the Sun?
- How large would Vision's mass need to get before he became a black hole?
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