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Saturday, June 22, 2024

How Much Power Does Batman Need for His Ascender Gun?

Everyone has the same comment about Batman: He's cool because he's just a normal dude, but he's also a superhero. It’s true, he doesn't have superpowers. However, what he does have is a combination of skills and equipment.

In the movie The Batman, we get to see him use one of his "toys"—his ascender gun. (It's also known as a grapple gun or a grappling gun.) Batman uses it to launch something like a grappling hook, which is connected to a cable. Once it attaches to a high point, an electric motor within the gun winds the cable, pulling Batman up.

In this scene, Batman is in a building with a bunch of Gotham police officers who have detained him. He doesn't think that's such a good idea. After breaking free, he runs to an interior stairwell and shoots the ascender cable up near the stairwell’s top, then activates the motor to pull him up. Spoiler: He escapes. (But you probably knew that.)

Now for the physics calculations: What kind of battery or power source would his ascender need, and how much power would it use? Let's start with a bit of background on energy.

Energy for the Ascent

One of the key ideas for understanding energy is to define a system of interest, which is the collection of objects we want to study. (Of course, the most complete system is the entire universe, but it’s not very practical to deal with the whole thing all at once. Instead, we want to isolate only the objects that we are interested in.)

Let's use the following system: Batman plus the ascender (and its battery) and the Earth. (You might think the Earth is a weird thing to add to the system, but just hold on. We’ll get there.)

Once we have a system, we can use one of the most important concepts in physics: the work-energy principle. This says that the work done on a system is equal to the change in the system’s energy. But what the heck is energy?

That's actually a hard question, but here is my best answer: Energy is not a real thing, but rather a way to keep track of different interactions. Energy also comes in different forms. For example, kinetic energy is associated with the motion of objects, and potential energy is the kind that depends on the position of objects.

So work is a way to add or take away energy from a system. In terms of forces, we define work as the following:

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In this equation, F is the applied force and Δr is the distance over which the force pulls (or pushes) an object. However, only the component of the force in the direction of the displacement matters—that's what the cos(θ) term is for, where θ is the angle between the force and the displacement.

Honestly, the best way to understand energy is with an example, and Batman using an ascender is a perfect situation to demonstrate the work-energy principle. So let’s consider the work and changes in energy as Batman zooms up to the top floor of the building. The first thing we need to do is define our system of interest. This is actually a pretty important step—by defining the system, we can figure out which interactions we can represent as "work" and which ones as "energies."

I'm going to start with a force diagram showing Batman ascending the cable at a constant speed. (Although the Earth is part of our system of interest, I'm just going to show Batman, since the Earth is rather large.)

Here you can see there are two forces that act on Batman: the downward-pulling gravitational force (mg), and the upward-pulling force from the tension in the cable (T). Even though these two forces pull on Batman over some distance, neither one does any work on the system. The tension in the cable doesn't do any work because the cable doesn't actually move; Batman and the ascender move instead. Since the cable doesn't move, its displacement (Δr) is zero, so the work is also zero.

We could consider the work done by the gravitational force—but we won't. Forces are an interaction between two objects, in this case, Batman and the Earth. Since the Earth is also part of our system, we can't consider work done by this "internal" force. Instead of work done by this force, we will use a potential energy. You can think of a potential energy as energy stored in a system. In this case, we will call this "gravitational potential energy." It looks like this:

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Here, m is the mass of the object (Batman plus his stuff), g is the gravitational field (9.8 newtons per kilogram), and y is the vertical position. The awesome thing is that you can measure the position from any reference point you want, since really it's just the change in Batman’s position that is going to matter.

But now we have a problem. Our work-energy equation looks like this:

We already said the work was zero—but the change in kinetic energy is also zero if Batman is moving up at a constant speed (since the speed doesn't change).

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The change in gravitational potential energy will be some positive value, since he is moving up and his y value is increasing. But that means the left side of the equation is zero and the right side is zero plus some positive value. You don't have to be a "math person" to see that something is missing.

That missing energy is the chemical potential energy in the battery. That battery energy decreases (or gets used up) as the gravitational potential energy increases (since Batman’s y position increases). So we actually have this equation:

How much energy is needed for a battery-powered Bat-ascender? You just need to know the mass of Batman (m), the gravitational field (g), and the change in height (Δy).

Let's make some estimates. I have no idea about the mass of Batman's suit and stuff, so I'm just going with 100 kilograms. For the change in height, it’s sort of difficult to see, but it's clearly more than five floors and probably fewer than 15. Let's say 10 floors. Since a story is about 4.3 meters, this puts the change in height (Δy) at 43 meters. Putting it all into the equation above, the decrease in battery energy would be about 42,000 joules.

But what does that even mean? Let's start with what a joule is: a basic unit of energy named after James Prescott Joule. (Remember, if you do cool stuff, they might name something after you too.) If you pick up a physics textbook from the floor and put it on a table, that would require about 10 joules of energy.

How big a battery would you need to pull off this stairwell escape? Well, that depends on the kind of battery you use. The iPhone 13 uses a lithium-ion battery with a capacity of 3,227 mAh, or milliamp-hours. If you know the battery voltage (it's 3.7 volts), then you can convert this energy capacity to joules. Guess what? The energy in an iPhone 13 battery is 43,000 joules. That's enough for Batman to escape—and maybe have a little extra left over so he can post his epic feat on his TikTok channel.

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But what if he used AA batteries? Different brands of batteries have different amounts of energy, but Batman would only get the best AA batteries, with an energy of around 10,000 joules. So he would only need four or five of these batteries to get him to the top of the building.

I know that seems surprising, and it is—because there's more to this calculation.

Estimation of Power

We can’t only consider the energy needed to get Batman to the top of the stairwell. We also have to factor in how long this move takes. We have a quantity to describe how fast the energy of a system changes—it’s called the power.

If the change in energy (ΔE) is in joules and the change in time (Δt) is in seconds, this will give a power in units of watts. Since I already know the change in energy (that's the 42,000 joules), I just need to find the time it takes for Batman to travel the 10 floors in the video.

As is usual for a movie, the film cuts between shots and switches camera angles, so the timing is not completely clear. But as a rough estimate, I'm going to say that it took him 6 seconds to go from the ground floor to 10 stories up. That puts the power required from the battery at 7,164 watts.

Is that a lot of power? I guess so. If you wanted to compare this to the energy needed to power a car, you’d have to convert it to horsepower, and you’d get a pretty low figure: just 10 horsepower. Most cars have at least a 150-horsepower engine.

However, this is pretty good for a person. A human in top shape could produce about 800 to 1,000 watts of power for 6 seconds, according to experimental data from NASA. But not 7,000 watts. That's just not going to happen, even if that person is the Batman.

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There's another problem with this ascender and its battery. Although an iPhone battery could have enough stored energy to power the ascender for a 10-storey climb, it wouldn’t be enough to do it in 6 seconds. Suppose you have a 3.8-volt battery with a power output of 7,000 watts. This would require an electric current of over 1,800 amperes. That's a super high current. In fact, it's so high that it would heat up (or melt) the wires going to the ascender motor, which would require even more power.

A normal iPhone battery can't do this. You would need a much larger battery with a higher voltage so that you could reduce the current. But bigger batteries are heavier. And that would mean you would need a larger ascender.

Just to be clear, electric-powered ascenders are possible—check out this DIY ascender gun from Hacksmith), which is pretty awesome and looks like it works. In fact, something like this is probably a more reasonable version of what it would take to pull a human up a stairwell. But notice that this version is both larger and slower than the ascender in the movie clip. The ascender gun from Hacksmith uses batteries small enough to fit in two pants pockets—but they also have electric motors that are much bigger than Batman’s gun.

That might be fine for mere mortals, but Batman wants his ascender gun to be small and concealed. (That way no one knows what he can or can’t do.) So in the end, The Batman has to cheat with special effects. I'm totally fine with that, though, because it still makes for a great physics problem.

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