 ## Batman

The comment you hear about Batman is always the same: people like him because in addition to being a superhero, he is also a normal man. Despite not having superpowers, however, Bruce Wayne's alter ego can count on remarkable skills and equipment.

In the latest film dedicated to the batman, The Batman, we see one of his at work "gadget", the grappling hook gun. Batman uses it to shoot a kind of hook, connected to a cable; once fixed in one place, the electric motor inside the gun rewinds the rope, hoisting Batman towards the ceiling.

In the scene you can see below, Batman is locked in a building by some police officers of Gotham. Determined to escape the police, after freeing himself, he runs to an internal staircase, shoots the cable near the top of the ramp, and then activates the engine of the gun to pull himself on him. Spoiler: he manages to escape (but you probably already knew this).

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Now let's get to the physical calculations: We tried to understand what kind of battery or power source Batman's gun would need and how much energy it would use.

## Energy for Ascension

One of the fundamental ideas for understanding energy is to define a system of interest, that is the set of objects that we want to study (of course, the most complete system of all would be the whole universe, which however would be impractical. In this case we will isolate only the objects that interest us).

We use the following system: Batman plus the gun (and its battery) and the Earth (it may seem bizarre to add the Earth as an element in the system, but trust us: we are back).

Once the system has been created, we can use one of the most important concepts in physics: the kinetic energy theorem. This principle states that the work done by a force on a system is equal to the change in the energy of the system. At this point, however, we need to define what energy is.

Explaining what energy is is not easy. My best definition is that energy does not represent a real thing, but rather a way to keep track of various interactions. Energy comes in different forms. For example, kinetic energy is associated with the movement of objects, while potential energy is what depends on their position.

Work, on the other hand, is a way of adding or removing energy to a system. In terms of forces, we define work as follows:

Rhett Allain In this equation, F represents the applied force and Δr is the distance through which the force pulls (or pushes) an object. In this case, however, only the component of the force in the direction of the displacement counts: this is the purpose of the cosine (θ), where θ indicates the angle between the force and the displacement.

As always, the mode better to understand one thing is to give an example, and Batman using a grappling hook gun lends itself very well to proving the kinetic energy theorem. So let's consider the work and energy changes as Batman hovers up to the top floor of the building. The first thing to do is to define our system of interest. This is a rather important phase: by defining the system, we can understand which interactions we can represent as "work" and which as "energy".

I will start with a free body diagram showing Batman getting carried away by the constant speed cable (as mentioned, the Earth is also part of our system of interest, which I will not show for obvious reasons of space).

Rhett Allain Here it can be seen that there are two forces acting on Batman: the gravitational force that pushes it down (mg) and the force that pulls it up due to the tension of the cable (T). While these two forces act on Batman for some distance, neither of them act on the system. Tensioning the cable doesn't do any work because the cable doesn't move, unlike Batman and his gun. Since the cable remains stationary, its displacement (Δr) is zero, as is the work done.

Here we will not consider the work done by the gravitational force. Forces are an interaction between two objects, in this case Batman and the Earth. Since the Earth is also part of our system, we cannot take into account the work done by this "internal" force. In this case, therefore, we will use potential energy. Potential energy can be thought of as energy stored in a system. Here, we will call it "gravitational potential energy", and we will present it like this:

Rhett Allain M indicates the mass of the object (Batman and his gun), while g is the gravitational field (9.8 newton per kilogram) and y the vertical position. It is possible to measure the position from any point of reference, since in reality it is only the variation of the position of Batman that counts.

Here, however, a problem arises. Our theorem looks like this:

Rhett Allain We have already said that the work is zero, but also the variation of the kinetic energy is zero if Batman moves upwards at a constant speed (since the speed does not change ).

The change in gravitational potential energy will be a positive value, as Batman is moving upward and his y-value is increasing. But this means that the left side of the equation is zero and the right side is zero plus a positive value. You don't have to be particularly good at math to understand that something is missing.

The missing energy is the chemical energy of the battery. Battery energy decreases (or wears out) as gravitational potential energy increases (as Batman's y position increases). This equation follows:

Rhett Allain How much energy does a battery operated grappling gun need? It is enough to know Batman's mass (m), the gravitational field (g) and the change in height (Δy).

Let's make some estimates. Since I have no idea what mass the Batman costume and all of that is, I will limit myself to 100kg. As for the variation in height, in the scene we are considering it is a bit difficult to calculate, but it is clearly greater than five stories and probably less than fifteen. Let's make ten. Since a storey is about 4.3 meters high, the change in height (Δy) is 43 meters. Putting everything into the previous equation, the decrease in battery energy would be approximately 42,000 joules.