The Wild Physics of a Firefighter's Window Catch

Are superheroes real? Maybe. In this recently released video, a firefighter in Latvia catches a man falling past a window. Let me tell you something. I have a fairly reasonable understanding of physics and this catch looks close to being impossible—but it's real.

Here is the situation (as far as I can tell). A dude is hanging on a window (actually, the falling human is only rumored to be a male) and then he falls. The firefighters were setting up a proper way to catch him, but it wasn't ready. Of course the only solution is then to catch him as he falls. It seems the victim fell from one level above the firefighter. At least that's what I'm going to assume. Now for some questions and answers.

How fast was the human moving?

This is a classic physics problem (I hope my students are paying attention). An object (or human) starts from rest and then begins to fall under the influence of the gravitational force. If the gravitational force is the only significant interaction on the human then that person will fall with a constant acceleration of 9.8 m/s². That means that for every second of free fall, the human's speed will increase by 9.8 m/s (hint: 9.8 m/s is fairly fast—about 22 mph).

If I knew the time the human was falling, I could easily determine the speed since it increases a set amount every second. However, I can only approximate the distance the person falls. Of course that is only a small stumbling block for physics. In fact there is a kinematic equation that gives the speed of an object with a constant acceleration after a certain distance (you can also easily derive this with the definition of average velocity and acceleration). But if the object starts from rest and moves a distance y, then the final speed will be:

Yes, the greater the fall, the greater the speed. In this case, I'm just going to guess the distance at about 3 meters (it's just a guess). That would put the speed of the faller (is that a real word) at about 7.7 m/s. Maybe it's a little bit shorter fall at 2 meters—that would give a window-level speed of 6.3 m/s. Either way, it's fast.

How hard would it be to catch this human?

It doesn't take a superhuman to fall but it might take superhuman strength to stop someone during a fall. The key here is the nature of forces. A net force on an object changes the motion of that object. In this case, there will be two forces acting on the falling human. First, there is the gravitational force pulling down. This force depends on the gravitational field (g = 9.8 Newtons per kilogram) and the mass of the human (which I don't actually know). The second force is that of the firefighter pushing up during the catch. The total force (sum of these two forces) must be in the upward direction so that the change in motion is also up. This means the human (during the catch) will be slowing down. That's what we want.

I can estimate the human's mass, but what about that firefighter force? There are two basic ideas that deal with force and motion. First is the momentum principle. This is a relationship between force, momentum (product of mass and velocity) and time. The second is the work energy principle. This deals with forces, energy, and displacement. So it comes down to this. Do I want to estimate the time it takes to catch the human or do I want to estimate the distance over which the human was caught? I think I'll go with distance and the work-energy principle.

Here is your super short intro to the work-energy principle. First, let's look at work. Work is a way to add or take away energy from a system. The work depends on both the magnitude of the force and the direction the object is moving. Let's say that the human travels a distance d during this catch. In that case, the gravitational force will do positive work (since it is pulling in the same direction as the displacement) and the firefighter will do negative work (pushing up in the opposite direction as the motion).

But what about the energy? For this system (of just the falling human), there is only one type of energy—kinetic energy. The kinetic energy depends on both the mass and the speed of the faller. The idea is to have the total work done on the human decrease the kinetic energy to zero (so that the human stops). Now to put it all together, it looks like this (yes, I skip a bunch of details).

I already have the estimated speed (from above) so I just need the human mass and stopping distance. Let's say this is a human that isn't super big—maybe 50 kilograms. For the stopping distance, it looks like the firefighter grabs the falling human and moves about 1.5 meters before coming to a stop. With these values, the force the firefighter needs to exert on the human would be 1,478 Newtons. For you imperials, that is about 330 pounds. It's a large force, but not impossible. Still very impressive for just one hand.

Oh, and don't forget that if the firefighter pulls on the human with almost 1,500 Newtons, the person pulls on the firefighter with the same force in the opposite direction. This means that the hero has to hold onto the window sill in order to not get pulled out of the building and fall along with the victim. Yes, there does appear to be a harness on the firefighter but it doesn't look like it has tension. Still a superhero in my mind.

I have one final comment. Since I used the work-energy principle to estimate this force it seems like this is a good time to add an important note about energy. Remember—energy isn't a real thing. It's just something that we can calculate the can be conserved in many situations. There. I said it.

SCIENCE