So far, I've only seen the trailer for the new movie Free Guy with Ryan Reynolds, but it looks like it's a lot of fun. All I know is that it’s about the adventures of some guy (whose name is Guy) in a video game world where nothing is real. What's the point of using real physics to analyze a movie about a fake world? Because it's cool.
Let's consider a scene from the trailer. For some reason, Guy is running on a tall building and leaps toward a hanging wrecking ball. Oops! He misses and starts to fall. But don't worry, suddenly some sort of inflatable cushion poofs up around Guy. He hits a parked car and bounces off, unharmed. I'm all about unrealistic stuff happening in a movie—I mean, it's not a documentary. But it also makes me wonder if this inflatable thing would actually save a real human.
I'm going to start by finding Guy’s velocity both before and after he bounces off the car. For this, I can use Tracker video analysis. With this program, I can look at the location of an object (I will use bouncing Guy) in each frame of a video. That will give me position vs. time data that I can use to examine his motion.
Here is a plot of his vertical position as a function of time, before, during, and after the bounce:
He's mostly moving straight down before he crashes—and the trailer just shows two video frames of his motion before impact. I can use those two data points to get his velocity before the bounce by fitting a linear function. From that, I get a speed of 17.5 meters per second (39 miles per hour).
OK, what about after the collision? In that case, there are actually three frames showing his motion. I can again fit a linear function to get the slope, which would be his vertical velocity. From this, his speed after the bounce is 10.8 m/s (24.2 mph).
But wait! He doesn't bounce straight up—there is also a horizontal component to his velocity after the collision. Here is a plot of his horizontal position as a function of time for that same scene:
The horizontal position before the bounce doesn't change, which means he has a zero horizontal velocity. After the “bounce,” he has a horizontal velocity of 11.4 m/s (25.5 mph). Honestly, this is all fine. Really, we can just consider one velocity that has both horizontal and vertical values—this means it's a velocity vector. With that, I can write the velocity before and after the bounce as the following:
A quick note: We like to put an arrow over a variable to indicate that it's a vector quantity instead of just a plain number. There are many different ways to write down the different components of a vector, but I like the notation above. For each velocity vector, the x, y, and z components are inside of angle brackets and separated by commas. (Yes, I included the z-direction. This would be a direction moving out from the screen towards the audience, and perpendicular to both the horizontal (x) and vertical (y) directions.)
Why do we need velocity vectors? Because the way we can find out if the bounce would be survivable by a real human is to look at the acceleration. Acceleration is also a vector, and it tells us how the velocity vector changes with time. We can define it as:
Here, the Δt is the length of the time interval it takes the object to change in velocity by the amount Δv. In general, the Greek letter Δ (delta) stands for the change in a variable.
Great, I have everything except for the time interval. Looking back at the video, it seems like the bouncy thing is in contact with the car for about 0.125 seconds. With this, I now have everything I need to calculate the acceleration.
The nice thing about writing vectors in component form is that it's super easy to subtract vectors, or to find the change in velocity. I just need to take the components of the final velocity (v2) and subtract the components of the initial velocity (v1). This gives the following acceleration vector:
If we want to know what would happen to a human crashing into a car while wrapped in a bubble, then the direction of the acceleration doesn't matter. When it comes to estimating possible injury to a person’s body, it doesn’t matter which way they bounce. What matters is the magnitude of the bounce’s acceleration. We need to know the total acceleration—we call this the magnitude of a vector.
Since the x and y components of the acceleration are perpendicular to each other, they form a right triangle with the hypotenuse being the magnitude. That means that I can square the components, add them together, and take the square root to find the magnitude of the acceleration.
It's common to put "absolute value" lines around a vector to show that you are using just the magnitude of the vector. But still, there is just one more thing to consider: Guy's acceleration is calculated in units of meters per second per second. (We write that as m/s2.) However, it's very common to talk about the acceleration of humans in terms of g's where 1 g = 9.8 m/s2. With this, Guy has an acceleration on impact with a value of 25 g's.
You already have an intuitive feel for the value of 1 g. It's what you experience every day due to your gravitational interaction with the Earth. (Unless you aren't on Earth—in which case, that's cool.) Yes, that force you feel pushing down on you as you are sitting on the couch is 1 g. It’s the same force that you feel as you are walking around town or eating ice cream. As long as you aren’t accelerating, you feel 1 g.
Why is gravity like an acceleration? It's complicated and rooted in Einstein's equivalence principle, but in practical terms it means that having an acceleration of 25 g's would be like sitting down with a force equal to 25 times your weight. Oof.
Here we are fortunate that NASA and others have experimentally determined the maximum acceleration a human can withstand—they call it g-force tolerance. It’s not a single number. The maximum tolerance also depends on the duration of the acceleration, the orientation of the person during impact, and even how quickly the acceleration increases.
Well, how about Guy’s acceleration of 25 g's? It seems that if this bouncing impact lasts a little over 0.1 seconds, then Guy might be in trouble. It’s very likely that Guy would be at least partially injured—maybe even critically injured.
But it’s difficult to say for sure, since the NASA data is based on experimental evidence. And on top of that, every human is different, with different tolerances. The orientation of the body during the acceleration also matters. Humans are most tolerant to an acceleration in an orientation called “eyeballs in.” This would be the position of an astronaut taking off in a rocket, lying back and looking up, such that the acceleration pushes the eyeballs into the skull. If, however, Guy lands on the side of his ribs, he could probably only withstand about 10 to15 g's.
Now let’s figure out how you would protect a falling human in real life. Suppose someone falls off a building and has the same downward velocity that Guy does right before impact (about 17.5 m/s). If you want that person to end up on the ground with a zero velocity, there are two things you could change that would make a significant difference. (Remember, the goal is to have an acceleration with a small enough magnitude that the person isn't injured. Maybe that’s around 10 g's instead of 25, although that would still be rough.)
First, you could arrange it so that the person hits something soft and stops instead of bouncing off. The acceleration depends on the change in velocity (vector). This means that going from a velocity of 17.5 m/s down to one of 10 m/s up is a 27.5 m/s change, since direction matters. However, if the person just stops and doesn't bounce, it would only be a 17.5 m/s change in velocity. With a smaller change in velocity, you would have a smaller acceleration—which means a lower g-force. That would make the collision more survivable.
The second thing to change is the time. If you increase the time over which the human stops, you decrease their acceleration. I'm sure you have been in a car that was moving at a speed of 17.5 m/s, which is 39 mph. When you stopped, it most likely didn’t cause you injury. That’s because a car brakes to a stop over a time interval of about 10 seconds—so the acceleration is quite small even though you would have the same change in velocity as Guy.
In real life, you can increase stopping time with something like a stunt airbag. These are large inflatable structures that collapse on impact and are used when filming action scenes in movies. The airbag in your car is based on the same principle to keep you safe—or safer—in a crash. By stopping a moving body over a larger distance, airbags increase the impact time. which decreases the acceleration. Both types of airbags deflate on impact to prevent the person from bouncing back. (Which, as I explained in the previous example, is bad.) Of course, an airbag wouldn't work for the scene in Free Guy—you would have to set it up before the fall and know where Guy was going to land.
So, bottom line: The inflatable cushion ring around Guy looks cool and creates a funny bouncing scene. But given his acceleration of 25 g’s, that landing is still going to hurt.
Unless Guy isn’t even real. In that case, he’s fine.