I'm all about Falcon and the Winter Soldier—the latest Marvel show on Disney+. Don't worry, I'm not going to spoil anything serious. I just want to talk about the wingsuits in episode 1. Sam Wilson (Falcon) is dealing with a hostage situation aboard a military aircraft. The bad guys grab their hostage and jump out of the plane wearing wingsuits. If you haven't seen these, they are basically skydiving outfits with extra material between the arms and legs to make it like wings—thus the name.
The hostage doesn't have a wingsuit, so they strap him on the back of one of the bad guy jumpers. After that, Falcon flies in pursuit and there is some action stuff—see, no real spoilers.
But really, this is just a chance to talk about some fun physics. So, let's consider the following two questions. One: How fast can a human fly with a wingsuit? Two: What would happen if you have an extra human (a hostage) on the back of a wingsuit jumper?
Free Fall
Let's start with something simple and then make it more complicated. (That's what we like to do in physics.) Suppose you jumped out of a plane and there wasn't any atmosphere. Yes, that would be super weird—but just imagine. For this case, there would just be one force acting on you—the downward-pulling gravitational force due to the interaction between you and the Earth. The gravitational force can be calculated as the product of your mass (in kilograms) and the gravitational field (we use g for this). As long as you are within about 100 kilometers of the surface of the Earth, the gravitational field is about 9.8 newtons per kilogram.
What does this constant downward gravitational force do in an airless world? That's where Newton's second law comes in. It gives the following relationship between force and acceleration:
Two important notes. First, both forces and accelerations are vectors. (That's why they have an arrow over them.) This means that both the magnitude and direction matters. Second, this expression deals with the net force (the total force). Since there's only the gravitational force, you would accelerate downward—your speed would just keep increasing for as long as you fall. But that's just pure falling and not wingsuit flying.
Let's add another force to a falling person—the air drag. This is a force in the opposite direction as the motion of the object. It's a result of air molecules colliding with the surface as something moves through the air. Suppose that I replace the air with large balls instead—oh, and these balls are just completely stationary before the interaction with a falling object. As the object moves down, there is a collision, and then the balls move off with different (but mostly downward) velocities. Here is a diagram to help you see this:
Each ball will have a change in momentum when the falling object hits it—where momentum is the product of mass and velocity. In order to change the momentum of an object, you need to exert a force on it. The magnitude of this force depends on both the change in momentum and the time over which this momentum changes. This force on the "air balls" is applied from the falling object. But wait! All forces are due to an interaction—this means that if the object pushes down on the air, the air must push up on the object.
Each collision between the object and the air balls exerts a tiny force pushing in the opposite direction as the motion of the moving thing. So, you can see that the total air drag force could depend on the following:
- The area of the moving object. A bigger object collides with more air balls.
- The speed of the object. Again, the faster it moves, the more collisions it will have and the greater the change in speed of the recoiling air balls.
- The density of the air. A higher density means more air balls to collide.
There is actually one other thing that matters: the shape. A cone-shaped object will be able to just push air balls to the side for a smaller change in momentum and thus a lower drag force compared to a flat object. We call this shape-based parameter the drag coefficient.
With that, we get the following model for the magnitude of the drag force on a moving object:
In this expression we have the follow: ρ is the density of air, A is the area of the object, C is the drag coefficient, and v is the velocity of the moving object with respect to the air. Why is there a 1/2 in there? I'm pretty sure it's because the drag coefficient is defined in some other problem with a factor of 2 and no want wants two different drag coefficients.
So, what does this mean for our falling bad guys? Let's say they fall out of a stationary flying plane. (Yes, I know that's silly—but it's easier to explain.) Since they start from rest, the velocity with respect to the air is zero and the drag force is zero. That means they will increase in speed as they fall. But an increase in speed means there will now be a drag force pushing up in the opposite direction as the motion.
Eventually, the falling people will reach a speed such that the drag force is equal in magnitude to their weight. The net force will be zero and the humans will stop increasing in speed. This means that for the rest of the fall they will move down at a constant speed. We call this terminal velocity. For a normal human (without a wingsuit) in a standard spread-eagle position, the terminal velocity is around 120 miles per hour (about 54 meters per second). With a wingsuit, the area for the air drag is much larger. This means you can get a drag force equal to the weight at a much lower velocity. But lower terminal velocities are not why people wear wingsuits—they wear them so that they can fly.
Flying (Falling with Style)
If you take that falling wingsuit and tilt it just a little bit, something cool happens. The collision between the air and the suit pushes the air down and to the side. Like this:
Since the air balls (or you can just call it air, if you like) are deflected to the right, the drag force on the falling object is somewhat to the left. With this left-pushing force, the falling object will increase its horizontal velocity. So, now it will be falling down and moving to the left. That's better than just plain falling.
Of course, now there is another problem. Since the object is moving to the left, it will also collide with air balls on the left side. This makes the force situation a little bit more complicated. It's actually easier to split this air drag force into two parts. For the part that is in the opposite direction as the velocity of the object, we will call this the drag force (like before). However, the rest of the interaction with the air must be perpendicular to the drag force—and we call this lift. Yes, drag and lift are two parts of the same interaction.
So, now let's say we have our wingsuit jumper moving both down and forward with some constant velocity angled θ below the horizontal. The forces would look like this:
For many cases, the ratio of lift and drag forces is constant. That's why it's called the lift-to-drag ratio and it's often represented with the variable L/D, but I think that's a confusing variable. I'm going to use the lift-to-drag ratio as K such that I can write:
Now for a little bit of math. If the human is moving at a constant velocity, then the net force in both the x (horizontal) and y (vertical) directions must be zero. If I break these forces into components, I get the following two equations:
If I replace the drag force (FD) with the lift force divided by K (FL/K), I get the following:
This angle θ is another way to think of the glide ratio. Since the wingsuit is not powered, it would have to keep moving down as it moves forward (assuming no updrafts). The glide is the ratio of the distance an object moves forward compared to the distance it drops. A wing suit can have a glide ratio of about 3:1. So, for every 3 meters it moves across it moves down 1 meter. With this, I can get a relationship between the glide ratio, the glide angle, and the lift-to-drag ratio.
But now for the real question: What would happen if you increase the mass of the flying object? In particular, what would happen to a wingsuit jumper with an extra person on top of them that essentially doubles the total mass? Well, according to this calculation the jumper would still have the same glide ratio—but that's only true if the lift-to-drag ratio stays the same. Let's just assume that it is indeed the same to produce the same glide ratio.
With a larger mass, both the lift force and drag force would have to increase in order to keep the jumper at a constant velocity. However, it would have to be a larger constant velocity for a wingsuit jumper without an extra person. The only way to increase the lift is to increase the velocity. (Remember that the lift and drag forces depend on the velocity.) So this means that the wingsuit jumper with the hostage on the back would have to move down and forward at a greater speed than the other jumpers. This would prevent all these bad guys from flying in formation—but that's just what we see in the episode of Falcon and the Winter Soldier.
Is there a way to make it actually work? There is one thing: If the jumper with the hostage had a suit with larger wings, it's possible that he could still have the same glide ratio. But how big would it have to be? For this calculation, let's just assume they are falling straight down. (It will be a little bit easier.) In that case, I will have the downward gravitational force and the upward drag force. For terminal velocity, these two must be equal in magnitude.
You can see from this that if you double the mass (m) and you want to have the same terminal velocity (v), then the area would have to also increase by a factor of 2. What would that look like? Let's say that a normal wingsuit jumper is a rectangle that is 1 meter by 2 meters (approximately). That's an area of 2 square meters. For the wingsuit with the hostage, you have to have a length of 2.83 meters by 1.41 meters, which gives an area of 4 square meters.
So, the guy would need a bigger wingsuit. Big deal, right? Well, it's not a big deal if you plan for it before you bring a hostage—and maybe they did. But there's a bigger problem with this larger suit. It looks funny. There is probably nothing worse that a bad guy can do than to look weird in front of the other bad guys. But I guess sometimes you just have to do what you have to do.