In this blog post, we’ll explore Bernoulli’s principle and the Magnus effect—two fundamental principles of fluid dynamics—through the mechanism by which a soccer ball curves as it spins.
It’s already been 24 years since the 2002 World Cup. The children born back then are now adults, and the excitement and passion of that time have become nothing but memories. Do you remember the South Korean national team’s final World Cup match? It was the third-place match against Turkey. Although South Korea unfortunately lost 2–3, the spectacular free kick taken by Lee Eul-yong in the first half remains etched in people’s memories. The ball, struck with his left foot by Lee Eul-yong, curved from left to right, narrowly avoiding the defenders’ heads and flying perfectly into the corner of the net. It was a goal that left a deep impression on many people at the time.
In soccer, it is common for a ball struck with spin to curve as it flies. In these kicks—commonly called “banana kicks”—the ball curves from left to right or right to left, bypassing defenders trying to block it and finding its way into the net. But why does the ball curve as it flies? At first glance, the only thing touching the ball in the air is air. Therefore, for the ball’s direction to change, the air must be doing something. What exactly does the air do? In fact, it is not the movement of the air itself, but rather the pressure difference created around the ball by the airflow that generates the force causing the ball to change direction. Understanding this pressure difference requires “Bernoulli’s Principle,” a fundamental principle of fluid dynamics.
What is Bernoulli’s Principle? Simply put, it is a principle that describes the relationship in which, for a steady flow of an incompressible fluid—where the effects of viscosity can be ignored and differences in height along the same streamline are not considered—static pressure decreases as the fluid’s speed increases and increases as the speed decreases. This principle is related to the law of conservation of energy. The law of conservation of energy states that while the state of energy may change, the total energy remains conserved. In middle and high school physics classes, students often learn about the relationship between potential energy and kinetic energy: when a ball is dropped from a certain height, potential energy is converted into kinetic energy, and the sum of these two forms of energy remains constant. Similarly, in fluid flow, the relationship between pressure-related energy, kinetic energy, and potential energy can be described, and this energy conservation relationship is expressed by Bernoulli’s principle. Of course, in real-world fluids, various factors—such as viscosity, turbulence, the rotation of fluid particles, and changes in the boundary layer—must be considered together; therefore, Bernoulli’s principle alone cannot accurately explain the motion of all fluids. In particular, when explaining the motion of an actual soccer ball, factors such as the boundary layer forming on the ball’s surface, the separation of the airflow, and the ball’s spin also have significant effects. Therefore, it is appropriate to view Bernoulli’s principle as a basic framework for understanding real-world phenomena.
So, how is Bernoulli’s principle related to the ball’s curving path? If we analyze the motion of a flying ball, we can broadly divide it into translational motion—moving in one direction—and rotational motion—spinning in one direction. Let’s first consider translational motion. When a ball flies forward through the air, it must cut through the air in front of it. In other words, as the ball flies, it collides with the air in front of it; from the ball’s perspective, however, one can think of the ball as being stationary while air flows toward it from the front. In other words, the situation where the air in front is stationary and the ball is flying can be considered equivalent—from the perspective of relative velocity—to a situation where the ball is stationary and air is flowing toward it from the front. That is, we can think of it as a wind blowing around the stationary ball in the opposite direction of the ball’s motion.
Next, let’s consider the ball’s rotational motion. If we set aside translational motion for a moment and consider only rotational motion, we can think of the ball as spinning in one direction while staying in place. To make this easier to understand, let’s imagine the ball spinning counterclockwise when viewed from above. In this case, the air surrounding the ball is affected by friction with the ball’s surface—that is, by viscosity—and moves in the direction of the ball’s rotation. This is the same principle as when you put your finger in water and spin it; the water moves along with your finger as it rotates. In actual airflow, this movement near the surface forms a boundary layer, and the airflow on either side of the ball changes depending on the ball’s rotation.
Now, let’s consider both the translational and rotational motion described earlier simultaneously. As the ball flies forward, air flows in the opposite direction of its forward motion at the front of the ball, and the airflow around the ball also changes due to its rotational motion. Assuming the ball rotates counterclockwise when viewed from above, we can reason that on the right side of the ball, the direction of the ball’s surface motion and the direction of the relative airflow cutting through the air are opposite, resulting in a lower relative airspeed; whereas on the left side, the directions of the two motions act in the same direction, resulting in a higher relative airspeed. In other words, as the ball moves forward, air flows toward it from the front, and the rotation also creates airflows in different directions on either side of the ball. When these effects are combined, the relative effect of the two movements partially cancels each other out on the right side, reducing the relative airspeed, while on the left side, the two movements act in the same direction, increasing the relative airspeed.
Let’s revisit Bernoulli’s principle. As mentioned earlier, while Bernoulli’s principle assumes ideal conditions, it provides a useful fundamental principle for understanding the different airflows around the ball. On the right side of the ball, the relative airspeed decreases, so the pressure is relatively higher; conversely, on the left side, the relative airspeed increases, so the pressure is relatively lower. When this pressure difference arises on both sides of the ball, a force resulting from the pressure difference acts on the ball. The ball does not simply “move” from the high-pressure side to the low-pressure side; rather, it is subjected to a force caused by the pressure difference—that is, the pressure gradient. Therefore, the ball is pushed from right to left by this force, causing it to curve to the left—this is the principle behind the so-called “banana kick.” Applying the same principle, if you kick the ball so that it spins clockwise, a force in the opposite direction will act on it, causing the ball to curve from left to right. In reality, a soccer ball’s trajectory is influenced not only by its rotational speed but also by its velocity, surface characteristics, airflow, and the state of the boundary layer; therefore, not all trajectories can be explained solely by the relationship between pressure and velocity.
This phenomenon, in which a spherical object moving through a fluid changes its direction of travel due to rotation, is called the “Magnus effect.” The Magnus effect occurs not only with soccer balls but also with other spinning spherical objects such as baseballs, basketballs, and golf balls. The Magnus effect creates an asymmetrical airflow around a spinning object; as a result, different pressures and aerodynamic forces arise on either side, causing the object to change its direction of travel. Bernoulli’s principle is one of the key principles used to understand these pressure differences, and to accurately analyze the Magnus effect, various fluid dynamics factors—such as the boundary layer, flow separation, turbulence, and the Reynolds number—must be considered together. Recently, computational fluid dynamics (CFD) simulations and wind tunnel experiments have enabled more precise analyses of the aerodynamic forces and trajectories of a spinning soccer ball, and the Magnus effect remains an important subject of research for understanding the motion of sports balls.
Let’s recall the free kick by Lee Eul-yong mentioned earlier. Of course, it is unlikely that Lee Eul-yong calculated the exact values based on Bernoulli’s principle the moment he kicked the ball. However, using the “feel” he developed through years of practice and effort, he applied the appropriate force and spin to the ball, causing it to change direction just enough to narrowly evade the defenders and find the back of the net. In fact, on June 29, 2002, during the third-place match against Turkey, Lee Eul-yong scored the equalizer with a free kick in the 9th minute of the first half. As this example shows, there are many instances where we can find engineering theories underlying phenomena we’ve simply taken for granted. Engineering isn’t something that’s far removed from our daily lives or limited to difficult concepts. If you take a moment to look around with a little curiosity, you’ll realize just how closely engineering is connected to our daily lives. The next time you’re watching a soccer game with friends, if you can explain why the ball curves, you’ll have taken a step closer to understanding engineering.
