How Can Soccer Players Bend Their Shots in Midair?

足球运动员是如何踢出“香蕉球”的？

With the talent on tap, World Cup 2026 is sure to serve up plenty of jaw-dropping kicks, like a ball that curves in midair to go around a defender, or a shot on goal that swerves away from where the keeper thought it was headed. How is this possible? What wizardry enables a striker to change the ball’s trajectory after it leaves their foot? 随着各路天才球员齐聚，2026年世界杯注定会呈现许多令人惊叹的射门，比如在空中划出弧线绕过防守者的球，或是让守门员判断失误、偏离预定轨迹的射门。这怎么可能呢？究竟是什么“魔法”让前锋能在球离开脚后改变其飞行轨迹？

It’s not magic, it’s fluid dynamics, the behavior of objects in a fluid—and air is considered a fluid, since it flows. (Kids, want to be a real-life FIFA hero? Take physics.) To really understand what’s going on, let’s model the motion of a ball, starting with the simplest and silliest scenario, then adding back elements of reality one at a time. 这并非魔法，而是流体力学，即物体在流体中的运动规律——空气由于具有流动性，也被视为一种流体。（孩子们，想成为现实生活中的FIFA英雄吗？去学物理吧。）为了真正理解其中的奥秘，让我们先对球的运动进行建模，从最简单、最荒谬的情景开始，然后逐一加入现实世界的物理要素。

Soccer in Space

太空中的足球

Why would you play soccer in space? Well, if you’ve seen the ticket prices for this year’s tournament, you might think it’s cheaper to go off-planet. Anyway, say we’re way out yonder where there’s no air or gravity. The ball is at rest, and then a player in a space suit gives it a kick. 为什么要到太空中踢足球？好吧，如果你看过今年锦标赛的门票价格，你可能会觉得去外星球反而更便宜。总之，假设我们身处遥远的太空，那里没有空气，也没有重力。球处于静止状态，然后一名穿着宇航服的球员踢了它一脚。

While the foot is in contact with the ball, it exerts a pushing force. The ball compresses and then rebounds, launching off the foot; all of this takes about a hundredth of a second, and a pro can easily fire the ball at 80 miles per hour. 当脚与球接触时，会施加一个推力。球被压缩后反弹，从而离开脚面；整个过程大约只需百分之一秒，而职业球员可以轻松地将球踢出时速80英里的速度。

So the applied force changes the velocity of the ball, but the thing to know is that once the ball loses contact with the foot, there is no longer any force acting on it. Which means the ball will keep moving in a straight line at a constant speed … er, till the end of time. You might recognize this as Newton’s first law. 因此，施加的力改变了球的速度，但需要注意的是，一旦球离开脚面，就不再有任何力作用于它。这意味着球将以恒定的速度沿直线运动……呃，直到永远。你可能已经认出这就是牛顿第一定律。

Of course, you’d lose a lot of balls this way in space, so maybe it isn’t very practical. Let’s move the action back to Earth, but to keep it simple we’ll first assume there’s no atmosphere. Back into your space suits! 当然，在太空中这样踢球会丢掉很多球，所以这可能不太现实。让我们把场景移回地球，但为了简化起见，我们先假设没有大气层。穿回你们的宇航服吧！

Soccer on an Airless Earth

无空气地球上的足球

Now there’s a new interaction involved—the planet’s gravitational pull. We can calculate this downward force as Fg = m × g, where m is the mass of the ball and g is the gravitational field on Earth (9.8 newtons per kilogram). By the way, Fg is what normies call an object’s “weight.” 现在引入了一个新的相互作用——地球的引力。我们可以将这个向下的力计算为 Fg = m × g，其中 m 是球的质量，g 是地球的重力场（9.8牛顿/千克）。顺便说一句，Fg 就是普通人所说的物体的“重量”。

What’s different about this force is that it’s still there after the ball is kicked. The ball is moving with some velocity, and the gravitational force continuously alters its motion. The rate of change in velocity is called acceleration (a). 这种力与之前不同之处在于，即使在球被踢出后，它依然存在。球以一定的速度运动，而重力会持续改变其运动状态。速度的变化率被称为加速度（a）。

We need one more thing—how about Newton’s second law? This says the acceleration depends on the net force (Fnet) and the mass (m) of an object. It’s usually written as Fnet = m × a, but we can rearrange it like this: a = Fnet/m. Combining this with our gravitational force, we get something pretty interesting: 我们还需要一样东西——牛顿第二定律如何？它指出加速度取决于物体的合力（Fnet）和质量（m）。通常写作 Fnet = m × a，但我们可以将其重写为：a = Fnet/m。将其与我们的重力结合起来，会得到一个非常有趣的结论：

Since both gravity and acceleration depend on the mass of the ball, the mass cancels. We find that any object on Earth has a downward acceleration of 9.8 meters per second per second (m/s2). This means that if you drop a bowling ball and a marble at the same time, they’ll hit the ground at the same time—even though the gravitational force on the bowling ball is thousands of times higher. Weird, right? 由于重力和加速度都取决于球的质量，质量项被抵消了。我们发现地球上任何物体的向下加速度都是9.8米/秒²。这意味着如果你同时扔下一个保龄球和一个弹珠，它们会同时落地——尽管保龄球受到的重力要大上千倍。很奇怪，对吧？

Anyway, now, in the presence of gravity, if you kicked a ball at an upward angle, it’s vertical velocity would slow, halt, and reverse, with the speed increasing as it falls. In other words, it starts accelerating in the downward direction as soon as it’s kicked, even while it’s moving upward. 总之，现在在重力作用下，如果你以向上的角度踢球，它的垂直速度会减慢、停止并反向，随着下落速度增加。换句话说，球从被踢出的那一刻起就开始向下加速，即使它还在向上运动。

What about the horizontal motion? Ah, since there’s no horizontal force after the initial kick, the ball continues traveling forward at the same speed, just like in space. People tend to think a ball falls because its forward motion slows, but actually it’s the opposite. Without air drag it doesn’t slow down at all. It only stops because the ground gets in the way. 水平运动又如何呢？啊，由于初始踢球后没有水平方向的力，球会像在太空中一样以相同的速度继续向前飞行。人们往往认为球落地是因为它的前进速度减慢了，但事实恰恰相反。如果没有空气阻力，它根本不会减速。它停止运动仅仅是因为地面挡住了去路。

So what we get for a trajectory is that familiar upside-down parabola, often called a ballistic trajectory because it’s the path of any unpowered projectile, like a cannon ball, a bullet, or a basketball. Any flying object for which gravity is the only (significant) force acting on it will move this way. 因此，我们得到的轨迹就是那个熟悉的倒抛物线，通常被称为弹道轨迹，因为这是任何无动力抛射物（如炮弹、子弹或篮球）的路径。任何只受重力（显著）作用的飞行物体都会以这种方式运动。

Soccer With Air

有空气环境下的足球

Happily, the Earth does have air. But it drastically changes the game. Now there is a continuous force acting horizontally, which we call air resistance, or drag, and it pushes in the direction opposite to the ball’s motion. 幸运的是，地球确实有空气。但这极大地改变了比赛。现在有一个持续的水平作用力，我们称之为空气阻力，它朝着与球运动方向相反的方向推动。

Think of air molecules as a bunch of tiny ping-pong balls. As a soccer ball moves through the air it collides with gazillions of these little air balls, and each collision exerts a backward-pushing force; all combined, this creates the total air-resistance force. The bigger the object, the more collisions it has to fight through. 把空气分子想象成一堆微小的乒乓球。当足球在空气中移动时，它会与无数这些小“空气球”发生碰撞，每一次碰撞都会产生一个向后的推力；所有这些合在一起，就形成了总的空气阻力。物体越大，它需要克服的碰撞就越多。

You also have more collisions with a faster-moving object. This means that if you’re just throwing a soccer ball in from the sideline, air resistance isn’t a factor, but on a hard kick, you can’t ignore it. In fact, doubling the ball’s speed quadruples the air resistance. Without air resistance, a goalie could kick a ball the length of the field and over the stands beyond. 物体运动越快，碰撞次数也越多。这意味着如果你只是从边线掷入足球，空气阻力不是一个因素，但在大力射门时，你无法忽视它。事实上，球速加倍，空气阻力会增加到原来的四倍。如果没有空气阻力，守门员可以把球踢过整个球场，甚至飞出看台。

Soccer With Spin

旋转的足球

But there’s another way a soccer ball is affected by air. If the ball is spinning, the tiny air balls don’t just bounce off; they also get dragged along in the direction of rotation. Here’s your fluid dynamics. This causes the path of the soccer ball to curve. In the picture below, the ball is moving to the right but spinning counterclockwise, which means it has a horizontal axis of rotation. 但足球受空气影响还有另一种方式。如果球在旋转，微小的空气分子不仅会反弹，还会被带入旋转方向。这就是流体力学的作用。这导致足球的路径发生弯曲。在下图中，球向右移动但逆时针旋转，这意味着它有一个水平的旋转轴。

As it spins, it drags some of the air from above the ball and pushes it back and under. But if the ball is pushing the air down, the air must push the ball up. Remember, forces always result from an interaction between two things—so the ball pushing on the air and the air pushing on the ball are equal and opposite forces. (Hat trick! Newton’s third law.) 当它旋转时，它会带动球上方的一些空气并将其推向后方和下方。但如果球把空气向下推，空气必然会把球向上推。记住，力总是源于两个物体之间的相互作用——所以球推空气和空气推球是大小相等、方向相反的力。（帽子戏法！牛顿第三定律。）

We call this the Magnus force, and its magnitude depends on the size of the ball, the type of surface (rough or smooth), the rotation rate, and the velocity. Yes, it’s complicated. 我们称之为马格努斯力，其大小取决于球的尺寸、表面类型（粗糙或光滑）、旋转速率和速度。没错，这很复杂。