The Complex Physics Behind Bending It Like a World Cup Player 弧线球背后的复杂物理

达尼洛·博格斯　易晨光

The gems of every World Cup are the impossible goals. Cristiano Ronaldo bending it in over the wall against Spain in the group stage. Benjamin Pavard bending a cross1 to pull France over Argentina in the group of 16. Takashi Inui sweeping in a perfectly spin-less goal to bring Japan up over Belgium， if only temporarily.

Its breathtaking to watch. Every spin of the ball moves air across the surface， pushing it into a bend. For a player at this level， bending the ball is an intuitive motion， just a kick to the edge of the ball to arc2 it in the right direction. For a physicist， modeling air flow around a bending soccer ball is a complex blend of forces all tied to the way the air moves across the surface.

When a soccer ball flies， the air forms a layer around the surface of the ball. As the ball spins， it deflects3 the air off to one side， says John Bush， an applied mathematician at MIT. This deflection of air pushes the ball in the opposite direction.

Benjamin Pavards cross starts with a strike on the outside of his right foot， which hits the left side of the ball， initiating a clockwise spin. That rotation flings the air off to the left， and the force created by the air leaving the ball pushes it to the right， explains Bush. Thus， a ball spinning to the right （thats clockwise） will also arc towards the right. This force is called the Magnus Effect.

One of the tricks in the World Cup is that the Magnus Effect is predictable. A soccer ball with a counterclockwise spin will always bend left， one with a backspin that goes under the ball gives it a bit more upward movement， and topspin causes a drop.

“It helps the goalies， because the goalies then see uniform curvature when guys are bending it taking shots at them，” says Tom Benson， a retired NASA aerospace engineer who designed software to model this problem.  “If he can pick up the spin right， its going to be the same amount of curvature， and he knows where to put his hands.”

This is in part why players are much more likely to take bending shots during free kicks when goalies cant see their kicks quite as well because of the wall of defenders， says Bush. There a player is trying to get the ball over the wall and down， like the gorgeous shot from Cristiano Ronaldo in Portugals group stage game in the first round of the 2018 World Cup against Spain.

A player can change how the ball spins by kicking with the inside or outside edge of their foot to place the spin， says Bush.

Its the texture and design of the soccer ball that keeps the air moving across the surface of the ball in a predictable way， and keeps the Magnus Effect from reversing. “If you have a perfectly smooth ball， the ball can bend the wrong way so to speak，” says Bush.

During the World Cup in South Africa in 2010， the ball was essentially too smooth and the Magnus Effect would flip4， bending the ball counter to the way players expected. “If you hit two balls exactly the same， one rough and one smooth， they will bend in opposite ways，” says Bush.

Back in 2010， the 10 World Cup arenas were at very different altitudes from sea level in Cape Town to just over a mile above sea level in Johannesburg. It meant that games were played with different densities of air. At the different air densities the ball was traveling at very different speeds， Benson says. Its much harder to get bend on a ball if the air is less dense because there is less pressure to push the ball in one direction or another.

If a ball isnt spinning， it does something called knuckling， where the air peels off the ball in random directions， causing it to bob and weave in the air unpredictably， says Taketo Mizota， a fluid engineer at the Fukuoka Institute of Technology in Japan.

This knuckling motion only happens when the ball moves without spin， and is usually achieved when a player manages a sharp， fast touch of the ball， typically right on the air valve where the ball is most firm， says Bush.

Takashi Inui managed to get a perfect spin-less goal when Japan played Belgium in the round of 16 on June 2nd， 2018. When you watch it in slow motion， you can see that the pattern on the ball barely moves at all as it floats into the goal. Its lack of spin kept the goalie， Thibaut Courtois， from being able to predict where it was going until it was too late. “All the goal keepers quiver before the kicker who can shoot these slow spinning soccer balls，” says Mizota.                                                      不可思議的进球是每一届世界杯的珍宝。对阵西班牙的小组赛，克里斯蒂亚诺·罗纳尔多一记射门绕过人墙，落入球门。1/8决赛中，本杰明·帕瓦尔接到队友绝佳的传中，射出一脚弧线，帮助法国力克阿根廷。乾贵士扫出一脚丝毫没有旋转的射门，帮助日本领先比利时，只可惜是暂时的。

世界波看起来十分惊艳。球的每一下旋转，都会带动球体表面的空气，从而将球的行进轨迹推成一道弧线。对于这个水准的足球运动员而言，踢出弧线球是一种本能的举动，他们只需瞄准球的边缘，即可让球飞往合适的方向。对于物理学家而言，若要为旋转中的足球周身的空气建模，则要涉及多种力的复杂结合，这些力都与空气在球体表面各处流动的方式有关。

麻省理工学院的应用数学家约翰·布什表示，足球飞出时，球体表面围绕着一层空气。随着球体的旋转，空气也被转向了一侧，而空气的流转则会推动球向相反的方向移动。

本杰明·帕瓦尔的爆射是从他的右脚外脚背击中足球左侧开始的，从而让球发生了顺时针的旋转。根据布什的解释，球的旋转带动空气向左转动，而空气产生的力则将球推向了右边。因此，向右旋转（也就是顺时针）的球还是会沿弧形轨迹向右行进。这种力被称作马格努斯效应。

看世界杯的窍门之一就在于，马格努斯效应是可预测的。逆时针旋转的球总是旋向左边，下旋的球会往上扬，而上旋的球则会朝下坠。

“马格努斯效应能帮助守門员，因为有人朝他们射出弧线球时，他们总能看到相同的曲线。”已退役的美国航空航天局航空工程师汤姆·本森说道，他曾设计软件来为这一问题建模，“如果他能正确分辨球的旋转方式，弧线的弯曲程度就会相同，他就能知道该往哪边扑。”

布什表示，罚任意球时，如果人墙遮挡了守门员的视线，罚球者就更有可能踢出弧线球，上面所说的就是部分原因。罚球者会尝试让球越过人墙然后下坠，就像葡萄牙的C罗在2018年世界杯小组赛第一场对阵西班牙的比赛中那记惊为天人的射门一样。

布什说，球员可以选择用脚的外侧或内侧踢球，由此决定旋转开始的部位，改变球的旋转方式。

正是足球的材质和设计，让空气以一种可预测的方式在球体表面流动，防止马格努斯效应产生相反效果。“打个比方，如果你有一个表面极其光滑的球体，那么球就有可能以反常的方式旋转。”布什说道。

2010年南非世界杯用球就太过光滑，以致马格努斯效应发生逆转，使足球偏向与球员预料截然相反的方向。布什说：“如果你用完全相同的方式踢两个球，一个粗糙一个光滑，它们会偏向相反的方向。”    回看2010年，从接近海平面的开普敦，到高出海平面一英里有余的约翰内斯堡，10个世界杯比赛场地的海拔天差地别。这就意味着，每场比赛的空气密度不尽相同。本森表示，在密度不同的空气中，球的飞行速度差别也很大。如果空气比较稀薄，那么踢弧线球就困难得多，因为将球推向各个方向的压力更小了。

日本福冈工业大学的流体力学工程师武人溝田表示，如果球不旋转，那就会发生所谓的变线，球体表面的空气会往随机的方向剥离，让球在空中以不可预测的形式飘摇摆动。

布什说，只有球以不旋转的方式飞行，才会发生变线运动，而且往往需要球员迅捷利落地将球踢出，触球点一般就是足球气阀处，那是足球最牢固的地方。

2018年6月2日，日本对阵比利时的16强比赛中，乾贵士射出了一记丝毫没有旋转的完美进球。慢动作观看这粒进球时，你可以看到在球飘进球门的过程中，球面上的纹路几乎没有动过。没有了旋转，守门员蒂博·库尔图瓦就无法预测球会飞向哪边，直到为时已晚。“面对能踢出这种慢速旋转射门的球员，所有的守门员都会颤抖。”溝田说道。

1 cross横传。  2 arc按弧形轨迹行进。

3 deflect转向。

4 flip翻转。