The Complex Physics Behind Bending It Like a World Cup Player　弧線球背后的復(fù)雜物理

達尼洛·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.? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? 不可思議的進球是每一屆世界杯的珍寶。對陣西班牙的小組賽，克里斯蒂亞諾·羅納爾多一記射門繞過人墻，落入球門。1/8決賽中，本杰明·帕瓦爾接到隊友絕佳的傳中，射出一腳弧線，幫助法國力克阿根廷。乾貴士掃出一腳絲毫沒有旋轉(zhuǎn)的射門，幫助日本領(lǐng)先比利時，只可惜是暫時的。

世界波看起來十分驚艷。球的每一下旋轉(zhuǎn)，都會帶動球體表面的空氣，從而將球的行進軌跡推成一道弧線。對于這個水準(zhǔn)的足球運動員而言，踢出弧線球是一種本能的舉動，他們只需瞄準(zhǔn)球的邊緣，即可讓球飛往合適的方向。對于物理學(xué)家而言，若要為旋轉(zhuǎn)中的足球周身的空氣建模，則要涉及多種力的復(fù)雜結(jié)合，這些力都與空氣在球體表面各處流動的方式有關(guān)。

麻省理工學(xué)院的應(yīng)用數(shù)學(xué)家約翰·布什表示，足球飛出時，球體表面圍繞著一層空氣。隨著球體的旋轉(zhuǎn)，空氣也被轉(zhuǎn)向了一側(cè)，而空氣的流轉(zhuǎn)則會推動球向相反的方向移動。

本杰明·帕瓦爾的爆射是從他的右腳外腳背擊中足球左側(cè)開始的，從而讓球發(fā)生了順時針的旋轉(zhuǎn)。根據(jù)布什的解釋，球的旋轉(zhuǎn)帶動空氣向左轉(zhuǎn)動，而空氣產(chǎn)生的力則將球推向了右邊。因此，向右旋轉(zhuǎn)（也就是順時針）的球還是會沿弧形軌跡向右行進。這種力被稱作馬格努斯效應(yīng)。

看世界杯的竅門之一就在于，馬格努斯效應(yīng)是可預(yù)測的。逆時針旋轉(zhuǎn)的球總是旋向左邊，下旋的球會往上揚，而上旋的球則會朝下墜。

“馬格努斯效應(yīng)能幫助守門員，因為有人朝他們射出弧線球時，他們總能看到相同的曲線。”已退役的美國航空航天局航空工程師湯姆·本森說道，他曾設(shè)計軟件來為這一問題建模，“如果他能正確分辨球的旋轉(zhuǎn)方式，弧線的彎曲程度就會相同，他就能知道該往哪邊撲。”

布什表示，罰任意球時，如果人墻遮擋了守門員的視線，罰球者就更有可能踢出弧線球，上面所說的就是部分原因。罰球者會嘗試讓球越過人墻然后下墜，就像葡萄牙的C羅在2018年世界杯小組賽第一場對陣西班牙的比賽中那記驚為天人的射門一樣。

布什說，球員可以選擇用腳的外側(cè)或內(nèi)側(cè)踢球，由此決定旋轉(zhuǎn)開始的部位，改變球的旋轉(zhuǎn)方式。

正是足球的材質(zhì)和設(shè)計，讓空氣以一種可預(yù)測的方式在球體表面流動，防止馬格努斯效應(yīng)產(chǎn)生相反效果。“打個比方，如果你有一個表面極其光滑的球體，那么球就有可能以反常的方式旋轉(zhuǎn)。”布什說道。

2010年南非世界杯用球就太過光滑，以致馬格努斯效應(yīng)發(fā)生逆轉(zhuǎn)，使足球偏向與球員預(yù)料截然相反的方向。布什說：“如果你用完全相同的方式踢兩個球，一個粗糙一個光滑，它們會偏向相反的方向。”? ? 回看2010年，從接近海平面的開普敦，到高出海平面一英里有余的約翰內(nèi)斯堡，10個世界杯比賽場地的海拔天差地別。這就意味著，每場比賽的空氣密度不盡相同。本森表示，在密度不同的空氣中，球的飛行速度差別也很大。如果空氣比較稀薄，那么踢弧線球就困難得多，因為將球推向各個方向的壓力更小了。

日本福岡工業(yè)大學(xué)的流體力學(xué)工程師武人溝田表示，如果球不旋轉(zhuǎn)，那就會發(fā)生所謂的變線，球體表面的空氣會往隨機的方向剝離，讓球在空中以不可預(yù)測的形式飄搖擺動。

布什說，只有球以不旋轉(zhuǎn)的方式飛行，才會發(fā)生變線運動，而且往往需要球員迅捷利落地將球踢出，觸球點一般就是足球氣閥處，那是足球最牢固的地方。

2018年6月2日，日本對陣比利時的16強比賽中，乾貴士射出了一記絲毫沒有旋轉(zhuǎn)的完美進球。慢動作觀看這粒進球時，你可以看到在球飄進球門的過程中，球面上的紋路幾乎沒有動過。沒有了旋轉(zhuǎn)，守門員蒂博·庫爾圖瓦就無法預(yù)測球會飛向哪邊，直到為時已晚。“面對能踢出這種慢速旋轉(zhuǎn)射門的球員，所有的守門員都會顫抖。”溝田說道。

1 cross橫傳。? 2 arc按弧形軌跡行進。

3 deflect轉(zhuǎn)向。

4 flip翻轉(zhuǎn)。