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The World Cup’s most magical goals rely on physics rather than defy them

A Professor of Applied Mathematics at MIT explains the subtle physics behind all those swerving, dipping goals that make you question the nature of the universe.

Portugal v Spain: Group B - 2018 FIFA World Cup Russia Photo by Dean Mouhtaropoulos/Getty Images

Cristiano Ronaldo does not have a Ph.D. in geophysics from Harvard. He received his training at Sporting Lisbon’s academy. He is also not a professor of applied mathematics at MIT. Nope. Actually, he’s a Portuguese striker who plays soccer professionally with Real Madrid and racks up goals at a prodigious pace for club and country.

But that doesn’t mean he doesn’t display a mastery of physics in his job.

John W.M. Bush is actually a professor of applied mathematics at MIT. He’s the one with that Ph.D. in geophysics from Harvard. His specialty is fluid dynamics, which also makes him an expert in aerodynamics. This all makes him the rare person who can understand exactly why all those swerving and dipping goals at the World Cup behave the way they do.

“The flow of air over a ball is in my locker,” he says.

An avowed Lionel Messi partisan, Bush has been watching the World Cup all along: noting the instep spin utilized by Coutinho of Brazil, the deft outstep touch of Belgium’s Kevin De Bruyne, as well as the way so many of us without PhDs tend to holler about physics online when we see an incredible goal.

“They shouldn’t say it defies the laws of physics,” he reminds us,” it just relies on relatively subtle physics that is well documented in the literature by now.”

Professor Bush talked with SB Nation about his favorite goals of the 2018 World Cup (so far), the various ways of striking a ball to produce spin, and why Ronaldo gets on his tiptoes before taking a free kick.

(This interview has been lightly edited for clarity)

What have been your favorite World Cup goals so far?

JB: As much as I loathe Cristiano Ronaldo, I have to concede that he’s a great free kick taker, he takes those better than anyone. It was originally Didi, who was a great Brazilian who played for Botafogo in Brazil. He called it folha seca, which is “dry leaf” — you know when a dry leaf falls it kind of flutters from side to side. These are the ones when you hit them with topspin and they wobble a little bit. Cristiano Ronaldo has really got that one down.

Watch Didi bend a free kick at the 1954 World Cup


OK, what makes Ronaldo so good at this?

JB: It’s just practice. You can go online and see how to hit those balls, and you can see kids hitting it consistently. Practice makes perfect. You have to put in the time; and he’s done that.

You can hit it one of two ways. You can either hit it with the top of your foot. You can see when he’s hitting these things he sort of gets up on his tip-toes on his left foot, and this is so he can strike the ball with the top of his foot.

The other possibility is hitting it flat-footed but with the side of your foot, which is what Drogba used to do. There you hit it with the side of your foot and get the same dip.

Let’s get a little bit more granular on each of those: What type of spin is a player going for if he or she uses the instep versus the top of the foot?

JB: So, for example, Coutinho wanted to bend it in such a way that the goalie couldn’t get to the ball. So he’s going for the right corner, and if he’d hit a ball in a straight line right into the top corner of the net the goalie would have gotten to it. But, as it was, it bent so basically the goalie couldn’t reach it. So it’s a matter of feet. Things will bend by up to a meter, their path will be deflected by up to a meter or maybe two for the long-range ones. The most famous was Roberto Carlos’ free kick.

More typically when you have free kicks, for example, you have to get around the wall. The wall will typically be set up so that it covers all the shots which would go into the net if hit along a straight line.

Cristiano Ronaldo’s [free kick goal against Spain] was particularly hard because he went over the wall. So it had to go up and down, as well as have a little bit of curve to the left. The topspin is the hardest thing to do, and the other thing is that there is a transition to it. There is a critical speed at which the drag goes up a lot, which is to say the thing slows down. That’s the best way to get the ball to go down because basically if it’s starting to dip and then it hits this so-called drag transition, this critical speed, then it will really drop quickly. And so that’s how you can maximize the bend on the ball.

But is Ronaldo really thinking about all of this when he’s setting up?

JB: When Cristiano Ronaldo hits that shot with the topspin, he doesn’t hit it as hard as he possibly can. He hits it in a certain range so that it will go through the drag transition before it gets to the net, and this he doesn’t know.

But he will know just from practice, right? I suspect he’s never heard of the drag transition. I played soccer myself. I was pretty good at free kicks, but what I did with the ball was independent of what I learned in books. So, there’s no one who’s going to sit there and study the aerodynamics of soccer balls and step onto the field and dazzle people. It really just takes practice.

You’ve described the instep and the top-of-foot strikes, but what about that pass from De Bruyne when he used the outside of the foot?

JB: De Bruyne to Lukaku! I saw that too. That was him with the outside of the foot. That was very nice, and he bent that to his right because he hit it with the outside of his foot. Again, the most famous shot ever was Roberto Carlos’ free kick in the Tournoi de France in the 90s. He hit this crazy left-footed shot with the outside of his left, and so that bent from right to left.

You could see the trajectory on that. It strayed from its original destination by about 15 meters. So, it’s really impressive. But that was a long-range shot, like I think he scored from 38 meters. But it was technically similar to De Brunye because he struck that ball with his right foot and he basically wanted to bend it so that it didn’t hit any of the defenders. If he’d hit with the instep of his right foot it would be curling from right to left and would have hit the defenders. So he hit it with the outside of his right foot so it bent from left to right, out of the path of the defenders and into the path of Lukaku.

Let’s go back to that Roberto Carlos shot. It was from long-range and he took a long run-up to the ball. How were those two things factors?

JB: First, the run gives you power, if you’re moving faster. That’s the first thing. The other thing is it gets you more time on the ball. So when you’re hitting a ball and you want to control the spin, you want to have it sitting on your foot as long as possible.

If you’re playing tennis, that’s why you have some give in the strings. If you ever get your tennis racket strung too tight, the ball will like ping off the racket, and you don’t get any grip on the ball. But if your strings are little looser, in the desired range, then the ball sinks into the strings and you have a fraction of a second where you’re basically controlling the spin on the ball.

Similarly, in soccer you’re going to have more control on the ball if it spends more time on your foot. So that’s another consideration. If you hit the ball very hard it will be you’ll be on your foot for longer, and that basically facilitates applying spin.

OK, but what about those knuckleballs?

JB: If you want to hit a dead ball — like one of these knuckleball-type shots, that’s another which Ronaldo is very good at that — you want to hit the ball as quickly as possible, have minimum contact time because you just want to set them into motion without applying spin. There have been some of these in the World Cup: I think Ronaldo’s second goal was hit very hard, and it just sort of went through [Spanish goalkeeper David] De Gea’s hands.

That also happened recently, I think, in the Champions League Final. It was the other Gareth Bale goal. Did you see that one? [Editor’s note: Yes, I did.] We’re now testing your soccer fandom.

Gareth Bale hit a really nice shot. He hit perfectly cleanly so it had no spin. It was sort of fluttering around, which is why it went through the goalie’s hands, which is really embarrassing for a goalie. But, in any case, that was an example of a ball hit with no spin.

The best way to do that is to minimize the contact time during impact. So these shooters like Roberto Carlos and others who are famous, like Cristiano Ronaldo, for trying to hit these knuckleballs from a free kick situation actually hit the valve. Because at the valve, if you look at the mechanics of a soccer ball, the ball is particularly stiff around the valve. So it’s as if the local pressure is higher there, so you hit it there and you get very little contact with the ball.

So when you see a player rotate and place the ball before a free kick are they potentially setting it up so they hit the valve?

JB: Yep, exactly. When they’re going for a knuckleball, they’ll hit the valve.

What does the ball itself have to do with all of this?

JB: It’s interesting because there is a lot of physics behind it. How it actually moves depends on basically the pressure distribution, the pressure being force per unit area over the surface of the ball. In order to calculate the force you have to integrate that pressure over the surface and that pressure depends on the details of the flow around the ball. And so you can’t actually calculate it because it’s a turbulent flow and the details of that flow depend on the details of the surface roughness on the panels and on the orientation of the seams.

You actually can’t predict it theoretically, even if you want to. So they actually do aerodynamic tests, and every World Cup I’m sent the latest paper. There are people who take soccer balls and they put them in wind tunnels and they look at their aerodynamic behavior, which will ultimately tell you how they’re going to move in the air. But it’s a black art.

Denmark v Australia: Group C - 2018 FIFA World Cup Russia
The Telstar Adidas match ball used during the 2018 FIFA World Cup.
Photo by Stu Forster/Getty Images

Historically, the ball was paneled because they used to be made by stitching bits of leather together. Because of the behavior of those balls, we are accustomed to them spinning so that if you hit it with your instep on the right foot, like Coutinho’s shot, it bends from the right to the left. But you could now manufacture balls without stitching so you can make them perfectly smooth, but those would actually bend in the opposite direction. So if Coutinho took his shot it would have actually bent in the opposite direction. They basically manufacture now so they move in the way that we’re used to.

OK, knowing all this, what type of movement — or lack thereof — is the most difficult to achieve?

JB: The high-to-low drop, I think that’s the hardest. But also the knuckleball is very hard. I’m actually not convinced one is harder than the other. They’re both extremely difficult because they require a very clean strike and a quick release. Those kind of can be blurred because oftentimes when you try get a little bit of topspin it can turn into a knuckleball. So, I kind of put those in the same category. I would put Ronaldo’s free kick in that category. I mean that’s just brilliant and is very hard to do.

The more standard shots bending from side to side is easier because you can apply the spin with your instep our outstep, which a larger area. So there it’s quite easy to get along contact time and apply the spin. A good player should be able to do that, but there are very few people who can hit those topspin shots. Like there is a countable number in the world at any given time.

And Cristiano Ronaldo, as painful as it is for me to say, he’s the best at it right now.

Portugal v Morocco: Group B - 2018 FIFA World Cup Russia
Cristiano Ronaldo celebrates after scoring Portugal’s goal against Morocco at Luzhniki Stadium in Moscow.
Photo by Maddie Meyer/Getty Images