This is Why … The Beautiful Game is Beautiful Physics

(Image of football/soccer player in the rain under stadium lights, from Shutterstock.com)

A few days ago, my football-obsessed cousin, James, commented on Twitter on a video of an incredible header by superstar athlete Cristiano Ronaldo, concluding that the “laws of physics don’t apply to [Ronaldo]”. It wasn’t intentional, but a statement like that is a bit like waving a red flag at a bull and I was definitely intrigued. So here’s the full clip and let’s just stop for a moment and admire the INCREDIBLE skill on display:

Ronaldo scoring the game-winning goal with an incredible header in a 2-1 win over Sampdoria on Dec 18, 2019

What I also love about this clip is that it shows three great examples of projectile motion:

  1. the kick of the ball from the left wing, crossing to Ronaldo
  2. the perfectly timed leap of Ronaldo to meet the ball in mid air
  3. the arc of the ball from Ronaldo’s head to the back of the net, with the keeper looking on in dismay

Projectile motion is what physicists call any instance in which an object is moving through the air without any propulsion system, so my older daughter and her horse going over a jump, a dolphin leaping out of the surf, or my younger daughter diving into the pool. It does not, however, describe the motion of objects that have engines like airplanes or drones.

(from Shutterstock.com)

The common element is that the path is an arc shape – mathematically we call it a parabola. As it reaches the very top of the arc, the object is travelling horizontally for just a moment before falling back down to ground level. The eye-catching thing about Ronaldo’s jump is that he seems to travel horizontally at the top of his arc for an insane amount of time. Is he truly superhuman?

I had to find out. Using freeware called Tracker (https://tracker.physlets.org/), I can tag an object and look at its position in both horizontal and vertical directions in each frame of the video. When I tracked the motion of Ronaldo’s nose, I got this weird plot that suggests that he can defy gravity! Instead of an arc that is symmetric around the highest point, Ronaldo’s path appears to begin to drop and then miraculously rise up again before he heads the ball into the net! What????

I thought maybe Ronaldo was doing something weird with his head in preparation for contact with the ball, so I then tracked his torso (the white circle inside the P at the end of his logo, to be exact). Same thing …. how could there be two peaks in his path?? Unless he actually IS superhuman?

Nope – it’s all a trick of the eye/brain! The issue here is that the camera is moving both in the horizontal (x) and vertical (y) directions while it tracks this amazing play. There is also relative motion between the camera and Ronaldo in the forwards/backwards direction (z), and the problem is that we see the motion of Ronaldo relative to the camera instead of relative to the ground. To test my hypothesis, I tracked the “motion” of one of the metal bars at the top of the barrier at the sidelines – a stationary object that is in view most of the time. The resulting curve tells us that the camera was moving upwards as Ronaldo was moving up and then begins to move down as Ronaldo also moves down. Because the camera operator does such a great job of following Ronaldo’s motion, it appears as though he is hovering in thin air while patiently waiting for the cross to arrive. So, when I corrected Ronaldo’s tracking data for the motion of the camera, this is what I get:

The double bump has disappeared! It’s not exactly what we expect to see since it’s still a bit flat on top and not completely symmetric, but that’s because I have no way of correcting for the camera’s motion towards/away from Ronaldo. To properly track his jump, we need a video taken from a stationary camera that is always looking at Ronaldo directly from the side, like this:

Our daughter, Mara, enjoying the waterslide on our rented houseboat last summer

Because of how we filmed this clip, we can analyze it quantitatively using the known slide height to determine the scale of our measurements. Based on her steadily increasing vertical speed, we can fit a straight line to these data and determine the slope, which tells us the rate at which Mara’s speed is increasing, or her acceleration.

The circled slope of “-10.12” tells us that Mara’s speed is increasing by ~10 m/s every second, or an acceleration of 10 m/s2, which is awfully close to 9.8 m/s2, the value we expect for acceleration due to gravity!

Now let me be perfectly clear, this little bit of video analysis is in no way meant to detract from Cristiano Ronaldo’s incredible skills. This header was scored in the 45th minute just before half time to put his team, Juventus, ahead 2-1, and ended up being the game winner in a home match against Sampdoria back in December 2019. Ronaldo finished the season last year as the second highest goal scorer, with a pretty insane total of 31. He is arguably one of the top two players in the world, but even Ronaldo plays by the rules of physics!

Published by joanneomeara

Professor, Department of Physics, University of Guelph

One thought on “This is Why … The Beautiful Game is Beautiful Physics

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