# The physics behind freekicks and passing

Writing about the physics of shots is always a good excuse to mention some of the greatest free kicks of all time. Roberto Carlos 1997 goal against France in 1997, where the ball bent round the wall and curved its way in from 35m out. Mikael Nilsson Champions League free kick for IFK Göteborg that sent the keeper PSV Eindhoven first one way and then the other, before flying past him. In more recent times there are the goals of Dimitri Payet and Willian that entertain us by spinning the ball in to the back of the net.

NASA already has a research team dedicated to explaining these free kicks. They have even created an online shot simulator, where you can enter the position, direction, forces and spin on a ball and calculate whether or not it will hit the target. Below I recreated Dimitri Payet’s long range effort for France against Russia from last year using the simulator.

Playing around with the simulator, which is freely available online, gives an idea just how much advanced physics is involved in the perfect free kick.

At the MIT Sloan Analytics conference, physicist William Spearman unveiled his solution to an even more difficult problem than tracing the path of free kicks: describing how players pass and receive the ball during open play.

The model he uses for ball flight is simplified, and ignores the so-called Magnus effect, which Roberto Carlos and Mikael Nilsson both exploited to curve their free kicks in to the goal. But his model does account for drag and gravity, the two other forces that determine how a ball flies.

William’s model, created together with his colleagues at Hudl, works out the probability of a ball being received by its target and the probability of it being intercepted. He does this by working out how fast a player can move towards the ball’s trajectory, given their current speed, direction and the time they have to react to the pass. An example from William and his collaegues research paper is shown below.

In the example, the pass trajectory goes past three players. The nearest player has the shortest time to react and thus only has a 17.78% chance of getting the ball. The second nearest player has more time and has a larger chance (56.02%) of getting there. The third player will only collect the ball if the first two miss it.

I my last article, I argued that the biggest challenge to understanding tactical performance in football is in measuring which teams control which areas of the field. Since William’s model can calculate the player who will get to a ball first, it allows him to measure who controls which area of the pitch. An example video is shown below.

In this clip the blue team score a goal with a pass from 24 through to 28. William demonstrates how the pass might have been prevented, by moving the defending red team players around. As he moves the players the areas they control changes. In the match, player 24 did make the pass (which according to William’s model had a 26% chance of success) and it cut through the defence. Player 28 went on to score.

Putting physics in to ball and player movement adds a whole new dimension to football analytics. William Spearman told me that he is, “extremely excited about the advent of tracking data in football because it gives us an objective dataset that can be used to analyze space, control, and counterfactual scenarios.”

His work with Hudl is amongst the most advanced currently available and it will be interesting to see how and when clubs integrate these tools in to post match analysis. When I asked him how he saw the future, he told me that he believes that new tactics will start emerge from a simple set of rules that describe player and ball dynamics. It’ll be interesting to see if he is right, and one day tactics will be designed by computer simulation.

This article originally appeared on the Nordic Bet blog.

Professor of Applied Mathematics. Books: The Ten Equations (2020); Outnumbered (2018); Soccermatics (2016) and Collective Animal Behavior (2010).

Professor of Applied Mathematics. Books: The Ten Equations (2020); Outnumbered (2018); Soccermatics (2016) and Collective Animal Behavior (2010).