Let’s Use Physics to Model a Curving Soccer Ball


Paul Kane/Getty Images

It’s that World Cup time of the year—so that means it’s also time to talk about soccer physics. What about the impossible kick? The “impossible” kick has a ball leave the ground and then take a curved path while in the air. Of course it’s not actually impossible, but it is difficult to pull off.

I’m not going to actually kick a soccer ball, but instead I want to use basic physics to model the motion of a ball. Let’s start with the simplest possible case—a soccer ball kicked without the influence of air. Yes, that is unrealistic, but it’s helpful to start with an idealized situation and then make it more complex after that.

Here is a ball at some point after it was kicked.

In this case, there is only one force acting on the ball after it was kicked—the gravitational force. I have also included a vector to represent the momentum of the ball at this instant where the momentum is the product of the mass and velocity. The momentum is important because the nature of a net force is to change the object’s momentum.

Since the force is constant in both magnitude and direction, the motion of an object with only the gravitational force isn’t so difficult to model. The trajectory of the ball can be determined using basic kinematic equations that you would find in your introductory physics textbook. But let’s do this a better way—using a numerical calculation.

A numerical calculation takes a problem (like the motion of the soccer ball over the whole trajectory) and breaks it into many smaller problems. During each of these smaller problems, we can make some approximations to make the calculation simpler. These smaller problems are easier to solve than the whole trajectory, but simplicity comes with a cost. Instead of one difficult problem, you end up with a whole bunch of simple problems. The nice thing is that although there many problems to solve, they are so easy that even a computer can do it.

Here is the numerical recipe that we will use to model the motion of this ball.

  • Start with some initial values for the position and momentum of the ball.
  • Break the motion into small time steps (something like 0.01 seconds).
  • During each time step, calculate the following:
  • Calculate the total force on the object (in this case it’s constant and doesn’t change).
  • Use this total force and the momentum from the previous step (or the initial momentum) to calculate the momentum at the end of the step.
  • Use the momentum to find the new position of the ball.
  • Keeping doing the above steps until you get bored or the computer revolts and decides to not work for you anymore.

That’s the basic process. You can do this…

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Peter Bordes

Exec Chairman & Founder at oneQube
Exec Chairman & Founder of oneQube the leading audience development automation platfrom. Entrepreneur, top 100 most influential angel investors in social media who loves digital innovation, social media marketing. Adventure travel and fishing junkie.
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