You can call it football if it makes you happy. Anyway, this is a popular story going around. The physics of the magic curving soccer kick. Here are two ends of the spectrum.

First, there is the lower, easier to consume version from io9.com

Physics forced to come up with whole new equation to explain “impossible” soccer kick

I will summarize this article for you:

“Have you seen these crazy soccer kicks where the ball curves? It happens because the ball spins and physics. Here is a video”

Oh, and they have a diagram – which doesn’t seem to come from the original paper and they also have some nifty real-life soccer videos. I think this story is a little too light on the details. They could have done just a little bit more to make this a much better article. Essentially they said that the ball curves because of magic (but magic is physics).

Then, there is the original article on the motion of spinning objects (which talks about soccer at the end) from the New Journal of Physics – IOP:

The spinning ball spiral – Guillaume Dupeux, Anne Le Goff, David Quéré and Christophe Clanet

Let me select one tiny part of the paper to show you: (they used pictures for some of the variables, so some of this might not appear exactly as the author intended – but you will get the idea):

“The motion of the sphere of mass M is described in the Serret-Frenet coordinate system introduced in figure 2. We first focus on the direction . The Reynolds number Re = ρU0 R/η is of the order of 104, which implies a drag F1/2ρU2πR2·CD, with CD0.4 [28]. The equation of motion along thus is written as”

They lost me at “Serret-Frenet” coordinate system. So, this doesn’t appear to be consumable for the more general audiences.

**Update:** While looking for some soccer ball data, I found a third article. The first was too cold, the second was too hot, but this one was just right for Goldilocks. This is from physicsworld.com.

Like I said, I think this last article gives a better mix of understandablity and physics.

### The missing parts

I am going to try to fill in the middle between the io9.com article and the original article. I might fail, but I am going to try. (even though the third article did a pretty good job)

So, you kick a ball. What forces act on the ball? Well, the easy thing is to say “gravity and stuff that touches the ball”. In this case, the only thing the ball touches is the air. The air does indeed exert a force on the ball. The force the air exerts on the ball is ultimately due to collisions with the air particles and the ball. If the ball is spinning and non-smooth, there can be complex interactions. For this case, I will break this air force into two components.

- Air drag. If you have read this blog, you should be familiar with this model of air drag that says the force is proportional to the magnitude of the velocity squared and some other stuff (density of air, cross sectional area, and shape of the object).
- Magnus force. This is the force exerted on a moving and spinning object in a fluid or gas. Wikipedia’s page on the magnus effect is pretty ok.

There is also the gravitational force. But, let me just look at the ball from the top view. The key point of all of this is that if there were no spin effect or air drag, the ball would just move in a nice parabola. From the top, this would look like a straight and constant speed trajectory. If you exert a force perpendicular to the direction of motion, the ball will turn. If you exert a force in the opposite direction of the motion, the ball will slow down. These two things together make the ball do what it does.

Here is a force diagram of the ball as seen from the top (so you don’t see the gravitational force):

Why does this spinning cause a sideways force? Well, the idea is that the rough surface of the ball moves air near its surface. This means that on one side of the ball, the air is moving faster than the other side. On the faster moving air side, the air is moving more in a direction parallel to the motion of the ball. This means that an air particle is less likely to collide on the side of the ball and push it that way. The result is that there is more collisions on the slower side of the ball.

### Modeling air interaction

Here is the model that is commonly used for the air drag force:

Where the v-hat is a unit vector in the direction of the velocity of the ball. This along with the negative sign means that the air drag force is in the opposite direction as the velocity.

The magnus force can be written as:

S is some constant for the air resistance of the ball (a basketball and a soccer ball would have different values). The vector ω is the vector representing the angular velocity of the ball. For the diagram shown above, the vector ω would be perpendicular to the plane of the computer screen and coming out of the computer screen. The mangus force is related to the cross product of ω and the velocity. (here are some cross product tips).

Why don’t you always notice these forces? If the speed is slow and the mass is large, then the air drag and magnus forces will be small compared to the gravitational force. The motion for these cases will be dominated by the gravitational interaction. But with a high speed kick from a soccer ball (that has a relatively low mass) with a high angular spin, the effects can be noticed.

Let me model a high speed soccer ball in vpython. The original research paper gives some nice parameters that I will need for a soccer ball.

- Radius = 0.105 meters
- density = 74 times the density of air (if I understand the table correctly)
- S = 0.21 – I am pretty sure the S in this paper is the same S in the magnus force described above. – forget this S

After playing around (and finding that third article) I am pretty sure the S above is not the same S as in the wikipedia page. The physicsworld article gives the following useful info:

- Ball speed = 25-30 m/s
- angular velocity = 8 – 10 rev/sec
- Lift force (magnus force) of about 3.5 N
- horizontal ball deviation of about 4 meters
- ball mass of 410-450 grams (which agrees with my previous density)
- ball acceleration of about 8 m/s
^{2}– not sure if this is just the linear acceleration or the total magnitude of the acceleration and at the beginning or average?

If I assume the magnus force is S times the cross product of the angular and linear velocity, I can work backwards to find S (from the physicsworld data) in the case that the velocity and angular velocity are perpendicular.

Now for some python (here is my sloppy code –

Here is what I get for the trajectory of the ball (as seen from above).

That is more than 4 meters deflection – but maybe they are assuming you aim to the left a little or something.

How about a plot of the total acceleration (magnitude) as a function of time.

This gives an acceleration of around 8 m/s^{2} around the end of the motion. Maybe this is what the physicsworld author meant. Oh well, that is enough for this. I know there is one problem. I assumed a constant coefficient of drag, but it seems that this might not be true.