I’m not normally much of a soccer fan, but the World Cup doesn’t happen every day and it’s pretty interesting to see all the excitement and high level of play. I personally think the rules need a little tweaking to reduce the tendency toward 0-0 and 1-1 ties, but I suppose the sport couldn’t have so many billions of fans without doing *something* right.

In honor of the World Cup, let’s do a quick example of just how tough the game can be. In soccer of course the players are generally prohibited from touching the ball with their hands – hence, *football* in most of the world. But in fact the players can and do use anything but the arms to manipulate the ball during play. If you watch the game, you’ll notice the players using their heads to hit the ball in mid-air when kicking is not possible.

This can be a pretty traumatic thing – as in many contact sports the cumulative effect on the brain can be serious in certain cases. Let’s do the math and calculate roughly just how much of a whack heading the ball can produce.

In 1-d accelerated motion, the position as a function of time is given by:

We’d like to find a, the acceleration experienced by the ball. We don’t know x, the distance over which the ball decelerates against the player’s head, but I think we can estimate that it’s about equal to the radius of the ball, which happens to be about ~~35~~ 11 centimeters. [I originally used a wrong value here, which I've noted via strikethrough in the rest of the text.] We also don’t know the time t over which the ball decelerates. But we can estimate that too – since we know that the ball has to decelerate from its initial speed to a stop in the most dramatic case, we can use the equation relating velocity, acceleration, and time:

Substitute that back into the first equation and we can find the acceleration without directly using the time:

I’ve played kind of sloppy here and dropped initial velocity and position terms in the above equations with the understanding that really we’re really working with the change in velocity and position. If you’re currently in a physics class learning this for real, you might want to grab a pen and paper to verify to yourself that this is legit.

Right off the bat we see the acceleration is proportional to the velocity squared, so heading a faster ball will pack a much harder wallop. According to Google, 70 miles per hour is pretty typical of a hard kick. Plugging in that figure and the ~~35~~ 11 cm value for d (mind the unit conversions!) we get an acceleration of ~~1399~~ 4451 m/s^2. To find the force which the head applies to the ball to produce this acceleration, we just multiply by the 420 gram mass of the ball. I get ~~587~~ 1869 newtons, or about ~~132~~ 420 pounds force.

Which is quite a bit to have applied to your head, even for a few milliseconds. Now this is in some sense a worst-case scenario. Most of the time players are not heading high-speed balls directly opposite the original direction of motion. But it does give an idea of the possible hazards of the sport and why sports medicine will always be a booming business.