Built on Facts

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.

Comments

#1 Rory Kent
June 24, 2010

“[...] about equal to the radius of the ball, which happens to be about 35 centimeters”

Are you sure about this figure? That seems like an awfully big ball to me.
According to Wikipedia, the ball should be 70cm in cicumference, which would give a radius ~11.1cm.
#2 Jim Atkins
June 24, 2010

I think you have the radius of the ball wrong- as a red-blooded Amurrican, I have no use for effete foreign sports, but a football 70 cm in diameter seems awfully large! Looking on some WikiUselessTrivia site, it appears that 70cm is the circumference.
#3 Matt Springer
June 24, 2010

I pulled the number from Wikipedia, and it looks like I misread “circumference” as “diameter”. I should have donw what I always tell my students to do – consider the reasonableness of the answer. Ouch for me!

I’ll get it fixed pronto.
#4 dWj
June 24, 2010

I’d have used Fd=1/2 mv^2, which you can divide by md to get your final equation, or just by d if what you’re ultimately looking to calculate is force.
#5 Rob Monkey
June 24, 2010

Perhaps this is the force on the player’s head not when they use their head to hit the ball but when they headbutt the ref for being an incompetent wanker? The reasonableness checks out for me
#6 Rory Kent
June 24, 2010

Or when they headbutt other players for insulting their mothers.
#7 Tercel
June 24, 2010

Is this really that much, when applied briefly? Certainly it can hurt, but the pain is related to the pressure, not the force. Presumably if the ball is slowing down over a distance of one radius, it is also getting squished pretty thoroughly over the footballer’s forehead, thus reducing the pressure.

In terms of the brain trauma this can induce, that would be related to the acceleration of the skull. Given that the head is significantly more massive than the ball, and is being supported by neck muscles that cannot be neglected in this situation, the acceleration is not necessarily excessive.
#8 Karen
June 24, 2010

Footie (or soccer if you prefer) is definitely a dangerous sport, even if it can’t live up to the hazard level of something like American football, or the extreme hazard level of rugby.

These numbers are very interesting to me, since I actually received my very first concussion (at the tender age of 10) when I got hit in the face by a football kicked at point-blank range. Threw my head back and clicked my teeth together hard enough for brain trauma.
#9 Karen
June 24, 2010

Tercel is generally correct, though: the goal when heading the ball is to keep your head relatively still and then nod forwards slightly, using your neck muscles to keep your head pretty steady and avoid injuries like the one I suffered.
#10 Flavin
June 25, 2010

Tercel:
Is this really that much, when applied briefly?

If the momentum change happens briefly, it actually does more damage than a momentum change that takes place over a longer time. From the wikipedia article on impulse:

A small force applied for a long time can produce the same momentum change as a large force applied briefly, because it is the product of the force and the time for which it is applied that is important.
#11 Cait
June 25, 2010

Karen is right about the neck muscles; they make all the difference in the world. I don’t know why it works, but being sure that your head is braced and, in fact, moving forward slightly, makes the impact considerably less painful.

Also, the rule our coaches always gave us for heading (I played varsity at high school and university) was this: open your eyes, and close your mouth. Leaving the mouth open leads to the possibility of the tongue hanging out, and the impact is more than enough to snap the jaw shut in a hurry, which can be quite unpleasant for the tongue. Keep the eyes open helps to ensure that the ball is properly steered onto the forehead, and not (far more painfully) landing on the crown.
#12 Tercel
June 25, 2010

@Flavin

Yes, that’s correct. For a given change in momentum, the force will increase as the time for the change decreases. I, however, was talking about a given force applied over a short time, for a correspondingly small change in momentum, and therefore a small acceleration.

You assumed I was confusing force and impulse, which I was not.
#13 red pepper
June 26, 2010

I don’t know why it works, but being sure that your head is braced and, in fact, moving forward slightly, makes the impact considerably less painful.
#14 hf
June 26, 2010

“I personally think the rules need a little tweaking to reduce the tendency toward 0-0 and 1-1 ties”

Sorry, but you’re wrong there.
#15 CCPhysicist
June 27, 2010

The reason the neck muscles matter is because they act to make the head part of the body. Thus, when you go to calculate the acceleration of the head (which is what matters for brain trauma that results as the skull accelerates into the brain), you divide by the mass of the upper body (or some similarly large effective mass) rather than just the mass of the head.

With just the mass ratio of 10 to 1, the head would have an acceleration of 45 Gs for the numbers in your example. I couldn’t find a value for concussions, but that is getting close to the fatality level used by NHTSA.

BTW, I don’t think a typical header involves a ball going 70 mph (103 ft/s or 31.3 m/s) unless it involves deflecting a point-blank direct kick. Even a long goal kick will be slowed considerably by air drag before it gets headed.

It would, however, be interesting to consider the problem of an inelastic collision with a soccer ball in the case of a shot like the one where I saw the goalie duck rather than stop a point-blank shot with his face.
#16 Cait
June 29, 2010

Oh, thanks, CCPhysicist, that makes perfect sense – I’ll share that with my teammates.
#17 Ian Preston
July 3, 2010

@CCPhysicist:

This article puts the typical acceleration at about 10g and compares that to “minimum values for the development of sport related concussion” of 40-60g.

Footballs are much lighter now than they used to be though. Before the 1970s people were regularly heading heavy rain-soaked leather balls and there’s concern about whether that has had long run consequences.