Martial arts and personal injury seem to go hand in hand. In my decade or so of jujitsu training, I broke two toes, dislocated my wrist, endured countless bruises and sprains, fractured my right elbow, and had my back thrown seriously out of whack thanks to a powerful guillotine choke administered by an over-enthusiastic ex-Marine. Most notably while practicing self-defense techniques against a long bo (actually a stickball bat) for my black belt test, I miscalculated during a duck-under technique, came up too soon, and received a nasty thwack! on the forehead that dropped me to one knee. It didn't actually hurt; I was mostly stunned. Then the blood began to gush, and I realized mat time was over, and I'd be spending several hours in the ER instead. The slash went clear down to the skull cap, and required 14 stitches to close. I still have a jagged scar across my forehead, although few people notice unless I point it out. (Also? No need for future botox treatments in that area, since the tiny muscle that causes frown wrinkles got sliced clean through. I literally cannot frown in that portion of my forehead.)
And I still loved every minute of my training. It's just part of the rite of passage when one is seriously studying the martial arts, but to outsiders, it can seem a bit, well, extreme. (A friend of mine became so upset at the perceived brutality of my black belt test, he literally had to leave the room at one point.) I was reminded of my halcyon days sweating and bleeding with my fellow jujitsu practitioners when I received an email from my friend Jim D., who started training in Tae Kwon Do a few years ago with his teenaged son, and recently passed his brown belt exam. Jim fractured his wrist this past week when he agreed to hold six pine boards while his instructor attempted to break them with a kick. Apparently, the instructor missed the central target area and the full force of his kick landed off to the side, so all that kinetic energy (or should one say momentum? Terminology can be so confusing!) went into Jim's wrist instead of into the board. Ouchie! At least he has a very impressive looking cast with which to impress the laydeez:
Anyway, Jim took the injury in stride, like any respectable martial artist. But he's the curious sort, so he emailed me asking if I knew anything about what went on from a physics standpoint to bring about his injury. I've done lecture/demos on the topic, focusing on broad concepts as opposed to specific calculations, so I knew a little, even though I never featured board-breaking in any of my lectures. Frankly, I've never understood the point of such an exercise. I'm an adherent of the Bruce Lee philosophy, immortalized in Enter the Dragon: "Boards don't hit back."
Except in the strictest physics sense, they kinda do. Per Newton's third law, momentum is conserved, and that translates into the well-known maxim of equal and opposite reaction. Many of us remember this from our introductory physics classes (or the equivalent thereof): If an object exerts a force on another object for a specific length of time, the second object will react by exerting an equal but opposite force for the same amount of time. So a board does "hit back" in that sense. The force generated by Jim's instructor created a reaction force in the opposite direction when his foot made contact with the boards; the boards gained exactly the amount of momentum the instructor's foot lost, or almost as much; some would have been lost via conversion into heat or noise energy, for example. The boards accelerated in the opposite direction in response to the kick.
A board will break when the part that is hit -- ideally the center -- is infused with more energy than its structure can handle, causing it to crack and/or break. But not every part of the board accelerates uniformly. The part that took the brunt of the kick -- again, ideally the center, although unfortunately for Jim, in his case it was off to the side -- accelerate much more than the surrounding piney-parts. This produces a localized strain, and if the strain becomes too great, the board will crack in that locale. As for how much force went into Jim's poor wrist, I could only offer the grossest generalities. For someone weighing 140 pounds, traveling at a final velocity of about 10 MPH when s/he hits the target, that person's body would have about 504 joules of energy. But that's assuming a full-tilt run and putting one's entire body mass behind a kick. Chances are, only a portion of one's body mass will be used -- although a TKD instructor, one assumes, would have excellent technique, and would therefore employ a greater percentage of his/her overall body mass than the average untrained kicker.
I told Jim if he wanted a truly thorough answer to his question, rather than the generalities I could offer, he should contact Jearl Walker, physics professor at Cleveland State University in Ohio, longtime contributor to Scientific American, and general all-around daredevil scientist, trying his hand at firewalking, lying down on a bed of nails, and investigating the physics of the martial arts. (He has a book -- and a Website -- called The Flying Circus of Physics detailing various real-world illustrative examples of physics concepts, as well as a blog.)
Per Walker, the force required to break a standard single 3/4-inch pine board is about 3000 Newtons; the force required to break a solid pine block of the same thickness as six stacked standard pine boards is astronomically higher: six times higher, as one might expect (6 x 3000 Newtons), [CORRECTION: I didn't read my hastily scrawled notes correctly: that should be 6 CUBED x 3000 Newtons. Yowza! 6 pine boards would be about 18,000 Newtons and change.] although there are numerous variables, such as whether the boards are warped, how many have knots (which make them harder to break), and how much space is between each board. Still, that's a pretty good ballpark figure. To phrase it in slightly different units, it takes about 5 joules of energy to break one board, and about 30 joules to break six. And a large fraction of that energy went into Jim's wrist instead of into the board. (Also per Walker, it was the focused shock wave that broke Jim's arm, not a static wave of energy.)
I think it would be a bit more difficult to determine the force of impact of my old head injury, although if any of you are bored over the weekend and care to give it a shot, we'd all be interested in hearing what you came up with. The mass of the stickball bat could probably be estimated, along with the respective body masses of me and my friend Jordan, who was swinging the stick. Then we'd need to estimate the speed of the swing (Jordan's pretty big, and strong, and to his credit, respected me enough to not pull his punches, so to speak), and how fast the stickball bat was traveling when it struck my head. The hardness of my head might also be a factor; all materials have their own varying degree of elasticity, after all. Complicating matters is the fact that both Jordan and I were moving when the injury occurred, and because of that, it was more of a sharp, glancing blow that slashed across my forehead at a downward angle. That's probably why I escaped without a concussion or more serious skull fracture.
Of course, like most head wounds, it bled like crazy. There are a lot of arteries, veins and capillaries in the head, since the brain requires a constant supply of oxygen- and glucose-rich blood to function properly. Also, I'd been exercising for a good hour by then, so the blood was really pumping. Funny side story: My chief instructor was chatting with a visitor to the dojo when my head injury occurred, his back to the mat. The visitor, gazing at the gushing blood in horror, mentioned that I'd been hurt, and really, oughtn't someone to do something? My instructor was used to people worrying about my welfare (there were very few women in my chosen style, and a hard fall from, say, a judo throw can look much worse than it really is). So he just waved it off and said, "It's okay -- she gets back up." Quoth the guest, "But... but... she's bleeding all over the mat!" That got his attention. He swung around, and immediately warned, "Don't you bleed on my mat!" Too late!
Ironically, when I examined my gi upon getting home, there was hardly any blood on it at all, just a bit of staining around the collar when the ER doc rinsed the matted blood from my hair. It had spurted outward in impressive gushes, hitting pretty much everyone in the vicinity except me -- just like the Black Knight's arterial sprays in Monty Python and the Holy Grail. ("It's just a flesh wound!") That's not what I would have expected. Granted, we aren't talking about arterial spatter of the sort one sees routinely these days on C.S.I. Still, the blood pumping through my head at that point was clearly moving at a high enough pressure to cause an arterial-like spurt. Which meant I didn't need to soak my gi in bleach for two days, like a few of my unfortunate fellow students. (One guy -- who kindly administered pressure to my wound to stem the bleeding while waiting for the ambulance to arrive, and thus had my blood all over him -- just gave up bought a new gi. Thanks, Vito!)
The C.S.I. franchise certainly has its critics when it comes to how it depicts forensic science, but in fact, there is such a thing as bloodstain pattern interpretation that can be used to piece together the events that gave rise to a particular pattern of spatter or bloodstains. Experts trained in this approach consider such qualities as the viscosity of blood, the specific gravitational forces acting upon it, and the role of surface tension. (For instance, a bit of blood that falls off a pricked finger will round out into a sphere because surface tension acts to reduce surface area to the absolute minimum possible.) Here's some fun facts I learned about blood spatter (illustrated by some handy photos) from this excellent Website:
* Blood cast from a moving source will make smaller droplets than blood cast from a stationary source.
* Blood follows the same basic laws of physics as any projectile in motion. (This means it should be possible, in principle, to calculate the trajectory of the blood spurting from my head wound and predict where it would land on the mat, so everyone could steer clear of that spot.)
* The terminal velocity of a falling blood drop depends on its size: smaller drops have a lower terminal velocity and reach that point after a shorter fall distance that larger droplets (which accelerate over a greater distance and thus reach a greater terminal velocity).
* The shape of the blood spot depends in part on the texture of the surface on which it calls. If it falls on smooth glass, it will be circular and fairly uniform in shape. If it calls on a textured surface, such as paper, or wood (or a judo mat), the shape won't be nearly as regular. In general, the harder the surface, the less spatter there will be. If a blood drop hits a surface that is both hard and smooth, it will break apart upon impact into smaller droplets -- and those offspring droplets will continue to move in the same direction as the parent drop.
* The angle of impact also determines a blood drop's final shape. For instance, a vertical drop onto a smooth target tilted at 90 degrees results in a circular stain, and as the angle decreases, the stain becomes more elongated, and its length-to-width ratio increases accordingly.
* Finally, blood spatter patterns are classified according to the velocity with which the blood struck a given surface. For instance, spatter patterns occur when blood is projected at a velocity greater than the force of gravity, such as what occurs when blood is cast off a weapon. (Per the site, "The direction and origin of the backswing is often clearly discernible.") Low-velocity blood spatter is basically what happens when the stuff just drips downward from a cut. A blow with a baseball bat would constitute medium-velocity blood spatter, producing spots of about 4 mm in diameter, while a gunshot will produce high-velocity spatter and a "fine mist" of spots less than 1 mm in diameter. Arterial spurting is a category all its own.
The Black Knight would have been fascinated, I'm sure, to hear King Arthur ruminate on these matters (far more interesting than determining the air speed velocity of an unladen sparrow, both African and European varieties). There's much, much more to do with the science of blood spatter, and the physics of the martial arts (judo throws are a specialty all their own), but I suspect I've grossed everyone out enough for one day.