Simple Physical Considerations

by Mark Raugas 
 

People seem to give kendo a hard time for having sharp percussive strikes. The argument goes that the motion has developed out of using the shinai, and is somehow a degeneration of a proper cutting action and that a similar motion performed with a sword would not be effective. I always thought it funny, because in the Kenjutsu I am learning, I've been told that the sword is essentially a percussive weapon, and the cutting action itself, although present and necessary, is secondary to the damage provided by the initial impact. So I was wondering if the arguments against kendo in particular by certain people, on that front, might be misplaced. 
 

After all, have you ever seen the Jigen Ryu kata where Shidachi finds an opening and starts bashing the Uchidachi repeatedly on the hand or head? The one Kim finds funny? On one level it looks weird, but on another, it isn't so out of the ordinary--more indicative of teaching people how to be axe murderers, along the lines of "once you get him once, keep bashing his head until he stops moving". Probably not the greatest for fighting multiple opponents, but it looks like it would get the job done. Those strikes looked very percussive to me. Same thing with Katori Shinto Ryu's vertical cut, chambered off the left shoulder, the one they do repeatedly in their beginning kata. 
 

Lots of people like to make fun of Kendo, and I practice an obscure Ryuha that (God forbid. . .) didn't make it into Watanabe's Bugei Ryua Daijiten, but here are some rather well-known people doing what look like to me very similar motions. So I resorted to some classical physics and tried to get a quantitative feel for a men's kiri cutting action, as well as its cousins, Kiri Oroshi and Keisa Giri, to see what the deal might be. 

I model a cut executed in a purely whipping action--rotating around the right hand, with no forward translation of the right hand at all. I assumed also that the cut stopped at the target, that is, there was no follow-through. So imagine a very focused rotating cut in the vertical plane, stopping after ninety degrees of arc. Some rough measurements put the duration of that kind of motion at no more than 0.25 seconds. Assume you have a blade that is 0.7 meters long, that weighs no more than 1 kilogram. 
 

A rotating body possesses a certain amount of energy and angular momentum. Angular momentum L = Iw where I is the moment of inertia of the body and w is the angular velocity, i.e. w = v/R where v is the tangential velocity to the circle and R is the radius of the circle. So given a fixed angular velocity (e.g. RPM), the farther out on the circle you are, the faster you will be moving. In our case, the sword rotates through 90 degrees in? a second, so it moves through 360 degrees in a second, which makes w = 2 x PI, where PI = 3.141 So w = 6.28 radians per second. 

The energy at the point of impact is E=(0.5)mv2, assuming it is all transferred to the target. So the energy imparted to the target is directly a function of the velocity, and also the radius, since we are dealing with a fixed angular velocity w. In our case, the energy is (0.5) x 1kg x (6.28 radians / second ) x 0.7 meters x (6.28 radians / second) x 0.7 meters =9.67 Joules. The momentum in the tangential direction is, in contrast, P = mv = mwr = 1 kg x 6.28 radians / second x 0.7 meters = 4.4kg m /s. 
 

To find the force acting on the target, we use the relation that FDt=mDv. Assume the penetration takes place over 1/100 of a second, that is over 1/25th of the cutting arc, which for a sword of 0.7-meter length is 6 centimeters. We are modeling a cut that stops not that deep in the skull. So F = 100mDv = 100 x 4.4 = 440 kg m/s2 = 440 Newtons. Pressure is force per unit area, so assuming the Kissaki of the blade had a surface area of .03m x .001m = .00003 m2, then the pressure exerted on the skull would be about 14.6 million Pascal. 
 

For comparison purposes, a 100-gram bullet fired at a 400-meter-per-second velocity (over the speed of sound), with a diameter of 1 centimeter, has a momentum associated with it of 40 kg m/s, and an energy of 8000 Joules. Assuming the impact times were the same, the surface area of the bullet head (assuming it flattened like a .45ACP round), is 0.0007 m2. The pressure exerted by the bullet is then 5.7 million Pascal. 
 

So assuming the person doesn't know how to cut "for real", and just swings the sword very fast in a rotating manner at your forehead, and doesn't follow through, it still carries the same pressure of a .45 ACP slug shot point blank at your head. Of course, the overall momentum is only one-eighth of the value of the .45 ACP slug. The energy of the bullet is 1000 times the energy of the sword cut--assuming the bullet comes to a complete stop within the same distance. So the initial striking motion might not be able to translate enough energy into the opponent to do much beyond breaking the skull and damaging the forebrain. That might be enough, however, to dissuade an attacker. 
 

So much for a purely circular cut. Extending the elbows and tossing the sword forward a bit as you cut, can add more velocity to the weapon. I don't have any references, but using the same guidelines, if you can extend your arms by half a meter in 1/8th of a second, then you have a speed of 4 meters/second. You get at most 4 kg m/s extra momentum from just the motion of the sword. That puts your momentum at 8.4 kg m/s. The energy goes up by 8 joules to 17.67 Joules. If you are at a running pace--i.e. springing forward very fast, your velocity be increased by another 1 m/s, for a total momentum added another 1 kg m/s to the cut, to a total of 9.4 kg m/s momentum and 18.17 Joules of energy. 
 

So how does one increase energy and momentum by a substantial amount? In the last two paragraphs, I've made the tacit assumption that the effective mass of the sword remains at 1 kilogram. When looking at the contribution from the different motions (stepping or sliding forward, extending the arms a bit, rotating the sword), I held the mass of the sword in the equations at 1 kg. I in effect assumed that at the moment you hit the object, you let go of your sword. No one really does that. Instead, people seem to try to lock the sword with their bodies and use the power of their legs hips, and waist to give a more solid cutting action. Now, what happens if we use the total mass of the system (person + sword) in some of the previous equations? Notice, the sword is still taken to not penetrate deeply into the target--purely a surface-level cut. 
 

If we use the mass of the person in the contribution from extending the arms, assuming only half of the mass of the person can be put ideally into that motion, by twisting the hips and extending the arms and rocking slightly, we increase the momentum by 25kg x 4 = 100 kg m/s and the total energy by? x 25 x 4 x 4 = 100 Joules. 
 

If we use the mass of the person in the contribution from stepping or sliding forward, assuming the person is 50 kilograms, we increase the total momentum instead by 50kg x 1 m/s = 50 kgm/s and the total energy by ?x 50kg x (1m/s) x (1m/s) = 25 Joules. Dropping the center of gravity adds energy to the cut, for example, dropping the center of gravity by 10 cm results in a 50-joule increase in energy for someone who has a mass of 50kg. The momentum increases by 75 kg m/s. Then the total numbers for the sword become 230 kg m/s momentum and 210 Joules. This is now about 6 times the momentum of the bullet, but only about 3% of its energy. 
 

If we use the total of the sword + person in the rotational cutting component, we would increase again by 220 kg m/s to a total momentum of 550 kg m/s, and a total energy of about 450 Joules. The problem is, that I am not sure if I can justify using the total mass of the person in the rotational cutting action. I imagine it would factor in the ability to keep the cut on its path and stop correctly in the target, so take all these numbers as an upper bound, given the assumed velocities. The speed of the weapon is very important as far as energy is concerned since it scales like velocity squared. A cut moving twice as fast would yield four times as much energy. So the faster the cut, and the more you can bring your body mass into the cut, the more energy and momentum you wind up bringing to bear on this simulated target. This idea of bringing the body mass into the cutting action may very well be what people refer to as "cutting action." 
 

Also, assuming a 1cm wide wound channel along the length of a Keisa-giri, for example, cutting only with a 20cm long monouchi, along 90 degrees of arc, would yield .0002 meters cubed of tissue displaced. Versus a 1 cm diameter bullet traveling through someone, at most 40cm lets say (horizontally through the torso), giving a wound channel of .00005 So the wound channel of the sword has four times the volume as that of a bullet. If the bullet flattens to four times its size, lead being soft, the numbers on that end of things could be more or less equal. However, the cross-sectional area of the cut is much larger than that of a bullet. A greater number of physiological systems could be affected by a single cut transversing through the arteries at the base of the neck and downward through the lungs and diaphragm into either the liver or the spleen, than a single bullet. So those ideas of total system shock might very well make up for the smaller energy associated to a single sword cut. 
 

"Cutting action" might also serve to pull the blade out of the opponent so that it does not become stuck. Having a blade get stuck would not be the best thing to happen to someone on the prototypical medieval battlefield. Or, in the case of a men-kiri, or Kiri-oroshi, the downward weight displacement might serve to drive the dead opponent downwards in place, as opposed to letting him fall forward onto you. The large momentum of those actions, akin to a pushing action as opposed to a slapping action, might serve to do that. 
 

In conclusion, factors for a "good" cut include primarily the velocity at which the mono uchi is moving and the amount of body mass one can link with the cutting action. Assuming you can link your body with the cut, any forward motion and downward drop of the hips will add substantial amounts of momentum and energy to the cutting action. However, a purely naive cut without any of these amplifying factors still can exert pressure onto a target on the order as that an average bullet might. Which all might very well be nothing new to those who read this, in that case, look at it as a complementary description of what ki might mean in Newtonian terms.