# What was that?! Quantifying the impact of a martial arts strike

Measuring things like a martial artist's strike can be challenging because it is a complex motion and common vernacular will use imprecise or incorrect terms. This question is an attempt to identify the applicable measurements (quantification) and the correct units behind those numbers.

Background:
In a recent question of mine, I asked for sensors that could be used to measure a martial artist's strike, such as a kick or punch. My primary intent was to demonstrate to the student how their technique improved their strike. The answers there are great, and definitely addressed my question.

What led to this question is that I received a follow-up, clarifying comment asking if I wanted to measure force, energy, or power. To which I responded that I simply needed something measurable to show the difference or improvement to the student.

There was a subsequent comment* that got me thinking:

From what you write here, I would like to add that it is neither force nor pressure nor energy that you actually want to measure.

*The comment has since been deleted, but I started this question shortly after it was left.

What got me thinking about that second comment is while I've seen many references using units of force, there are other ways of quantifying a martial arts strike.

For example:

• This reference prefers to use 'force' and condemns using 'pressure'
• This reference also uses force, but mixes up their terminology when describing acceleration and velocity, so their usage of terminology is suspect.
• This reference focuses on velocity, momentum, and energy.
• This reference uses $lb_F$ for their measurements. Coincidentally, this is the series that started me down my original line of thought with the other question.

My question:
What is the correct terminology for measuring (quantifying) the magnitude of a martial artist's strike versus another strike?

The measurement should be applicable across a variety of strikes (punches, kicks, elbows, etc...) and valid for different practitioners. Likewise, I'm not worried about defining good versus bad; I want to be able to quantify the strike. This ties into basic process improvement - we measure the process we want to improve and compare the measured results.

• Force is something defined at an instance in time. In your original post, I assumed that you would be most interested in the maximum force during a given strike. However, I think impulse may actually be the most relevant quantity. The integral of force over the time period during which it is delivered. Feb 23 '15 at 12:52
• It would be nice if there were a "one size fits all" answer to your question, but i agree with olin in that the best metric really depends on the objective of the strike.
– Paul
Feb 23 '15 at 16:52
• ...and ultimately, the question should really first be asked on a sports site - it's really a martial arts question foremost, biotechnology second...
– SF.
Feb 24 '15 at 23:19
• @SF. - Obviously I'm biased, but this is a question about measurement first and domain or application second. Terminology behind a measurement is very much apropos to Engineering.
– user16
Feb 25 '15 at 1:32
• @GlenH7: Establishing what to measure is more of the original domain though. Engineering can tell how to measure it.
– SF.
Feb 25 '15 at 12:35

It all comes down to force.

We all know Newton's second law: $\vec{F}=m\vec{a}$1. It says that the greater the force that is applied to an object, the greater its acceleration will be (and acceleration is really just the second derivative of position with respect to time).

How do you measure force? One common method, used to measure the weight of an object, is to use a spring scale. You can either place an object on top of the spring or hang it from a spring. Be careful, though - when hanging it from a spring, be aware that the scale directly measures tension. Move in it some kinky direction and you'll have weird readings.

The equation for the force on a spring is $\vec{F}=k\vec{x}$, where $\vec{x}$ is the displacement2 and $k$ is the spring constant of the spring. So if you push against a spring, you can figure out the force applied.

You mentioned momentum. If you want to be snarky (but accurate), use impulse - the change in momentum of an object (or system). Impulse can be written as $\vec{I}$ of $\vec{J}$. I've seen both conventions used. Regardless, though, impulse is defined as $$\vec{I}=\vec{J}=\vec{F} \Delta t=\Delta \vec{p} =m\vec{v}_f-m\vec{v}_o$$ A greater force means a greater change in momentum. A force applied for a greater time interval means a greater change in momentum. By the way, divide all the terms by mass and you get that familiar kinematics equation: $$\vec{a}t=\vec{v}_f-\vec{v}_o$$

Energy is a weird choice. Unless you're punching your opponent straight into the air, you'll be transferring some chemical energy into kinetic energy. If you want to ignore your body's energy conversion, just say that the force does work on the opponent's body.

Back in high school (i.e. algebra-based) physics, work was defined as3 $$\vec{W}=\vec{F}s$$ However, your fist is most likely not applying the same amount of force while it travels and is in contact with your opponent. So we have to use calculus: $$\vec{W}=- \int_{x_o}^{x_f} \vec{F} dx$$ Mind the minus sign.

So the energy transferred is equal to the work done by the force.

Power is simply energy (here it's work) divided by time: $$\vec{P}=\frac{\vec{W}}{t}$$ The relationships there are implied. But here, energy is equal to the work done. And that's really just a function of force, $F$.

So impulse, energy and power are all measures of force. Same goes for pressure. Take the proper derivatives and you'll see: $$\frac{d\vec{p}}{dt}=\frac{d\vec{W}}{ds}=\frac{d^2\vec{P}}{dt \cdot ds}=A\vec{P}=\vec{F}$$

I'm not sure how well that addresses your question, though. All of these things are different entities, and what you measure depends on how you measure it. If you measure how fast your opponent moves after the punch (and his mass), you can measure impulse or work4.

There's no "right way" or "wrong way" to measure the magnitude of the strike. In all these cases, the greater the quantity, the greater the force that has been applied. The only difference, for practical purposes, is that the force is divided amongst something (area), applied for a given amount of time (impulse), applied over a given distance (work), or applied over a given distance in a certain amount of time (power). That's it. Call the quantity $\vec{Q}$, and you get $\vec{Q} = \vec{F} \cdot d\text{whatever}$5. Just pick your $d \text{something}$.

1 Also written as $F=m \frac{d^2x}{dt^2}$
2 Displacement is a vector and thus force is a vector. Same goes for acceleration.
3 I'm ignoring the $\cos \theta$ I really should throw in there, for simplicity.
4 $\vec{J}=m\vec{v}_f-m\vec{v}_o$ and $KE=\frac{1}{2}mv^2$. Take your pick.
5 Yeah, yeah. Force divided by $d \text{something}$ in the case of power, but multiplied by $d \text{something else}$, too.

• I apologize for all the $\vec{}$s floating around, but I wanted to make a point about what things are vectors and what are not, to make my point clearer. Feb 22 '15 at 22:20

The basic problem is that you haven't defined what a "good" kick or punch is. Is the purpose make the opponent lose his balance? To inflict the maximum pain? To break bones? To hit a particular part of the body to disable him? Shatter the most wooden boards stacked on top of each other? Something else?

Each of these different outcomes will weigh various physical properties of the strike differently in coming up with a single number of effectiveness. For example, getting someone to lose balance, even fall over, is mostly about impulse, which is force times time, but as long as the time is short enough. Breaking bones is more about maximum force. Inflicting a concussion (as is the goal of a knockout in boxing) is probably more about overall momentum, but also targeted in the right place.

In short, your question can't be answered without a definition of how "good" is measured.

• All of the chastisement for not defining 'good' seems silly since your middle paragraph explains what 'good' would be in a number of different situations. Why not just expand on that? Feb 23 '15 at 18:22
• @Chris: I give a few examples of different desired outcomes to show that different metrics are required. I don't want to answer for every possible outcome, with one of them being what the OP wants. It makes more sense for the OP to clarify what the desired outcome is. Feb 23 '15 at 19:34

If I were tasked with measuring this I would also want to know what the purpose or desired outcome of each strike was. This is because you may want to measure different things for different types of strike.

• A strike that is designed to cause damage must focus the force on a small area in order to maximize the pressure at that point - so for that type of strike you would want to measure the pressure.
• A strike that is designed to impart a force on the target (for example to wind them) - you would need measure the force applied at the area of impact.
• A strike that is designed to push the target away - you should measure how much of the force was transferred to the target and how much back into the striker (kinetic energy before and after the strike).
• A strike designed to startle and confuse the opponent may rely on speed.

There may be other scenarios too, but the main ones will be interested in force and pressure. In the third bullet above, you would want to measure the kinetic energy of both parties before and after the strike, this could be done with a high-speed camera.

So the confusingly contradicting references you cited may not be contradictory at all, they may just be focusing on measuring different things because of different types of strike (or different desired outcomes for the same strike types, in which case I imagine the technique may be different as well as what you want to measure).

With a strike or kick, you move (more or less) you whole body, connect a hand or foot or whatnot to your target and if your technique is good, the target will deform far more than your body. I think this thought is crucial, as there are a lot of joints and squishy things between the part of your body hitting the target and your center of mass. This also means that a strike that may apply a lot of energy to a light focus mitt may be less effective when applied to a heavy bag, because in the latter case the squishy bits squish more. As we all know, the idea with striking someone is to have only little total and no plastic deformation in our squishy bits, while we want to plastically deform the target and move it someplace. Sometimes we will emphasize one over the other.

On the target side, you have some deformation and some movement. The exact composition will depend on the speed of the strike and the weight of the target. Of course you could argue that the deformation is actually a form of movement or vice versa. Here, by movement I mean the target actually moves as a whole, and by deformation any 'internal' movements of the target.

Deformation
There are probably elastic materials that yield a quantifiable result in how much you deform them. One rather cost effective way to show your students how much prgress they make with their strikes (or not) would be to stack up boards and see how many they can break. I'm not sure Karate is ready for such a novel idea.
Another way would be to have a spring and measure its elastic deformation. If I were to design such a system, my first start would be a robust air baloon and a pressure sensor, rigged so that you get the pressure spike. Compressing gases is not ideally elastic (some energy gets lost as heat). But you could mount an air cushion on the 'student' side of your mitt. I think any classical mechanical spring will need an aditional cushion, and that would add uncertainties. I think in Karate there's a practice of sticking boards in the ground and punchiung them, the teacher checks how much the board 'gives' by holding a hand a few cm behind it. If the board doesn't reach the hand, still much the student has to practice.

Movement
You could build a pendulum and measure how far it swings. Or maybe accelerometers and som complex integration that lets you arrive at an energy. Or you could mount your spring onto a fixed object, for example by sticking a board in the ground. Then you eleminate the movement term and can focus on deformation.

A martial artist's live strikes can be quantified between practitioners using force applied as pounds per square inch. The basic premise is (of course) force equals mass times acceleration. The importance is pairing with the area of contact used with which it is delivered - inverse square law. These two are the main pair.

There are of course wild complications in live striking - like transverse compression waves and [insert just about anything here]. In basis, transferring force can be qualitatively assessed through trained observation and the particular study of change of momentum, relative position and angle of attack relative to centre-mass. (Radial and longitudinal effects should be gestured rather than demonstrated).

I believe that there are some factors (not including deformation and shear stress) which cannot be accurately repeated in simulation and do require quantification/demystification - but is much harder to do so.

Ultimately, practitoners seek the obvious quantification of a strike: timing and accuracy. It should take into account the effectiveness of a technique arising as a result of competing motions. A trained student's degree of automaticity should also be assessed.