Saturday, April 21, 2012

ma ≠ F Clarified

I received the following comment with regards to my previous blog: F = ma, ma ≠ F.
Hey John, you lost me on this one. I am sure your article will clarify it (hopefully you will be able to post it here?). "It does not say that if you increase the mass or acceleration of that arm or leg that the force will be greater." But it does say that! It says that an object whose mass or acceleration is increased has greater force. If I punch with greater speed (all other things being equal) it will have greater striking force. I must be missing something...
One of the reasons I'm writing this blog is that it makes my writing better. Thank you for the comment because (a) it forces (pun intended) me to test my understanding, and (b) it informs me about the clarity of my argument.

Many in the martial arts literature interpret F = ma as the comment above does. Force (F) equals mass times acceleration, so if you increase mass (m) and/or acceleration (a) in a punch or kick then you increase the force of that punch or kick. A logical interpretation of F = ma, if F = ma is taken at face value.

F = ma arises out of Newton's second law of motion. It is commonly referred to as the law of acceleration. It states: 'The change of motion of an object is proportional to the force impressed, and is made in the direction of the straight line in which the force is impressed.' More simply stated, this law says that if a force is exerted on a body or object, that body or object will accelerate in the direction of the force, and its acceleration will be directly proportional to its mass. This law explains what happens if a force acts on a body or object. It does not explain what happens to a force.

F is not comprised of m and a; F makes m and a happen. Momentum (M) is calculated as the product of mass (m) and velocity (v): M = mv. M is comprised of m and v. If you increase m and/or v you increase M. Not so with F = ma.

'The action of a force causes a body's mass to accelerate' (Hall 2007: 63). F = ma does not say that in order to increase the force of a body or object you increase the mass and/or acceleration of that body or object; as so many in the martial arts literature suggest. So what biomechanical concept do we refer to in order to explain the understanding that increasing mass and/or speed of a strike or kick increases the force of that strike or kick? Now that is a question.

Recall that I have no interest in the science for science's sake. I'm only interested in the science in so far as it facilitates the understanding and study of techniques taught in the martial arts or used in violence generally in a practical way. So, the question that is raised so often by me when reading the science used to explain striking and kicking techniques in the martial arts literature is, 'So what?'

A strike or kick possesses momentum and kinetic energy. So what? If you increase mass and/or velocity in a strike or a kick you increase its momentum and kinetic energy. So what? If you apply a force over time (impulse) you transfer momentum from one body or object to another. So what? If you do the same over distance (work) you transfer kinetic energy. So what? As Carr (2004) suggests, many biomechanical (and martial arts) texts fail to relate good technique to mechanics meaningfully for coaches and athletes (and martial artists).

I need my explanations simple in order for them to help in understanding and studying technique. So, being frustrated in my search for an understanding of the science behind striking and kicking techniques, I asked the question, 'What causes an injury?' A mechanical answer to that question would provide the information I was looking for. That's when I discovered injury science; the science that studies injuries and their causes.

William Haddon, the founder of injury science, divided an injury event into three temporal phases: pre-event, event, and post-event. We can use this division to analyse striking and kicking techniques. With regards to an impact the phases are pre-impact, impact, and post-impact. Nakayama used the same division when analysing stances in Dynamic Karate. He referred to pre-execution, execution, and post-execution of a technique.

Injury science defines injury in terms of exposure to energy. So with striking and kicking techniques we are interested in kinetic energy (KE = 1/2mv^2). Forget momentum (M = mv). But KE does not cause things to change. KE is a property of a body or object. So KE can be used to understand striking and kicking in the pre-impact/pre-execution phase of a technique. What is being done in terms of mass and velocity in a strike or kick? This determines the potential for a strike or kick to apply a force and change the motion or shape (injure or damage) of a body or object. Nice and simple; only two variables to consider in order to understand and study all striking and kicking techniques taught by all martial arts or used in violence generally.

A force in mechanics is a specific thing. It involves the interaction of at least two bodies or objects. Forces cause change. Thus, forces can be used to understand and study the impact/execution phase of a technique. But F = ma only tells us that if you apply a force it can accelerate a mass. A force is applied to an arm to accelerate the mass of that arm in a punch. When the punch impacts a body or object it can cause the mass of that body or object to accelerate. So what? That might be useful when considering striking or kicking techniques that are designed to change the motion of an opponent, but what about when injury is the aim of technique?

Another comment I received was about the mechanical concept of power: 'are force and power the same thing? Or is power something you get out and force something you put in? Define power (as in 'a powerful punch')...(please)!'

Power (P) is calculated as work (U) divided by change in time (t): P = U/t (sorry, can't get the triangle change symbol to go with t). Have your eyes started to glaze over yet? Stay with me. Work is the means by which KE is transferred. KE is the potential to cause a change in motion or shape (injure) of an opponent that is realised when a force is applied. The smaller the time period (t) the same amount of KE has to be absorbed by a person's tissues, the greater the amount of KE per unit of time that has to be absorbed and the greater the potential for injury. F = ma offers very little by way of facilitating the understanding and study of striking and kicking techniques, but P = U/t would appear to offer more (despite the jargon), specifically during the impact/execution phase of a technique.

So, the power of a strike is dependent upon its KE and the time over which that KE is transferred to another body or object. More powerful strikes possess more KE and/or transfer it to an opponent or object in less time. More powerful punches apply more force because work is calculated as force applied over a distance (U = Fd) which can be inserted into the power formula: P = Fd/t.

I say 'appears to' above because I'm in the process of writing the impact/execution phase of a technique section of my article. In order to obtain mechanical concepts that could be used to understand this phase, when forces are applied, I have had to study forensic science.

In the end I hope to have a model that enables us to understand and study all striking and kicking techniques over the three phases in the execution of a striking or kicking technique using as few mechanical concepts as possible. I've already started to use this approach in teaching and which is proving successful as explained briefly in my blog concerning one-inch and three-inch punches.

Hope this clarifies things.


McGinnis, P.M. 2005. Biomechanics of sport and exercise. 2nd edn. Campaign, Illinois: Human Kinetics.

Hall, S.J. 2007. Basic biomechanics. 5th edn. New York: McGraw-Hill.

Carr, G. 2004. Sport mechanics for coaches. 2nd edn. Campaign, Illinois: Human Kinetics.


  1. Striking is much more in my area of knowledge. Have you looked at sequencing, and sequencing training for your article? You may find that quite interesting. Sequencing at the micro level is one of the ways of explaining why similar people can achieve widely different results, especially in striking. You’ll find mma coaches using cross fit type techniques to improve sequencing in their athletes if you look hard enough.
    One of the laurels I have to give to mma is that the money has made people truly professional, and specialist and knowledgeable coaches and scientists are being employed by them to deliver real results.
    From a Karate perspective, many of these things can be compared to traditional exercises, but people now understand more of what is happening and how to get the most out of them.
    This is where the gap of understanding the physics and being able to apply things appears. I might understand the macro-mechanics of Judo techniques, but a child who actually practices Judo can apply them better than myself, without ‘understanding’ anything.
    This leads me to worry a bit about you moving into striking after only looking at it on an intellectual level – besides the quoted bit of Silat, have you done much practically in terms of striking arts (western or eastern)?

  2. Hi Paul. Thanks for the comment.

    Never come across sequencing training so I had a bit of a look at it. It appears to be a training method. My work is more about understanding what makes techniques work rather than training methods. Having said that, once you understand what makes a technique work, you can then target your training accordingly.

    To clarify further. In the case of a punch; how does a punch injure a person? Injury science tells us its through energy exchange, in this case kinetic energy. KE is one half of the product of mass and velocity squared. This then tells us to look for what is being done in terms of mass and velocity in a punch. Ok, so now we look for mass and velocity in a punch action. Using a wing chun or power punch as an example, the distance these punches travel is small so their velocity and KE and therefore their capacity to injure an opponent is relatively small. But then you see different ways they increase the mass behind their punches in order to increase the forces applied when impact is made. Now you understand what makes those punches work.

    When you appreciate the role of KE in a punch, what do you orient your training around. Velocity has an exponential contribution to KE, so, like virtually every author who refers to it in the martial arts literture, you should orient your training around increasing velocity in your punches. How do you do that? There are a number of ways. However, studies hidden away in academic journals consistenly show that the difference between experienced and inexperienced practitioners, and even nonpractitioners, is that experienced practitioners exert more force on a body or object through increasing the mass behind their strike. Many studies show the difference in velocity is negligible, it is mass that is the difference between the different type of practitioners. This then suggests where the focus of training should be.

    That is the objective of my articles, and work, at this stage. It also produces a teaching method. Teach the student what is important to look for in a technique. They then know what is important, and why it is important. This should increase the rate of learning of those students.

    I agree with you totally. A practitioner of a particular movement can perform it better than one who only understands how it works. In fact, this is the norm in the martial arts. Most practitioners can do it and have no need to understand why it works. They understand the how and don't understand the why. This becomes very evident when they try and explain the why of the how. Most of them should stick to explaining the how and not attempt the why.

    The thing that is common to all techniques in all of the martial arts is the science. For instance, all striking techniques contain KE which is the potential to cause a change in the motion or shape (injure or damage) of a body or object. So, you can analyse familiar and unfamiliar striking techniques, and their teaching methods, and understand what they are trying to do and why. This then should increase the rate of learning and improved performances.

    With regards to my striking experience, the jujutsu I studied included a lot of striking and kicking, although in a more karate fashion. I've also trained silat quite a bit, different styles, and less so boxing, kung fu, and a couple of styles of karate.

    The science is the common element to all these techiques in all these arts. The science enables us to understand the techniques, but you still have to train them in order to become proficient at them. No doubt. Understanding the science offers the tantilising possiblity of increasing the rate of learning and proficiency.


  3. Not to put too fine a point on it, but that's just basic physics. What you need to do is sit down with a physicist and get them to explain force, power, and KE and their inter-relationships.

    I would...but it's half a lifetime ago since I looked at that.

  4. Not to put too fine a point on it, but what you said? That's just basic physics. What you need is to sit down with a physicist and get them to explain force, power, and KE, and explain their inter-relationiships.

    I'd help...but it was half a lifetime ago that I last looked at all that.


Your comments make my work all the more relevant as I use them to direct my research and theorising. Thank you.