Wednesday, September 10, 2008

Hitting harder: physics made easy

Introduction

Most martial artists share the goal of “hitting harder”. This is usually expressed in colloquial terms as hitting with “more force” or “more power”. But even a basic knowledge of physics will tell you that “force” and “power” are not the same thing. Which is it that makes you hit “harder” – force, power or both? And is it more helpful to talk in terms of something else such as momentum?

Understanding force

Many people think of “force” in very nebulous terms (perhaps explaining such misappropriations as “The Force” in Star Wars). So what is “force” as understood in physics?

Force is something that enables you to cause an object with mass to accelerate.

In other words, it is Newton’s famous formula: f = m x a.

Using this formula people commonly argue that in order to hit “harder” they need to –

1. maximise their mass; and
2. maximise the acceleration of their attack.

For the most part this is true: if you have a big mass and accelerate that mass well, you will maximise your chance of applying a greater force to a target. But it is important to note that this argument refers to force you are applying to your own body. It does not refer to the force you are applying to the target.

Put another way, in this argument the equation f = m x a assumes that m = your body mass, not the mass of your target (as it should for the purposes of calculating the force on that target!).

So instead of looking at the force applied to your body, let’s examine the force applied by your body to a target.

For example, consider a stationary object weighing 1 kg. For simplicity’s sake, let’s say you hit/push it for one second causing it to reach a velocity of 1 m/s. We can use the following formula to determine the accleration of the object:
v = u + at
or

where a = acceleration, v = the final velocity (1 m/s), u = the initial velocity (0 m/s) and t = time.

Accordingly we can determine the acceleration of the object as 1 m/s2.

Now in order to determine how far the object will move in that one second, we can use the formula:

or

where s = distance.

From this formula we can determine that the object will move 0.5 m. Put another way, if we cause a stationary 1 kg mass to move 0.5 m in one second we’ve caused it to accelerate at 1 m/s2. By reference to the formula f = m x a you’ve applied a net force of 1 N or kg/m2 to effect this result.

Clearly the more force you apply, the farther you will move it in the same time (by accelerating it to a higher velocity). So imagine you have managed to force the 1 kg object 1 m in one second (ie. you’ve managed to accelerate the object from 0 m/s to 2 m/s). In that case you’ve applied a force of 2 N.

So far so good. At this point you might be forgiven for thinking: “If I can knock my opponent across the room with my punch, surely I must be hitting very hard?”

Well yes and no. Let me explain by asking a question: Does “hitting harder” really mean moving something farther? If it does, a strong push or shove arguably constitutes the “hardest hit”, because a strong push is likely to move something farther than a punch or kick. I’m sure you’ll agree that this is an extraordinary proposition (for more on this topic see my article "Visible force vs. applied force"). The former is designed to move, the latter to hurt (with or without moving).

It seems that “movement”, and hence the acceleration of your target, might not be the most significant indicator of “how hard you have hit” after all.

Furthermore, you might hit a brick wall with your fist as hard as you can. You’d be unlikely to move the wall at all. Does this mean you haven’t hit it “hard” (you have) or with any force (you have, except that the wall has exerted an equal force on your fist as per Newton’s Third Law)?

Theoretically if you were heavy enough you could push the brick wall over (ie. move it) – but we’re back to “pushing”. Theoretically if you punched fast enough your peak force1 would be sufficiently high to break through the brick wall without “moving” it (eg. a high velocity bullet can pass through objects without displacing them).2

So what distinguishes a strike from a push? It is your deceleration before hitting the wall. The more you decelerate before you reach your target, the slower the velocity at impact. If your velocity at the point of impact is low, you’ll effect more of a push. If your velocity at the point of impact is high, you’ll effect a blow.

By now you should be starting to see the limitations of using f = m x a as a means of determining “how hard you hit”. The equation might tell me how to increase the force applied to my body (which is necessary in order for me to apply force to someone/something else), but it doesn’t tell me how to apply my force.

Rather than talk about the acceleration of your mass or the mass of the target, it seems to be more useful to consider the velocity of your mass at the moment of impact.

This is momentum.

The importance of momentum and its transfer

In physics momentum is defined by the following equation:

Momentum (p) = mass (m) x velocity (v)

Obviously a one tonne (m) car travelling at 60 km/h (v) will do some damage if it hits you. A fist, while not so destructive, obeys the same physics.

It appears that a well-thrown fist reaches its maximum velocity when the arm is about 80% extended. Accordingly if your punch covers say, 60 cm from a fully chambered position to full extension, then your punch reaches its maximum velocity at 48 cm.

This speed is generated by moving a number of body parts toward the target; eg. the whole body (by stepping or lunging), hips, leg, shoulder, upper arm and lower arm. If the full range of movement is used, and the body parts act in a staged way to transfer momentum in a whip-like sequence from larger to smaller body parts, then the fist will be accelerated to the fastest possible speed. A reverse punch is biomechanically better suited to transfer momentum than in this way than a leading arm punch.

So far so good. But doesn’t this come to more or less the same thing as previously argued – in other words to hit “hard” you want a mass travelling at a high velocity? Again, close. But there is more. You not only want your body to have momentum; you want to transfer it effectively.

Now, momentum transferred is called "impulse". The equation for impulse is as follows:

Impulse = force x time

In other words force = impulse / time

With impulse a fixed quantity, force and time are necessarily inversely proportional. In other words, one can deliver a given amount of momentum by:

1. transferring a large force for a short time; or
2. by transferring a smaller force over a longer time.

So, the longer it takes to transfer momentum, the less force is applied. If you want to maximise your force, you must ensure that as much of your momentum is transferred on impact as possible. This is true for both focused punches and follow-through punches used by boxers. The only real difference is that punching with boxing gloves increases the time it takes to transfer momentum which reduces the force compared to a similar punch with bare knuckles. In other words, when punching with gloves the momentum transferred will be the same as bare knuckles but there will be less impact force and instead the target will feel more of a push.

"Power punching" vs. traditional punching

When you throw a cross punch (see my article “Chambering punches”), the punch travels a longer distance than a straight punch – particularly if (as is inevitably the case) the cross follows a curved path to some extent. The further the punch travels from the chambered position to the target, the more time it has to accelerate and the faster its maximum speed will be (at 80% extension). This is why the cross punch carries more momentum than any other punch.

If this is so, why wouldn't one always use a boxer's cross punch? Why would one ever stop one's punches at a pre-determined point rather than use a follow through? The answer to this lies in the inherently conservative approach of traditional strikes - ie. they are part of a civilian defence strategy that puts "not being hit" ahead of "hitting with maximum force". A traditional martial artist focuses a strike so as to stop it at a pre-determined point in space in order to avoid over-committing (eg. if the target was to duck or evade) (for more on this topic see my article "Stopping strikes at a pre-determined point").

However it is not true that the traditional strike is "weak": it achieves the maximum possible force that is possible if one doesn't use a follow-through. It does so by stopping the punch as quickly as possible at an optimum predetermined point just beyond the target. This is the definition of "kime" or focus, a concept central to the study of karate and many other traditional Far Eastern fighting disciplines - see my article "Kime: soul of the karate punch".

As with striking a makiwara or pad, the strike will be stopped earlier than the predetermined focus point by the target if it does connect. But if the strike does not land, the traditional martial artist's positioning, balance and defensive capacity won't be compromised. For more on this topic see my articles "Stopping techniques at a pre-determined point" and "Karate punches vs. boxing punches".

The role of power

That a competitive boxer can apply a staggering amount of force with a gloved cross compared to, say, a suburban karateka performing a bare knuckle reverse punch, does not detract from the fundamental observation I have made above: a gloved punch has more “push” and less “hit” than any bare knuckle variant. The competitive boxer is big enough, and is moving fast enough, to compensate for any “diffusion” of his/her force resulting from the wearing of gloves. He or she might knock you across the room – and hurt you at the same time.

However this in no way invalidates a karate punch (ie. a straight punch) in which the impulse is maximised. Put another way, while the karate punch might not carry the same momentum as a boxer's cross, it can still achieve an effective result - without necessitating the same amount of work as one sees in a boxing contest.

By “work” I mean, of course, the concept in physics defined by the following equation:

Work (w) = force (f) x displacement (d)

In other words, work is done when a force acts upon an object to cause a displacement. Because of the diffusion of force caused by gloves, a boxer's gloved cross punch will cause more displacement than a bare knuckle punch thrown with the same force. This means that the gloved boxer has necessarily done more “work” than he or she might have had to had he or she been fighting without gloves.

Work can also be described as the amount of energy transferred by a force. In this case the gloved cross punch, using the same force, transfers less energy than it would were the boxer not wearing gloves.

As I have said above, the boxer will usually more than compensate for any diffusion caused by gloves. He or she will do so by adding more “power”. However this also means that a great deal of boxer training is concerned with power, given it's central role in the art of fighting with gloves.

In physics the equation for power is as follows:

Power = work / time

To compensate for the diffusion of his/her force because of gloves, the boxer has to work more in a shorter time. This means he/she needs more power and will train accordingly.

Don’t all martial artists need power? Of course they do. Everyone will be subject to the same laws of physics. However, remember the purpose of this article: to determine how one can “hit harder”. Power and displacement are directly proportional. Yet “hitting harder” doesn’t require greater displacement - rather, greater displacement is (at least to some extent) a foreseeable by-product of gloved fighting.

For more on the issue of displacement, see my article "Visible force vs. applied force".

Accordingly, while boxing training and methodology is eminently applicable in self defence, it is a mistake to assume that it is perfectly suited to a civilian defence system - or that other strategies designed with civilian defence specifically in mind are less effective for their purpose just because they don't develop strikes that are as "powerful" as those in combat sports.

Conclusion

If you want to “hit harder” you should look to maximising your momentum by increasing your velocity and/or your mass. But you also need to look closely at how you transfer that momentum.

Transferring momentum effectively is a question of maximising your impulse; ie. applying a large force over a short time. The alternative (a small force for a longer time) produces a push – not a “hard hit”.

The smaller your impulse (ie. the longer you take to apply a force), the more power you will need to compensate.

It is my view that a martial system that focuses on maximising power is what is classified in China as “external” or "hard" (waijia). On the other hand, a system that focuses on impulse is what is classified in China as “internal” or "soft" (neijia) – see my articles “Internal vs. external martial arts” and “Understanding the internal arts”. As I have noted in that article, no art is purely "external" or "internal" – just as no strike can be effected without some impulse and no technique can be effected without power.

A good “civilian defence system” will attempt to find a pragmatic mix of both – or ideally adopt a sequential system of teaching progressively more “impulse-oriented” techniques and less “power-oriented” techniques – something I call “sequential relativism” in martial arts training (see my article “My quest for the martial ‘holy grail’”).

Footnotes

1. It is important to note that when I refer to maximising “force” I am of course referring to the peak force – force is never applied evenly but is applied in a curve. It is the peak force that is in issue when it comes to “hitting hard” – not the average or median force etc.

2. In this article I have not attempted to address factors such as stress and pressure. In the case of the former we are all familiar with the idea that we can increase damage by targeting softer areas. In the case of the latter we also know that we can increase damage by applying force over a smaller surface area – ie. increasing pressure. These are topics unto themselves which I hope to address at another time.

Copyright © 2008 Dejan Djurdjevic