*“When this baby hits 88 miles per hour,*

* you’re going to see some serious sh!t…”*

* Emmett ‘Doc’ Brown*

**Caveat: **If you don’t care about maths of physics at all, stop reading now, however a better understanding of some simple physics will improve your understanding of which cut to apply when. Not to mention the fact that this physics analysis is pitched at around a grade 9 level of education.

When many people begin martial arts (particularly when they are young) they fully expect to learn “secret techniques” which harness unknown energies and are steeped in mysticism. This culture of what the philosopher Daniel Dennet calls “Deepities”, these are gnomic utterances which in one sense are true but trivial, and in another they sound profound but are essentially false and meaningless. To the extent that deepities are true, they don’t matter, and to the extent that they would matter, they aren’t true. Consider people saying “everything is connected”, trivially speaking this is true (we are all connected by the abstraction that we exist in this universe), but in fact the saying itself seems profound, but is in fact false; everything is demonstrably not connected in any meaningful way. It evokes a feeling of energies connecting us all, but is in fact a semantically impoverished statement.

The movie “Mystery Men” made excellent satirical use of the deepity with in the character “The Sphinx” whose power is entirely undefined, but we know he is “terribly mysterious”. Throughout the movie he utters Yoda like sayings typically of proponents of mysticism in the martial arts (“when you doubt your powers, you only give power to your doubts”). So it goes with martial arts, deepities abound, giving rise to legends of miraculous abilities verging on the magical.

The reality of martial arts is much more mundane, yet at the same time beautiful in its complexity. Each motion of the sword is a complex chain of muscle contraction and momentum creating looping movements of amazing complexity and mathematical beauty.

In such mathematical terms even the simplest cutting motion of a blade is quite complex; an elliptical path through three dimensional space with multiple forces acting on it. If we wanted to fully discuss the physics of cutting we would have to devote several chapters to it and still only cover the basic groundwork. However we can perform some simplifications which make the science easier, and also allow us to elucidate some important points about the use of the sword.

There are many aspects of the cut we could examine here, but for now we will talk about momentum, impulse, kinetic energy, leverage and their cumulative effects on parrying and cutting.

### Sword Physics for Beginners

There are many factors peripherally affecting a sword’s impact on its target, but in many ways we can simplify much of the detail into four variables:

- Mass (m)
- Velocity (v)
- Distance (d)
- Time (t)

Given only these starting parameters we can begin to derive some interesting relationships based on simple physics:

TimeTime is a function of velocity and distance: t = d/v Where t = time, d = distance, v = velocity. From common sense, the faster a thing moves and the shorter the distance it must travel, the lower the time taken. |

MomentumWe are all familiar with momentum – it is the propensity for a moving object to keep moving in the same direction it is currently travelling (we use the related term inertia when we discuss the effort needed to start an object moving from rest). The formula for momentum is: p = m x v Where p is the momentum, m is the mass, and v the velocity (the SI unit for this is kg per metre per second) |

Kinetic EnergyThe amount of “moving energy” expressed by the sword. This can be thought of in many ways as the ability to inflict damage and to cut. An attack with higher kinetic energy is going to be more penetrating than a similar attack with the same weapon, but lower kinetic energy. This is calculated with: Ek = ½mv² Where Ek is the kinetic energy, m is the mass, and v the velocity. |

These formulae show a link between mass and velocity for both momentum and for kinetic energy, and a link between velocity and time for both momentum and kinetic energy.

This is good news; we can improve both our momentum and our kinetic energy by increasing mass and velocity: we need a massive sword moving very fast. Simple! If we’re simply looking for cutting potential as kinetic energy we see that velocity is considerably more important; the kinetic energy formula uses the square of the velocity in its calculations. If we have a 1 kg sword travelling at 3 metres per second The momentum of the weapon would be 3 whereas the kinetic energy would be 4.5 (we can ignore units for now). If we up the velocity by a factor of 10 to 30 metres per second, the momentum jumps to 30, but the kinetic energy increases to 450.

The the disproportionate difference the velocity makes to the equation is considerable, so if we want to achieve high kinetic energy we should maximise velocity, while for momentum mass and velocity are equally important.

Relating to velocity for both is time. Time has a great impact on getting to the target first. As such cuts which are quicker usually traverse less distance than slower cuts (though are usually less powerful, as we’ll discover). This gives us a rule of thumb:

HEURISTIC 1 - Short cuts will get there first: since time to impact is based on velocity and distance, shorter cuts will get to the opponent more quickly.

HEURISTIC 2 - Maximise Kinetic Energy by Increasing Velocity: the velocity parameter is squared for calculations of kinetic energy, so it is much more important than mass. If a particular technique requires kinetic energy, look to maximise velocity rather than mass.

HEURISTIC 3: Maximise Momentum with Mass and Velocity: As both are equally important, if we’re looking to cut through the enemy’s defences or knock their sword aside, we need mass and/or velocity.

Since all three are velocity linked, let’s investigate velocity further.

Cutting Radius and VelocityOne of the easiest ways to achieve a high velocity impact is through an increased radius cut. Swords cut along a circular path, and given the fact we know that speed is a function of distance and time: s = d/t s = speed, d = distance, t = time. Consider a spinning wheel. The centre of the wheel and the outer rim of the wheel are both moving at the same number of revolutions per minute, but the circumference of the motion changes as we move toward the outside of the wheel. For example: The outer circle has a radius of 3m, thus a circumference of 18.85, while the inner circle with radius 1m has a circumference of 6.28. If we have a wheel rotating at 1 revolution per second Inner circle speed = 6.28 m/s Outer circle speed = 18.85 m/s A point on the outer extremity is travelling three times faster than a point one third out from the axle. |

Clearly the outer point of a circular motion is significantly faster than the inner point. If we apply this to a sword it means that we want a large radius which impacts on the outer area of the cut. This is why Meyer’s fast long cuts strike with extended arms; maximising the circumference covered in the same time.

HEURISTIC 4: Cuts Move Faster at the Tip: All other factors being equal, a cut with the weak has significantly higher kinetic energy than cuts further down the blade.

Acceleration defines how quickly we get the blade up to the desired velocity, and it is dependent on two things, mass and velocity, in the following relationship. a = F/m Where a = acceleration, F = force, m = mass. So more force and less mass means greater acceleration. v = ut + ½(at²) Where v = end velocity, u = starting velocity, t = time, and a = acceleration |

The amount of time is tremendously important (it is squared in value), as is the starting velocity. From these previous formulae we see several new heuristics relating to velocity and acceleration.

HEURISTIC 5 - Short cuts will not inflict as much damage: As they will not have the same acceleration time, their final velocity will not be as great.

HEURISTIC 6 - Maintain Acceleration through Long Cuts: Long cuts allow greater acceleration time, and hence will strike with greater velocity.

HEURISTIC 7 - Time and Velocity Are a Trade Off: If your cut is longer in duration, it is going to take a longer time to reach its target. If a cut is shorter, it will strike with less force. Choose the correct cut for a given circumstance.

HEURISTIC 8 - Cutting to the Point Steals your Energy & Momentum: If you cut to an on-point position and stop, you are decelerating hard as you come to the point position. This robs you of the kinetic energy you generated in the cut at the very moment you may need it most.

HEURISTIC 9 - Maintaining Motion Maintains Energy & Momentum: We can if we don’t try to stop our cuts, but instead let them flow through, we can retain a great deal of the momentum and energy of the cut, and are also able to accelerate from a moving start for the next cut.

HEURISTIC 10 - Maximise Your Acceleration: The higher the acceleration the faster the final cutting speed, hence faster cuts, more momentum and greater impact energy. Our acceleration formula shows that we need to maximise force in order to achieve high acceleration, and hence high velocity. We could also minimise the mass of the sword (a sword half as heavy accelerates twices as fast under the same force).

However, we also know that mass affects momentum and kinetic energy. While a very light blade moving very fast would have a high kinetic energy, it would have very little momentum. Since both momentum and kinetic energy have their role to play, we have to compromise between the two.

There are ways, however, that we can increase the effective mass in an attack or defense without relying on a heavier sword. The easiest way to do this is to use our own body-weight. Consider a boxer striking with a right hook. The mass in the calculation of the momentum of the blow is not simply the weight of the fist or arm. A good boxer puts their whole body behind a strike, meaning that the momentum of the impact uses all of the boxer’s mass.

We can extend this to the use of swords; if we use our entire body weight behind the blade we tremendously boost the momentum without requiring a heavier sword. This is part of the reason we step with the strike; our whole body weight is moving with the attack, lending the sword additional mass for its momentum. Likewise the discussion of correct stance tells us we should move our body as a holistic unit; this too increases functional mass of the blow. There are some technical points worth considering here later, but for now we can create the heuristic:

HEURISTIC 11 - Increase Effective Mass by Using Body Weight: We can boost the effective mass of our cut by putting our own body weight behind the blow either by moving forward with the blow, or using rotational momentum of our body in the cut.

We can also attempt to improve velocity. We know this is particularly important for kinetic energy. Aside from mass, the acceleration of the blade is based on force. This is where we try to maximise force. There are several easy ways to help maximise the force imparted on the blade to make it accelerate.

HEURISTIC 12 - Increase Force by Getting Stronger: The effect of physical conditioning should not be underestimated. We should all strive to be stronger and fitter to improve our fencing. You may hear some people say that physical fitness and power is less important than skill. This is a fallacy of false dichotomy; the two are not mutually exclusive, and between two equally skilled fighters, the better conditioned generally wins. Work on fighting fitness and skill together.

HEURISTIC 13 - Increase Force by Recruiting Larger Muscle Groups: Cuts relying on small muscle groups will not be able to accelerate effectively. This is why wrist cuts are not as strong as full arm cuts. This is part of the motivation for a good stance and structure; we use our full body motion to launch cuts, letting our legs, hips, torso, and arms add to the force. Some guard positions allow for greater recruiting of large muscles. Compare zornhut and alber; the former provides far better muscle recruitment, and will result in more damaging blows.

HEURISTIC 14 - Increase Force by Borrowing from Gravity: a falling cut from above gets free bonus acceleration using gravity over a cut from below; if the same muscular force is in use of each of them, the falling cut will land with a higher velocity.

There is another principle we can investigate to boost force; the principle of mechanical advantage using levers.

In general the effort force can be expressed as Fe = Fl dl / de Where: Fe = effort force, Fl = load force (eg. weight), dl = distance from load force to fulcrum , de = distance from effort force to fulcrum |

Fundamentally the closer to the fulcrum the load is (ie. the weight) and the further away the force is, the greater the effective force which can be exerted.

Let’s apply these lessons to a longsword. When holding our longsword our lead hand is the fulcrum, the rear hand provides effort, and the load is the weight of the sword, or more properly the weight at the sword’s balance point. So visually:

Notice that the force is considerably further from the load (centre of mass). This means a force applied at the pommel would cause a large resultant force at the load point, and this is exactly why the sword is balanced this way. If we compared this with an axe and tried to use the same close grip that we do with the sword, the mechanical situation would be reversed. The heavy head so distant from the front hand would make moving the axe difficult (and that is why axe use has a much wider grip). This affects both momentum and kinetic energy, and which longer hilts were developed to begin with, and why we discussed the levering of the sword hilt in the cut in the section concerning how to grip the sword. So our new heuristic is:

HEURISTIC 15 - Increase Force through Mechanical Advantage: use a levering action around the lead hand fulcrum, along with the other principles to maximise force, and hence acceleration and velocity.

The downside to this is that leverage is a double edged sword (pun intended). We can use leverage to make faster and stronger cuts, but our opponent can use our own sword as a lever against us. If an opponent can take advantage of the length of our sword, and manipulate the weak of our blade they have a very long lever indeed. For example, If an opponent parried our weak, or cuts hard to our weak, they are able to use the whole length of our blade as a lever against our own actions. Likewise if we parry with the weak of our blade the opponent will easily move our blade aside. Exactly how much advantage will they have? Let’s throw some numbers at the problem. Consider the following situation:

Fencer A is in a bind with fencer B, who is trying to force A’s sword down with the cross-guard. Fencer A has a fulcrum-lever distance of 20cm, fencer B is pressing down on the weak of A’s sword and the 80cm mark.

Looking at the numbers we can see that fencer B only needs 25% of the effort of fencer A to control A’s blade.

In this way we see that while many of the principles we have discussed so far refer to the effectiveness of the cut, they also have application to defensive handworks and parrying, effectively you can move someone considerably stronger than yourself simply through correct application of leverage.

HEURISTIC 16 - Decrease Your Opponent’s Leverage, & Maximise Your Own: We should strive to avoid parrying our opponent’s blade with the weak of our own, or spend two long with our weak near their strong in the bind because that grants them mechanical advantage. Either take away your blade from theirs, or move to use your strong, all while decreasing their mechanical advantage.

Another observation from the material presented is that velocity (and hence momentum and kinetic energy) are all vectors. A vector is a measurement which has magnitude (which we’ve talked about so far), and just as importantly, direction. Consider the angles of basic cuts.

Each cut also has a vector associated with it as you can see here. This means the energy and momentum of that cut is travelling only in the direction of that arrow. This becomes important if we wish to set aside a cut, or avoid having our own cut set aside. The oberhau and mittelhau are the simplest cuts to deflect. Their vector travels vertically, or horizontally, and as such the oberhau has no momentum or energy in the horizontal direction, and the mittelhau has none in the vertical direction. This means that we have two choices.

First we could parry in direct opposition to the direction of travel (so for an oberhau, for example, we would block with an equal and opposite force in the vertical direction. While this is certainly effective, and we can use the heuristics already seen to do so (maximising our leverage by blocking with the strong to their weak, using our own inertia in opposition to the blow, and so on). A more efficient option, however, is to deflect the blow with a force perpendicular to the vector of the attack.

Notice that for a downward attack such as the oberhau in the diagram above that there is no momentum in the horizontal direction. This means that we don’t need a particularly large horizontal vector of our own to deflect the blow off course. Likewise a horizontal blow can be deflected downward easily, and a diagonal cut such as the zornhau in the diagram could be taken offline with a perpendicular force from the upper right side cutting down to the lower left. By removing the need for high momentum in setting aside, we can now look to our rules of thumb.

HEURISTIC 17 - Deflect the Line of Motion With Low Energy Perpendicular Movements: If we oppose a vertical cut with horizontal force, or a horizontal cut with vertical force, we can change the course of our opponent’s attack with minimal effort of our own.

We can also cutting behind their blade to increase its speed and fling it down or to the side along the line of motion itself. The risk here is that they will flow around in the cut, which is what we should do whenever our own blade is deflected.

HEURISTIC 18 - Borrow Momentum From Deflections to Drive New Attacks: For example, if our mittelhau is knocked downward, using this new momentum to drive our cut around and over for a follow-up zornhau. In this way we accept our opponent’s energy as a gift, and use it against them.

The alternative to deflecting is that we must oppose the opponent’s cut on the line of attack directly by using an equal and opposite force.

HEURISTIC 19 - Oppose the Line of Motion by Maximising Opposite Force: There are two ways to do this easily, both of which should use mechanical advantage by using our own strong on their weak. We can oppose with a cut (using velocity to gain momentum), or with stationary inertia. The first is easy; block an incoming uperhau with a cut up. The second is to get behind our blade and use the mass of our whole body, along with rigidly held arms and legs, to stop the blade on its path (from rules 10, 11, 12, & 15)..

It is also useful to note that unterhau and zornhau cuts have vectors which have both a horizontal and vertical component. The advantage of this is that if the momentum is robbed from a single direction (horizontal or vertical) using a parry or deflection we still have the momentum continuing along the other axis. It also means that these cuts can be used as a generic counter-cut against horizontal or vertical cuts.

HEURISTIC 20 - Diagonal Cuts are More Difficult to Parry Simply: The opponent is required to account for both the horizontal and vertical components of the cut, or risk follow through attacks with the remaining directional energy/momentum.

HEURISTIC 21 - Diagonal Cuts are an Effective Generic Counter: Because they have both vertical and horizontal momentum components, diagonal attacks can set aside vertical and horizontal cuts, and counter opposed diagonal cuts effectively.

So How Hard do we Need to Hit?

Now that we have 21 observations concerning maximising blade mechanics for several different parameters, energy, momentum, time, and so on, so which should we use and when? Logically it depends on the circumstances, your style of fighting, and the style of fighting the opponent is using. Since everything comes down to damaging our opponent, at some stage we need to face the questions of how much momentum, force, speed, energy, and so on, do I need to use? The answer depends on the outcome we’re looking for.

In an ernstfechten scenario, damaging an unarmoured opponent enough to take them out of the fight actually takes very little force. A short fast cut using good pommel leverage is sufficient to cause a debilitating wound to an opponent. We don’t need to buffet them with massive momentum to knock them, or their sword, aside by relying on our mass alone. In fact logically then we need not hit with very much momentum at all if we’re correctly applying leverage. This applies even more so in sporting bouts such a schulefechten; in these bouts we really need very little force to achieve our desired result.

Of course if we’re aiming to displace forceful attacks, or facing a heavy-handed opponent we may need to invest our cuts with slightly more momentum and force, but once again we don’t need to use heavy handed blows to achieve this; bodyweight, leverage, and vector of attack, are all our allies in this action.

The same advice applies when we are in the bind. Certainly great physical strength might give a force advantage on both the hew and the bind, but the clever fencer knows not to resist force with force. Instead they yield when the opponent is strong, and use force when the opponent is weak.

The conclusion to all of this is that in fencing (and especially during non-lethal or sporting events) there is no point in hitting excessively hard; it demonstrates nothing of a competitor’s martial skill, and in blossfechten is completely outside what is required to disable or defeat our opponent. Furthermore it expends excessive energy (tiring you more quickly), opens us up to obvious counters, robs the fencer of control, and generally makes us a poorer at our art.

The evidence bears us out in this matter; the design of swords themselves is light and thin toward the point optimise speedy cuts with the weak over massive force and momentum; the person who desires heavy strikes uses an axe or hammer instead. The images from historical sources show cuts in blossfechten that don’t favour heavy-handedness, and the texts themselves describe more “light and fast” cuts than they do heavy smashing blows (as well as admonishing us against the “buffalo” – the person who charges in with heavy cuts.

As such we can add a final heuristic to our 21; the “one rule to rule them all”:

PRIMARY HEURISTIC - Strike Only as Hard and as Fast as You Need To: An effective cut requires very little force in the cut; using extra force simply tires you out quicker, endangers your partner, and makes you less able to defend yourself effectively. It robs you of options, and lessens the quality of your fencing.

### Let’s apply what we have learned to the images from Meyer, to elucidate the possible intent of the various cuts.

### Case Studies

Having provided at least a basic groundwork in the physics of sword motions we can use this groundwork to assess the illustrations in Meyer’s text as a way of elucidating Meyer’s broader style of fighting with the longsword. We will consider a handful of representative examples, though the same analysis applies to all of the images from Meyer.

#### Case Study 1: Oberhau

The figure is shown at the maximum extent of a cut from above, hands outstretched and leaning slightly forward. We can combine things we know about the cut and heuristics to make an overall assessment of this cut. A priori knowledge concerning this cut:

- The cut started in vom tag and is continuing all the way through to wechsel.
- The cut was accompanied by a passing step with the right.

From this we can conclude:

- The cut cut was performed with significant force, though more by gravity assistance, and a levering “push-pull” hand action, rather than from large muscle groups, which couldn’t effectively be used in vom tag (from H13, H14, H15)
- Therefore the cut had a high acceleration rate (from 1, and, H6, H10).
- Therefore the tip of the blade is moving very fast based on acceleration (2) , tip cut with a large radius (H4),
- Therefore the tip has high kinetic energy (H2).

We can also conclude:

- This cut has a vertical cutting line, and so could easily be deflected horizontally of course (H17).
- The cut has limited momentum as there is no body weight behind the cut at this point (H11)
- The fencer has little leverage over the sword at this point in the cut (the fencer’s arms are outstretched, making the entire arm-sword structure a lever, also muscle groups controlling the hilt at this extension are weak (H16), so would not control the bind well.
- The low momentum and poor leverage mean that the blow could be parried easily with a forceful cut from below or deflected to the side (2, 3, H19).

**Conclusion:**

### Case Study 2: The Unterhau

The figure is shown at the maximum extent of a cut from below, hands high and uncrossed, with the hilt held reasonably close with arms bent. A priori knowledge concerning this cut:

- The cut started in nebenhut and is mostly finished its motion at this point.
- The cut was accompanied by a passing step with the right.

From this we can conclude:

- The movement was performed with significant force, owing to large muscle groups, gravity assistance (from H13, H14)
- This means this cut would be good for opposing strikes from abot (5, H19)
- However the cutting to the point also implies that most energy is spent at this point in the cut (H8)
- Likewise the cut is not especially fast at the tip as it does not take advantage of a leverage based “push-pull” rotation with the hands on the hilt in the cut, and the cut doesn’t have a the extended radius of the oberhau (H4),

We can also conclude:

- This cut has a diagonal cutting line, and so would not be as easily deflected horizontally of course as the oberhau (H17).
- The cut has good momentum owing to bodyweight & body structure, and there is considerable force/stability behind the cut (from 5, H11)
- The fencer has excellent leverage over the sword at this point in the cut as the arms are bent and the fencer can push and pull effectively using body eight and large muscle groups (H16), so would control the bind well.

**Conclusion**:

The unterhau is a medium range, high force/momentum cut which is not simple to deflect or parry. It makes an excellent defensive cut against strikes from above, and provides an ideal entry into the handworks.

### So What Have We Learned?

We could (and should) go through all of the basic cuts and guards with this same lens of physics. By doing so we can identify the strengths and weaknesses of each of the guards, cuts, and even handworks. While this kind of explicit knowledge will not make us a better fighter by itself (after all, we hardly have the opportunity to deploy our higher level explicit knowledge and reasoning in a fight situation), it should be used to guide our training and to drill our techniques with the right mechanical context in mind such that when our opponent attacks or defends we are able to immediately respond with automated responses which have been tailored to optimize for physics through this improved practice framework.

I like this Post: Another!

This is a very good way to teach to folk like me, who want to know why certain cuts work and why to use them in certain circumstances.

Cheers