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By Charles Rhodes, P.Eng., Ph. D.

Karate is a martial art that involves development of the capacity to deliver energy to an opponent at the right place and in the right direction at the right time using punching, kicking and other striking techniques. The object of these striking techniques is to disable the opponent.

A human being is an assembly of complex molecules. Each molecule in isolation has a potential energy component that is contained within a surrounding electric-magnetic field. When these molecules are adjacent to one another the surrounding electric-magnetic fields overlap. This field overlap causes a reduction in the total potential energy.

As the molecules assemble together during normal biological growth the law of conservation of energy causes the reduction in potential energy to become an equal increase in kinetic energy. This increase in kinetic energy lifts electrons to high energy states from which they naturally decay over time by emitting infrared photons. This infrared photon energy is lost to the environment as heat.

The net result is that within the human body the molecules have a lower potential energy than if they were free outside the human body. At body temperature the thermal kinetic energy of the molecules is not sufficient for the molecules to overcome the step change in potential from inside to outside the body. Hence the body remains an integral whole. The energy per molecule required to overcome this step change in potential is the Inter-Molecular Binding Energy.

A human can be disabled by instantaneously supplying sufficient kinetic energy from an outside source to add the Inter-Molecular Binding Energy to a large cluster of molecules. In effect the molecules are instantly transferred from the solid or liquid phase to the gaseous phase. The phase change causes bone and tissue damage and may disrupt the nervous system. In karate this energy supply is via an impact. Provided that the energy delivery is sufficiently concentrated in both time and space, the larger the energy delivery the more severe the damage. Hence karate is fundamentally about maximizing capacity for energy delivery via instantaneous impact.

An effective karate technique is not a push. A push develops a combination of elastic energy and CM kinetic energy in the opponent, neither of which does damage. An effective karate technique is more like a hammer strike that exceeds the opponents elastic limit and hence breaks inter-molecular bonds, doing damage.

Sensei Malcolm Fisher recognized that the primary goal of Shotokan karate is maximization of energy delivery to an opponent within an elapsed time of less than 0.5 seconds (< 0.5 s) from initiation of a technique to the instant of impact. The process of maximizing energy delivery within < 0.5 s uniquely defines the detail of a karate technique and leads to a series of improvements to Shotokan.

The law of conservation of energy requires that:
energy delivery
= bulk kinetic energy of attacker prior to impact
- bulk kinetic energy of attacker after impact
- bulk kinetic energy of opponent after impact
+ bulk kinetic energy of opponent before impact
- elastic energy in opponent after impact
+ elastic energy in opponent before impact

usually the bulk kinetic energy of the opponent before impact and elastic energy in the opponent before impact are approximately equal to zero. If the karate technique is a well focused strike rather than a push the bulk kinetic energy of the opponent after impact is also relatively small.

All karate techniques involve development of kinetic energy and non-elastic dissipation of most of that kinetic energy in an opponent. The attackers initial kinetic energy and the fraction of that initial kinetic energy dissipated determine the amount of energy delivery. Optimizing a karate technique involves maximizing the energy delivery.

Consider two masses Ma and Mb where Ma is initially moving with velocity Va and Mb is initially at rest. If mass Ma strikes and imbeds itself in Mb the velocity of the mass (Ma+Mb) after impact is Vb. Conservation of linear momentum requires that:
Ma Va = (Ma + Mb) Vb
Vb = Ma Va / (Ma + Mb)
The initial kinetic energy Eka is:
Eka = (Ma / 2) Va^2
The final kinetic energy is:
Ekb = ((Ma + Mb) / 2) Vb^2
= ((Ma + Mb) / 2) (Ma Va / (Ma + Mb))^2
= ((Ma^2) / 2 (Ma + Mb))Va^2

The energy delivery Ed is:
Ed = Eka - Ekb
= [(Ma / 2) Va^2] - [((Ma^2) / 2 (Ma + Mb))Va^2]
= (Ma / 2) Va^2 [1 - (Ma / (Ma + Mb)]
= (Ma / 2) Va^2 [(Mb / (Ma + Mb)]

Note that if Ma >> Mb as in a gyaku-suki attack to the opponents head the fraction of the initial kinetic energy dissipated is ~ (Mb / Ma) and the energy delivery Ed is limited to:
Ed ~ (Mb /2) Va^2

One method of achieving Ma > Mb on a strike to the opponent's body is to rigidly brace Ma to the ground at the critical instant of impact, which increases the effective mass of Ma. This method is used to by Fisher Shotokan to increase energy delivery.

Note that if Ma ~ Mb the energy delivery is about half of the initial kinetic energy. Hence if the attackers body is rigid at impact but not braced to the ground about half of the attacker's initial kinetic energy is actually delivered.

If a high velocity whip like attack is used in which the connection between the striking limb and the attacker's body is not rigid, then:
Mb >> Ma
[(Mb / (Ma + Mb)] ~ 1
which results in almost 100% of the striking limb's kinetic energy being delivered. However, the kinetic energy prior to impact is reduced because the striking mass is much smaller. To some extent the smaller striking mass can be offset by a larger striking velocity prior to impact.

Karate techniques naturally divide into two categories:
In one category are techniques in which Ma is greater than or approximately equal to Mb. These techniques rely on a rigid nearly straight striking limb transmitting full body kinetic energy into the opponent.
In the other category are techniques in which Ma is much less than Mb. These techniques rely on Ma acquiring a very high velocity prior to impact with the opponent. However, except for strikes to the opponent's head these high velocity techniques deliver less energy. These high velocity techniques rely on expertise in kyusho (knowledge of nerve centers and pressure points).

For techniques aimed at the opponents body trunk Fisher Shotokan focuses on techniques where Ma is greater than or equal to Mb because these techniques provide greater energy delivery.

For a big man the maximum available kinetic energy from a single leg compression is about 500 joules. Consider an olympic high jumper. A man with a mass of 100 kg raises his CM from 1.00 m to 2.02 m using energy from both loaded legs assisted by arm movement. The acceleration of gravity is 9.8 m / s^2. The gravitational force F on the man is given by:
F = mass X gravitational acceleration
= 100 kg X 9.8 m / s^2
= 980 newtons
The change in altitude is:
(2.02 m - 1.0 m) = 1.02 m
The initial kinetic energy Ek at takeoff that gets converted into gravitational potential energy is:
Ek = force X distance
= 980 newtons X 1.02 m
= 1000 joules
Hence, the maximum kinetic energy provided by each loaded leg compression is:
1000 joules / 2 leg compressions = 500 joules / leg compression

An olympic sprinter with a mass of 100 kg can reach a CM velocity of 11 m / s. At that velocity his CM kinetic energy Ek is given by:
Ek = (mass / 2)(velocity)^2
= (100 kg / 2)(11 m / s)^2
= 6050 joules
Absent air resistance the number of leg compressions assisted by arm movements required to reach 11 m / s is given by:
(6050 joules) / (500 joules / leg compression) = 12.1 leg compressions. Due to air resistance a few more leg compressions are required so that it typically takes a sprinter about 15 steps (15 leg compressions) to reach his maximum velocity.

For karate techniques in which Ma ~ Mb part of the attacker's body momentum is transferred to the opponent's body via a rigid nearly straight striking limb.
Oi-zuki: An oi-zuki (lunge punch) involves 2 leg compressions with matching arm movements that together provide a kinetic energy Ek of up to:
Ek = 2 leg compressions X 500 joules / leg compression
= 1000 joules
At impact the attacker's entire body is rigid. Because Ma ~ Mb, for a non-elastic impact about 50% of the attacker's kinetic energy causes energy delivery Ed to the opponent. Hence the maximum value of energy delivery Ed for an oi-zuki to the opponent's body is given by:
Ed = Ek / 2
= 1000 J / 2
= 500 J

Mae-Geri-Kekomi: A mae-geri-kekomi (front thrust kick) involves three leg compressions assisted by arm movements that together provide a kinetic energy Ek of up to:
Ek = 3 leg compressions X 500 joules / leg compression
= 1500 joules
Because Ma ~ Mb, about 50% of the attacker's kinetic energy can cause energy delivery Ed to the opponent. Hence the maximum value of Ed for a mae-geri-kekomi to the opponent's body is given by:
Ed = Ek / 2
= 1500 joules / 2
= 750 joules

Gyaku-suki: A compress-slide gyaku-suki (slide-in reverse punch) involves only one leg compression. Hence:
Ek = 1 leg compression X 500 joules / leg compression
= 500 joules
The attackers CM velocity Vcm reaches:
Vcm = (2 Ek / Ma)
= (2 X 500 J / 100 kg)^0.5
= 3.16 m / s
Because Ma ~ Mb, about 50% of the attacker's kinetic energy causes energy delivery Ed to the opponent. Hence the maximum value of Ed for a gyaku-suki to the opponent's body is given by:
Ed = Ek / 2
= 500 joules / 2
= 250 joules

For techniques such as a gyaku-suki to the opponent's head, in which Ma >> Mb, the maximum energy delivery Ed is given by:
Ed ~ (Mb / Ma) Ea
where Ea is the initial kinetic energy (500 J for a gyaku-suki).
For Mb = 10 kg and Ma = 100 kg the corresponding value of Ed is:
Ed = (Mb / Ma) Ea
= (10 kg / 100 kg) 500 J
= 50 J

Kinetic energy can be concentrated in the attacker's striking limb via a sequenced energy release that forms a shock wave.

The human body is a complex assembly of liquids and solids that under the appropriate circumstances can generate kinetic energy shock waves. Shock waves cause the striking hand or striking foot to have a high impact velocity. The shock wave is a zone of muscle energy discharge that moves outward along the attacker's striking limb causing the portion of that limb ahead of the shock wave to accelerate.

In a shock wave attack the striking hand or striking foot moves much faster than the attacker's body CM so that the striking limb absorbs CM momentum. This rapid movement is enabled by unfolding of joints such as the elbow, shoulder and knee. The striking hand or striking foot contains significant kinetic energy by virtue of its high speed. Energy propagates along the striking limb as a shock wave that grows as it progresses due to synchronized muscle energy discharge within the attacking limb and due to a decreasing mass / unit length along the attacking limb.

The primary advantage of a shock wave attack is speed of execution as compared to other techniques with comparable reach. Another advantage of a shock wave attack is that because Ma << Mb almost 100% of the kinetic energy contained in the attacking hand or foot becomes delivered energy. The major disadvantage of a shock wave attack is that the striking mass is reduced which reduces the initial kinetic energy. The reduced striking mass is partially offset by a higher impact velocity.

Correctly executed shock wave attacks transfer the body's linear or angular momentum to the attacking limb. These attacks include mawashi-geri (round house kicks), mae-geri-keage (front snap kicks) and kizami-zuki (jabs).

Shock waves are generated by appropriate sequential release of muscle energy over a short period of time. For a jab energy release starts with the rear leg, then the hips, then the back, then the striking arm's shoulder, then the striking arm's elbow. Part of the skill of an advanced karateka lies in his/her ability to precisely sequence discharge of his/her muscles to generate shock waves.

Most formal karate instruction refers to simultaneous contraction of all major muscle groups on impact (kime). That reference is appropriate for an attack such as an oi-zuki that relies on a rigid body at the instant of impact. However, in most formal karate instruction there is little or no explicit mention of shock waves or of the role of sequential muscle contraction to form shock waves. Most karate instructors know little about the mathematical formalism of wave motion. However, most senior karate instructors are aware that a good jab has a characteristic feel and causes a gi snapping sound similar to the crack of a whip.

In karate the development of shock waves is a physical skill that is learned through long physical practise. Some techniques are practiced many tens of thousands of times. These techniques become embedded in muscle memory so that the karateka can reflexively implement them while under adrenal stress. In daily practise the feedback mechanisms normally used relating to shock wave development are the feel and sound of the punch, kick or strike. Numerous karateka marvelled at the extraordinary whipping action developed by Sensei Assai, without thinking about the underlying physics.

Sensei Malcolm Fisher has indicated that mastery of whip like striking (shock wave development) comes from prolonged immersion in the JKA Instructor Training environment. Even in Japan the skill is communicated more by imitation, feel and sound than by explicit verbal communication. The JKA Instructor Program emphasizes learning by copying and by doing rather than by physical analysis.

Effective launching of a linear shock wave as in a jab involves a rear leg compression to initiate CM motion followed by a shock wave that originates from the body CM but contains the initial energy of the rear leg compression.

A whip has a decreasing mass per unit length between the handle and the tip. Shock waves in karate are enhanced by the decreasing mass per unit length along the arms and legs. The body geometry is complicated. However, from the perspective of shock wave development we can assume that the wave propagates along a limb with a cross sectional area Ao at linear position X = 0 and cross sectional area Al at the wrist or ankle where the linear position is X = L. In general Ao > Al. Assume that the striking mass (hand + forearm) or (foot + lower leg) is Ma. Let A(X) be the cross sectional area as a function of linear position X along the shock wave path.

A(X) = Ao at X = 0;
A(X) = Al at X = L;
Ma = mass of striking hand and forearm or striking foot and lower leg;
Rm = mass density.

As the shock wave progresses the moving mass M ahead of the shock wave is:
M = Ma + Integral from X to L of {Rm A(X) dX}

To achieve a constant acceleration Aa of the striking (hand + forearm) or (striking foot+ ankle) the force F(X) on the moving mass must satisfy:
F(X) = M Aa
= [Ma + Integral from X to L of {Rm A(X) dX}] Aa

Let Re = stored energy density in attacker's body at full muscle loading. Assume that the stored muscular energy discharges as the shock wave passes. Then conservation of energy gives:
F dX = Re A(X) dX

(Ma + Integral from X to L of {Rm A(X) dX}) Aa = Re A(X)
Differentiate with respect to X to get:
- Rm A(X) Aa = Re dA(X) / dX
(- Rm / Re) Aa dX = dA(X) / A(X)

Integrate from X = 0 to X = L to get:
(- Rm / Re) Aa L = Ln(Al / Ao)
Aa = (Re / Rm)(1 / L) Ln(Ao / Al)
This equation gives the acceleration Aa of the striking hand or foot in terms of parameters that can be numerically quantified.

The mass density Rm of the human body is about:
Rm = 1000 kg / m^3

Numerical substitution gives:
Re / Rm = [(1000 J / 100 Kg) X (1000 kg / m^3)] / (1000 kg / m^3)
= 10 J / kg
This is the available energy per unit of body mass.

For my body for an arm jab:
L ~ 1.5 m
Ao / Al = (Waist circumference / wrist circumference)^2
=(38 / 8)^2
= 22.6
Ln(Al / Ao) ~ 3.1
Aa = (3.1 X 10 J / kg) / 1.5 m
= 20.67 J /kg-m
= 20.67 m / s^2
This acceleration is with respect to the CM.

The shock wave propagation velocity is:
V = Vo + Aa T
Vo = initial velocity at CM established by rear leg compression
T = 0 when shock wave starts at CM.
T = Ti at instant of impact
V = Vo + Aa T
from T = 0 to T = Ti
L = Vo Ti + Aa Ti^2 / 2
Aa Ti^2 + 2 Vo Ti - 2L = 0
This is a quadratic equation with the real solution:
Ti = [-2Vo + ((2 Vo)^2 + 4 Aa 2L)^0.5] / (2 Aa)

Eo = M Vo^2 / 2
Vo = (2 Eo / M)^0.5

Due to compression wave starting at rear foot, at the instant that the compression wave reaches the CM, M involves only half the body mass.
Recall that Eo = 500 J, body mass = 100 kg
Vo = (2 X 500 J / (100 kg / 2))^0.5
= 4.47 m / s

Numerical evaluation of Ti gives:
Ti = [-2Vo + ((2 Vo)^2 + 4 Aa 2L)^0.5] / (2 Aa)
= [-2(4.47 m / s) + (80 m^2 / s^2 + 4(20.67 m /s^2)3 m)^0.5] / 2(20.67 m / s^2)
= [-8.94 m / s + 18.11 m / s] / (41.34 m / s^2)
= 0.222 s

The total execution time for shock wave type jab is the sum of the time required to execute the initial CM drop, the time required for the rear leg unloading and the time Ti required for shock wave propagation from the body CM to the striking hand.

The time T required for a 6 inch (15 cm) CM drop is given by:
D = A T^2 / 2
T = (2D / A)^0.5
D = .15 m
A = 9.8 m / s^2
T = (.3 m / (9.8 m / s^2))^0.5
= .175 s

Based on sprinter performance (~66 steps in 10 seconds) the time required for one rear leg unload is about 0.15 s.

Hence the total execution time for a jab is about:
.175 s + .15 s + 0.222 s = .55 s

Numerical evaluation of the impact velocity Vi gives:
Vi = Vo + Aa Ti
= 4.47 m / s + (20.67 m / s^2 X 0.222 s)
= 9.06 m / s

For my forearm Ma ~ 1.7 kg giving:
Ed = Ma Vi^2 / 2
= 1.7 kg X (9.06 m / s)^2 / 2
= 69.8 J

Hence hand jabs that rely on forearm velocity rather than CM momentum can potentially deliver more energy to an opponents head than other techniques but do not deliver enough energy to the opponents body to be considered finishing blows.

For comparison purposes consider the energy delivery of a professionally pitched baseball or a professional hockey slapshot. In professional play both projectiles reach speeds of 100 miles per hour.

A units conversion gives:
Va = 100 mph X (88 ft / s) / 60 mph X (12 X .0254 m) / ft
= 44.7 m / s

A 5.25 ounce (0.149 kg) baseball thrown at 100 mph (44.7 m / s) has a kinetic energy of:
(0.149 kg / 2) (44.7 m / s)^2 = 149 J

A 6 ounce (0.170 kg) ice hockey puck slap shot at 100 mph (44.7 m / s) has a kinetic energy of:
(0.170 kg / 2) (44.7 m / s)^2 = 170 J

Thus being hit by a 70 joule jab causes only about half the energy delivery of being hit by a baseball thrown by a professional pitcher or of being hit by a puck slap shot by a professional ice hockey player. However, the energy delivery due to a karate technique will greatly increase if the attacker and opponent are moving toward each other so that a technique that starts as a jab beyond the attacker's normal punching reach becomes a full punch due to closing of the distance between the attacker and the opponent.

Mathematical modelling of the shock wave impact associated with a karate technique is similar to mathematical modelling of water hammer in a steam heating pipe. Water hammer occurs when a slug of liquid water is accelerated by steam pressure in a straight section of pipe and then hits an elbow in the pipe. The impact of the high velocity slug of liquid water with the pipe wall at the pipe elbow generates a large localized pressure that can literally punch a hole in a steel pipe elbow. Even if the pipe elbow is sufficiently robust to resist damage, the noise caused by the impact of the high velocity slug of water with the pipe elbow is comparable to the sound of the pipe being struck with a sledge hammer. Hence the name "Water Hammer".

Find the impact pressure if the striking hand is soft or loose:
Consider an element of target area dA.
The density of water Rw is:
Rw = 1000 kg / m^3
Assume that the opponent absorbs all the impacting kinetic energy.
The mass flow of water hitting the area dA per unit time is:
(Rw Va dA)
The rate of change of water momentum at area dA is:
(Rw Va dA Va)
Hence the impact pressure P for a non-elastic impact is given by:
P = (Rw Va^2 dA) / dA
= Rw Va^2
= 1000 kg / m^3 X (9.06 m / s)^2
= 82.08 X 10^3 Pa
= 82.08 X 10^3 Pa X 14.7 psi / (101 X 10^3 Pa)
= 11.95 psi
Note that this impact pressure is proportional to the square of the striking velocity.

Find the impact pressure if the striking hand is rigid:
If the impacting hand is rigid instead of being loose the impact pressure is much larger but the time over which that pressure is applied is much shorter. However, the larger pressure exceeds the elastic limit of the opponent's tissue and hence causes much more damage. Recall that the primary object of Fisher Shotokan is to maximize deliverable energy. Hence the impact should be non-elastic. Hence the attacking hand or foot should be rigid at the instant of impact. A karateka with a rigid fist and forearm can easily deliver a 100 pound force punch distributed over about 1 square inch of knuckle area for an impact pressure of about 100 psi.

There are some techniques such as a compress-slide jab which, depending on the arm position at the instant of impact, may satisfy either Ma << Mb or Ma ~ Mb. If the circumstances of a jab to the body are Ma << Mb then the appropriate action is an immediate follow-up reverse punch that satisfies Ma ~ Mb.

For rigid body attacks the amount of energy delivery is limited to a maximum of about half the kinetic energy that can be developed. Shock wave attacks (whip like striking) reduce the technique execution time but result in a further reduction in the amount of energy delivery. The amount of impact damage is also dependent on the rigidity of the impacting hand or foot as that rigidity is required to maximize impact pressure and hence minimize formation of elastic energy that reduces energy delivery.

This web page text last updated May 19, 2012.

Contents Blogs Introduction Fisher One Page Contacts Links