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This is the first of a series of articles on the physics of martial arts. This first article is on the physics of breaking boards (or bricks or ice blocks). On and off in future releases of this website there will be articles on the physics of other aspects of martial arts.

In the movie The Karate Kid, Part II, Daniel LaRusso was challenged in a bar/arcade to a large monetary bet by his rival Chozen to break a large block of ice with a single hand strike. With some last minute instruction on breathing techniques by his master/mentor Mr. Miyagi, Daniel was successful in breaking the large block of ice. Is that realistic from a physics point of view? Can someone like Daniel who previously hasn’t practiced that kind of technique break a large block of ice on his first attempt? The answer is highly unlikely to impossible. Only in a movie that allows dramatic exaggeration is that possible, because it requires an extremely large force to break a large block of ice with a single hand strike.

However, from a physics point of view, breaking a thin pine board (e.g., one inch thick) with a hand strike may be possible even for someone who is attempting that for the first time if that person is given some proper instructions. The key ingredient is that the person’s hand needs to move at a high speed at the point of impact with the board. In order for the hand to be moving still at a high speed when the hand hits the board, the person must imagine that the board is further down from its actual position, i.e., he should be aiming his strike at a point that is lower than the actual position of the board. Otherwise the hand would have a tendency to slow down when it reaches the actual position of the board. There is a couple of other ingredients which I will not bother to mention.

Of course, the amount of force or energy that is required to break a board depends on the type of board. Martial artists usually use one inch thick pine boards (the length of the board is around 12 inches, and the width of the board is around 6 to 10 inches). Such pine boards usually take about 5 Joules of energy to break (if the grain of the board runs parallel to the length of the hand strike, then it requires even less energy to break the board). To have an idea of how much energy is one Joule, let me give two common examples. One Joule of energy is equivalent to the energy that is needed to produce one Watt of electric power for one second, or equivalent to the kinetic energy of a 2 kilogram mass moving at 1 meter/second.

It is a fairly straightforward physics calculation to determine the minimum velocity (same as speed for our purpose in this discussion) of the hand/forearm at the time of impact with the board to generate the energy needed to break the board. Before we provide a quantitative explanation, we will first provide a qualitative explanation. If the board is going to stop the hand/forearm from going through it, i.e., reduce the velocity of the hand/forearm to zero, the board must apply a force to the hand/forearm to decelerate it (i.e., slow it down) to zero velocity. The larger the velocity of the hand/forearm, the larger is the force needed to stop the hand/forearm. By Newton’s Third Law, the hand/forearm will apply an equal and opposite force on the board. When the velocity is large enough, the force that the hand/forearm is exerting on the board will be large enough to break the board, i.e., the board will break before it can exert a large enough force on the hand/forearm to stop it. The amount of kinetic energy of the hand/forearm when in motion (or the amount of energy that the hand/forearm can impart to the board) is proportional to the square of its velocity; that is why the key ingredient to breaking the board is to have the hand/forearm moving at a high velocity.

Now we present the quantitative explanation. Let’s denote the energy needed to break the board by Eb. The calculation makes use of the conservation of energy and conservation of momentum, and a couple of simplifying approximations. If we denote the mass of the hand and forearm by m and the velocity of the hand/forearm just before its impact with the board by V, then the kinetic energy of the striking hand/forearm just before impact is:

½(mV2) , (Eq. 1)

Suppose the impact has broken the board, and we denote the mass of the board fragments (or board) by M, and the velocity of the board fragments and the hand/forearm after the impact by U (making the reasonable approximation that the board fragments and the hand/forearm are moving at the same velocity immediately after impact). We now make another reasonable approximation that the energy that was lost to heat from the impact is negligible; then conservation of energy implies

½(mV2) = Eb + ½(m+M)U2 , (Eq. 2)

Because of Newton’s Third Law, for every action there is an equal and opposite reaction, if we consider the hand/forearm and board as the whole system, there was on the whole system no net force that affected the impact. Therefore, there is conservation of momentum, which implies

mV = (m+M)U , (Eq. 3)

Equation 3 implies

U = mV/(m+M) , (Eq. 4)

Substituting U from Eq. 4 into Eq. 2 and solving for Eb gives rise to

Eb = ½(mMV2)/(m+M) , (Eq. 5)

Since we already know that Eb is about 5 Joules, we can substitute the mass of the hand/forearm (about 1.3 kilogram) and the mass of the board (about 0.5 kilogram) into Equation 5 to determine that V, the velocity needed at impact to break the board is about 5 meters/sec, or about 11 miles/hour.

High speed photography has shown that a non-martial artist who has been instructed to strike fast and aim at a position that is significantly lower than the actual position of the board can actually reach a maximum velocity of about 10 meters/sec, or twice the velocity needed to break the board, thus verifying our initial statement that a novice can break a pine board if he has been given proper instructions.

We now briefly discuss about breaking multiple boards or bricks (or ice blocks). When we break a stack of boards (or bricks/ice blocks), we have two ways of stacking: Pegged or unpegged. Unpegged stacks are stacks where multiple boards (or bricks/ice blocks) are stacked directly on top of each other. Pegged stacks are stacks where multiple boards (or bricks/ice blocks) are stacked with spacing items (such as pegs, nuts, coins, and pencils) between them.

Because a wood board would flex slightly before it snaps at the impact point. When unpegged, the entire stack of boards would flex together upon impact. When pegged (with a sufficient gap), the gap between the boards means that each board would flex and snap before the next board is reached. Thus the striking hand must physically touch every board in a pegged stack. That is why breaking a pegged stack of boards is more difficult than breaking an unpegged stack of boards.

Bricks, on the other hand, are ceramic, and snap (or shatter) upon a large enough impact force. Bricks don’t flex. When a stack of bricks is unpegged, the amount of force required to break all of the bricks increases with each additional brick. When bricks are pegged, the gap actually allows the bricks (which are heavier than the pine boards) to break each other, i.e., the force shatters the first brick, and the brick pieces falling downward through the gap will break the second brick, and so on. The larger the gap, the easier the following bricks will break, because the broken brick pieces will be falling for a longer time, picking up higher velocity and therefore leading to a larger impact on the next lower layer of brick. This means that breaking a pegged stack of bricks is significantly easier than breaking an unpegged stack of bricks.

Ice is also like bricks; ice doesn’t flex. Therefore, breaking a pegged stack of ice blocks is significantly easier than breaking an unpegged stack of ice blocks. That is why in most of the brick or ice block breaking demonstrations, the stacks are pegged with large gaps. With the large gaps, it might look impressive making the audience wonder how the hand can reach the bottom brick or ice block with such a large force. The answer is that it doesn’t have to; the falling broken higher layer brick or ice block breaks the layer below. Broken chunks (especially of the ice block) are so heavy that their falling momentum is large enough to shatter the next layer below. Therefore the more impressive the demonstration appears, the easier it actually is.

Some references for the subject matter discussed in this article include:

• Breaking (martial arts): http://en.wikipedia.org/wiki/Breaking_(martial_arts)
• Kung Fu Science: http://www.kungfuscience.org/access.asp
• Breaking Boards: The Physics of a Karate Chop:
  http://findarticles.com/p/articles/mi_m1511/is_5_21/ai_61692484
• Video of Dolph Lundgren breaking five ice blocks (notice that his hand may not even reach the lower ice blocks): http://www.youtube.com/watch?v=LF4BLBGRgqk
• Pine boards for martial arts: http://www.plymouthpine.com/
• National Geographic and Fight Science Lab:
  http://news.nationalgeographic.com/news/2006/08/060814-fight-science.html

In this first article on the physics of martial arts, we have discussed the physics of breaking boards or bricks/ice blocks. In future articles, we will discuss the physics of other aspects of martial arts.