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Impulse of a Taekwondo PunchFor example, consider the amount of force needed to break a board supported at both ends struck in the center by a force F. The downward force will be shared by the two supports, so each will supply an upward force of F/2. Imagine that the board is deflected downwards by the force, as shown in green. In this case the top surface of the board will be in compression, and the bottom surface will be in tension.This force will produce a torque about an axis through the middle of the board. To understand this, consider the forces on the left half of the board that are due to the right half. The top of the board is in compression, so the right half is pushing to the left. The bottom of the board is in tension, so here the right half is pulling to the right. These forces are indicated by the red horizontal arrows. This means the forces will produce a counter-clockwise torque about the center. The board will break if this torque is greater than:
tmax = 1/6 Wh2sb
where W is the width of the board, h is the thickness, and sb is the modulus of rupture of wood.Notice that the upward force from the left support will tend to produce a torque in the opposite direction from the torque due to the stresses. The magnitude of this torque is simply r*F (since r and F are perpendicular in this case), or F/2 * L/2. If this torque is less than tmax given above, then the board will not break. Thus, the minimum force needed to break the board can be calculated by setting these two torques equal to each other:
FL/4 = 1/6 Wh2sbF = 2/3 sb Wh2/L
Notice that the force is proportional to h2; boards get much harder to break as they get thicker. This is why breakers will break stacks of several thin boards rather than a single large piece of wood with the same total thickness. If we put in some typical dimensions for a board, such as L x W x h = 30 x 20 x 2 cm, and look up the rupture modulus, we get a force of around 711 N, or 160 lbs. This force sounds feasible for a punch to generate, but let us check to be sure.If we assume, as above, that we have a 7-kg arm traveling at 7 m/s then we have a total momentum of P ~ 49 kg m/s. We assume that the fist and arm come to rest during the blow, so this is the total DP in the collision. Since the force is DP/Dt, we need to know how quickly the fist stops. In a graph of the downward hammer fist strike,your will notice that the velocity starts out fairly constant (and negative) and then quickly changes. The maximum acceleration upon contact with the target is upwards of 350 g! By looking at the width of the peak in the acceleration curve, we may get an idea of the interaction time. It looks to be a little over 5 ms. Since this situation is a bit different from ours, we will be conservative and assume an interaction time of 10 ms. A change in momentum of 49 kg m/s in a time of 10 ms would require a force of 4900 N, which is more than sufficient to break the board described above.However, what about the breaking the cones in the hand? There is a bit of a safety margin sense bone is inherently stronger than wood (or even concrete). A torque analysis similar to the one above for a small, short cylinder of bone (similar to the bones in the hand) indicates that it should take a force of about 1500 N to break a hand bone. But the bones in the hand are protected because they are not rigidly supported like the board discussed above. The soft connective tissue (muscles, tendons, etc.) in the hand may absorb much of the energy of the strike, if the hand is held in the proper position. Also, recall that bone is much stronger in compression than in tension or torsion, so, if the hand is held in a position such that the bones are exposed to compression rather than tension, the bones will be further protected.So the board should break before the hand does. Still, the ~4900 N available in the punch is much greater than the 1500 N required to break a hand bone. So, even with the safety margins, there is still a risk. Most of the safety margin depends on technique; holding the fist properly, snapping it out with great speed, etc. If the punch is executed properly, the board will break; otherwise, the hand will break.