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Old 09-20-2007, 01:48 PM   #40
Erick Mead
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Dojo: Big Green Drum (W. Florida Aikikai)
Location: West Florida
Join Date: Jun 2005
Posts: 2,568
Re: Aiki-Boxing

Chhi'mčd Künzang wrote: View Post
An elastic collision preserves the most kinetic energy by conserving all of it.

The best you can do with an elastic collision is for the second ball to leave with double the speed the first ball arrived at -- but this is the theoretical limit as the ratio of the masses approaches infinity.
I know *you* know this Mike, more clearly I am sure than I do. I just threw it out there for the number geeks to chew on. If my math's wrong, please point it out -- but I think I got the equations right.
I don't quibble with the math -- I just quibble with the model of the problem. First of all it is even worse than you assume. Because the balls must hit in a spinning collision and here is always some offset in the eccentricity of the collision this results in an amount of the impact energy proportional to the eccentricity of impact being dissipated to increase spin in the target ball -- vice increase linear velocity -- which application of billiard English suitably leads into my main point.

Linear momentum -- p= mv is a proportional system -- mass times velocity, and the kinetic energy increases as half the mass times the square of the velocity 1/2mv^2. -- So only one square term operates in the linear momentum energy equation.

Angular momentum is a very different animal when it comes to concentrating and dissipating kinetic energy, because the inertial radius controls angular velocity, creating an additional effective square term.

Angular momentum (L) -- L = Iω where I is the inertial moment and ω is the angular velocity. Angular velocity is proportional to the inverse of the radius (the skater spins faster when tighter). But the inertial moment of the body also reduces proportional to of the radius, leaving the total angular momentu proportiaonl to the inverse square of the radius.

Thus, by simply reducing the radius of the rotation we add an additional square term to our angular velocity in the kinetic energy equation -- which is the term that is further squared to yield the effective kinetic energy (or by increasing radius to dissipate energy in terms of the inverse square of the radius). This energetic transformation does not operate in in the linear, non-rotational momentum scenario. In in fact as illustrated above induced rotations are a serious source of inefficiency of linear momentum transfer actual collisions. This degree of disproportion in ability to manipulate the energy positively or negatively though rotational transformation cannot be duplicated in linear momentum terms with normal human anatomy.

On the other hand, this rotational energy equation operates at every stage of reducing or increasing effective radius of rotation. You cannot move your body around without every component rotating about its own center as well as about the point of attachment to some other component. Either initially or ultimately it involves rotating the entire length of the body about its center of mass AND its point contact with the earth. (Bishop Berkeley said a pendulum also rotates with reference to the "fixed stars" so the "Ki of Heaven" is actually a bit of a cross- cultural image.)

Commencing a rotation, let's say, of 2 meters radius (exagerrating for simplicity's sake) of my full height and at its center of mass reduced by about 40 percent (Divine proportion) when I lock out the bottom part and then rotate the upper toso about its C.o.M., then again by roughly 40% at every successive limb component, results in a theoretical multiple of about 6.25 times the angular velocity transferred from the immediately prior stage -- at every stage of the progression.

From my whole body to my upper body/shoulder, to my upper arm, to my forearm, to my fist rotating about my wrist -- that is four conversion stages over the initial motion -- so the potential angular velocity delivered is about 6.25 * 6.25 * 6.25 * 6.25 or 6.25^4 = ~ 1500 times the angular velocity -- limited only by inherent inertia, efficiency losses due to asymmetry and the degree of stiffness in the connections. All of these are very large potential inefficiencies, and thus there is a wide field for training to improve on.

The first and second inefficiencies would be addressed by following a spiral symmetry in the whole movement, aligning every part of every limb to the same scheme of rotation -- which is the pattern of Aikido movement. The second inefficiency would be addressed by "softening" the linkages, as Amdur suggests was O Sensei's purpose in adapting the selection of waza for his training regime. That reduces stiffness in rotation for impulse delivery, and also allows increased sensitivity to and coordination of "viscous" coherence of simultaneous joint rotations across the whole body to impulse meant to to be dissipated.


Erick Mead
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