25. Constraints from m_cloud and v_cloud

For m_head = 5 lb

v_lurch, ft s^-1

PE, ft-lb

For m_head = 6 lb

v_lurch, ft s^-1

PE, ft-lb

For m_head = 7 lb
    

v_lurch, ft s^-1

PE, ft-lb

For m_head = 8 lb
    

v_lurch, ft s^-1

PE, ft-lb

For m_head = 9 lb

v_lurch, ft s^-1

PE, ft-lb

No solutions.

Summary table

Observations

• Increasing the mass of the diffuse cloud reduces the forward speed of the lurch or increases the rearward speed.
• Increasing the (forward) speed of the diffuse cloud has the same effect.
• Higher potential energies decrease the rearward lurch and tend to make it into a forward lurch.
• The lurches are all large. Except for the case where the head is 8 lb, which has two of its 10 values at 0.1 and 0.4 ft s^-1 to the rear, all the other values are at least 0.6 ft s^-1 in the rearward direction.
• The smallest lurches are associated with the lightest clouds, the smallest velocities of the cloud, and the highest potential energies. Restricting the mass of the cloud to values >0.1 lb (a mild criterion, since up to a pound of brain was lost), eliminating stationary clouds (speeds of 0 ft s^-1), and eliminating PEs greater than 600 ft-lb eliminates 111–117 of the 121 cases and eliminates all the positive lurches and the small negative lurches. Not a single positive lurch remains.
• In other words, the lurch can't have speeds greater than about 2 ft s^-1 to the rear because that requires negative potential energies. It can't have speeds less than about 0.5 ft s^-1 to the rear (or forward speeds) because that would require potential energies far in excess of anything reasonable and masses of the cloud that are too small. The result of satisfying these three constraints is a narrow band of lurches (all rearward), potential energies, masses of cloud, and speeds of cloud. There is surprisingly little flexibility left in the final solutions.

Simplest explanation (based on Lurch 7 Angular)
    Energy. The system begins with 1164 ft-lb of KE, all in the approaching bullet. The bullet smashes through the head, using about 200 ft-lb of energy to penetrate scalp and skull on both sides and plow through the brain tissue in between. That leaves 964 ft-lb of energy available. The bullet leaves the skull with about 14 ft-lb of KE. That leaves 950 ft-lb for the rest of the system (the body, the clouds, and the two large fragments). The cloud gets by far the most, about 772 ft-lb, which leaves 178 ft-lb. The two large fragments get 130 and 48 ft-lb, leaving 0 ft-lb. The body, a bit player in the energy equation because it is so heavy, gets only 0.1 ft-lb, which is in the noise of the other components. (Basically, it doesn't count for anything.)
    Momentum. The system begins with an angular momentum of 3.71 lb ft s, all in the approaching bullet (we will forget these unintuitive units for the rest of this paragraph). It ends (after the collision) with 4.68 units of momentum in the forward direction (the bullet, the cloud, and the two large fragments) and 0.97 units in the rearward direction (the body). The algebraic sum of these oppositely directly momentums is 0.97 - 4.68 = 3.71. Note that as with the energy, the lurching body only contains a minority of the total momentum of the system (only 20% of the forward momentum, for example). The cloud again gets the lion's share of the forward momentum (3.13 out of 4.68). This emphasizes the importance of the cloud to the final solution.
    The real role of the body. It is important to grasp the smallness of the role that the body plays here, for its dramatic motion in the Zapruder film makes it appear to be a much larger ingredient than it really is. The total energy of the system before the collision is converted into afterward components of 17% potential energy and 93% forward kinetic energy—the "huge" rearward lurch of the body accounts for a mere ripple—only 0.01%, or 1 part in 10,000, of the final kinetic energy. It is literally lost in the noise of the other components, which share 99.99% of the original energy. From the standpoint of energy, the rearward lurch is only an afterthought.
    Momentum holds the key to understanding the lurch. When the head exploded and hurled the diffuse cloud of fragments forward, the new forward momentum had to be balanced by an equal rearward momentum. That momentum was not provided by a rearward cloud because there was no such cloud—almost all the material from the explosion moved forward. That left only the body to move to the rear, which it then did. The body didn't have to move rearward very fast because the cloud and large fragments took only a little more momentum than the incoming bullet originally had (26% more, to be exact). The body could lumber backward at the relatively stately pace of 0.4 to 1.0 ft s^-1 because it was so massive.
    Constraints on the final motion. Of the twenty-some variables considered here, only a handful contribute meaningfully to the final solution. Even those are not as free as they might seem, for they are strongly constrained by the others. For example, the lurch can't be in the forward direction or less than about 0.4 ft s^-1 in the rearward direction because that would have to be caused by potential energies (of breaking the skull on both sides) that are much higher than anything reasonable. (It would also require excessively low velocities and masses of the cloud.) The rearward lurch can't be faster than about 2 ft s^-1 because that would require the potential energy to be negative (to add energy to the system rather than using it to break skull). 

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