30. Grand summary of constraints

We have previously identified the seven most important variables to the snap and lurch as mhead, dsnap, tlurch (these three for the snap), Θcl, mcloud, vcloud, and PE (these four for the lurch). We now summarize the findings for the constraints on each, and then assemble a grand summary of constraints on the lurch.

mhead — 5–7 lb (from "Preconstraints").
dsnap — >2.2 in (from basic measurements on Zapruder film).
tlurch — 0–25 ms (from Zapruder film).
Θcl — 0°–180°; probably about 30°–90° or so in practice.

mcloud — >0 to at least 1.0 lb (no effective constraints for our purposes).

vcloud — 200–600 ft s-1 (simulations and speed of sound).

PE — >0 to 600 ft-lb; 100–600 in practice.

The effect of these constraints on the lurch can be summarized in a table of vlurch for PEs of 100 and 600 ft-lb (the effective limits of PE) at angles Θcl from 0° to 180°, for dsnap = 2.4 in and heads of 5, 6, and 7 lb. No other values of dsnap need be shown because it has only a minor effect on vlurch for these allowed conditions, and the individual mcloud and vcloud need not be shown because they are not fundamentally constrained (mcloud is not constrained, and vcloud takes its values from mcloud). For all intents and purposes, the 29 previous sections come down to this single table. (Units are omitted to save space in the first row.)

 Mhead Θcl vlurch 100 vlurch 600 Mhead Θcl vlurch 100 vlurch 600 Mhead Θcl vlurch 100 vlurch 600 5 lb 0 -3.48 -2.18 6 lb 0 -3.45 -2.10 7 lb 0 -3.20 -1.62 10 -3.44 -2.15 10 -3.40 -2.07 10 -3.17 -1.60 20 -3.30 -2.06 20 -3.28 -1.99 20 -3.05 -1.53 30 -3.09 -1.91 30 -3.07 -1.85 30 -2.87 -1.42 40 -2.81 -1.72 40 -2.80 -1.66 40 -2.62 -1.28 50 -2.48 -1.48 50 -2.47 -1.43 50 -2.33 -1.10 60 -2.10 -1.21 60 -2.10 -1.18 60 -1.99 -0.91 70 -1.69 -0.92 70 -1.70 -0.91 70 -1.64 -0.70 80 -1.28 -0.63 80 -1.30 -0.63 80 -1.28 -0.48 90 -0.87 -0.34 90 -0.91 -0.36 90 -0.92 -0.28 100 -0.49 -0.07 100 -0.53 -0.10 100 -0.58 -0.08 110 -0.14 0.17 110 -0.20 0.13 110 -0.28 0.10 120 0.16 0.39 120 0.10 0.33 120 -0.02 0.26 130 0.42 0.56 130 0.34 0.50 130 0.20 0.39 140 0.61 0.70 140 0.54 0.63 140 0.38 0.49 150 0.76 0.80 150 0.68 0.73 150 0.50 0.56 160 0.85 0.87 160 0.77 0.79 160 0.59 0.61 170 0.90 0.91 170 0.82 0.82 170 0.63 0.64 180 0.92 0.92 180 0.84 0.84 180 0.65 0.65

Important as this summary table is, its essence can be shown much more clearly in a graph. The same data are plotted below, with comments below the graph.

First some general comments on how this graph is set up. There are two bold horizontal lines, one for a lurch of 0 ft s-1 (the boundary between forward and rearward lurches) and the other at -0.8 ft s-1, the value for the observed lurch. There are two bold vertical lines showing the most-likely range of Θcl, 30°–90°.
Next come some general comments on the shape of the curves. A common feature of all the curves is that they have strong rearward lurches for narrow clouds (small values of Θcl
) and weaker rearward or positive lurches with wider clouds (greater values of Θcl). Note the S-shapes for all the curves, meaning that the lurch changes slowly with narrow and broad clouds, and fastest with clouds of intermediate width. The lurches for PE = 600 ft-lb are always more positive than those for 100 ft-lb (for a given weight of head), but the difference lessens as the cloud broadens, and the two values merge at Θcl = 180°.
Another interesting feature of the lines is that all three for the same PE have a common crossing point at some value of Θcl, about 80° for 100 ft-lb and about 100° for 600 ft-lb. Near those places, the influence of the head on the lurch disappears. Since we use a default Θcl of 70°, which is near those angles, we can say that the mass of the head plays little role in the lurch.
Now to the major questions about the lurch posed at the beginning of this monograph. The first is whether a rearward shot from Oswald's rifle can produce a rearward lurch. The answer from earlier sections and confirmed by this graph is that it can, and with ease, for all values with angles less than about 105° are negative, and most below 90° are more negative than the actual lurch. The second question is whether Oswald's rifle must have produced a rearward lurch consistent with the one observed. The answer to that is yes for any cloud narrower than 90° or so. This is a remarkable finding, inasmuch as it is essentially independent of the mass of the head and the mass of the fragments. Since the great bulk of the material in the cloud was confined to the forward sector, a hit from the rear must have produced the observed lurch.

Confirming the critical role of the cloud
The previous sections have identified the major variables and noted that they had to do with the head, the cloud, and the time intervals. The grand summary in the paragraphs above have shown that the only thing required to produce the observed lurch was a forward-moving cloud of fragments. This conclusion strongly reaffirms the primacy of the cloud in producing the lurch, which was the original gut-level starting point.
There is a simple way to confirm the importance of the could to the lurchjust remove the variables individually from a representative negative solution and see what happens to the lurch. Unimportant variables will leave the lurch near its original negative values, whereas important variables will displace it farther. A variable that is critical to having a negative lurch will reverse the solution to a positive value when it is removed. The table below shows the results with SL7A and default conditions (including Θcl = 70°, dsnap = 2.4 in, and mhead = 7 lb) . The results are extraordinarily clear—deleting the cloud (by deleting any of its variables) reverses the lurch, from -1.09 ft s-1 to +0.61 ft s-1, but deleting any of the other variables increases or decreases the speed while keeping it rearward (speeds of 0.57–1.86 ft s-1). This confirms that the cloud alone is responsible for the rearward lurch.

 Variable removed from SL7A vbodyafter, ft s-1 None removed—all present -1.09 Variables critical to the rearward lurch mcloud +0.61 vcloud +0.61 mcloud, vcloud +0.61 Θcl +0.61 fxcl (with Θcl) +0.61 fkecl (with Θcl) +0.61 Variables that adjust the rearward lurch vbullet -1.86 Θfrag1 -1.22 Θfrags23 -1.23 fI (by setting it to 1) -1.21 fkefrag1 (with fkefrags23) -1.09 fkefrags23 (with fkefrag1) -1.09 fkebody -1.09 dtransit -1.07 ttransit (with dtransit) -1.07 Θ -1.00 mfrags23 -1.01 tdelay -0.99 vbulletafter -0.76 dsnap -0.66 tsnap (with dsnap) -0.66 mfrag1 -0.64 dfrags (with vfrags23, dfrags) -0.57 vfrag1 (with vfrags23, dfrags) -0.57 mfrag1, mfrags23 -0.57 vfrags23 (with vfrags23, dfrags) -0.57 mfrag1, mfrags23, vfrag1, vfrags23 (with dfrags) -0.57

Simplest description of the physics (based on SL6A)
Energy. The system begins with 1164 ft-lb of KE, all in the approaching bullet. The bullet smashes through the head, using 300 ft-lb of energy to penetrate scalp and skull on both sides and plow through the brain tissue in between. That leaves 864 ft-lb of energy available. The bullet leaves the skull with about 14 ft-lb of KE. That leaves 850 ft-lb for the rest of the system (the body, the cloud, and the two large fragments). The cloud gets by far the most, about 563 ft-lb, which leaves 287 ft-lb. The two large fragments get 209 and 77 ft-lb, leaving 1 ft-lb. The body, a bit player in the energy equation because it is so heavy, gets that remaining 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 3.80 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 7.61 units of momentum in the forward direction (the bullet, the cloud, and the two large fragments) and 3.82 units in the rearward direction (the body). The algebraic sum of these oppositely directly momentums is 0.97 - 4.68 = 3.79. Note that in distinction to the energy, the lurching body contains a much larger fraction of the total momentum of the system (a full 50% as much as the forward momentum). The cloud again gets the lion's share of the forward momentum (5.76 out of 7.61). This again 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 26% potential energy and 74% forward kinetic energy—the "huge" rearward lurch of the body accounts for a mere ripple—only 0.1%, or 1 part in 1000, of the final kinetic energy. It is literally lost in the noise of the other components, which share 99.9% of the original energy. From the standpoint of energy, the rearward lurch is only a twig in a forest.
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 had to move rearward with a reasonable speed because the cloud and large fragments "took" twice as much momentum as the incoming bullet had (201%, to be exact). But the great mass of the body allowed its speed to be the relatively stately 0.8 ft s-1, which then kept its KE small (because KE is proportional to v2, not v as with momentum).

Constraints on the final motion. Of the thirty-some variables considered here, only a handful contribute meaningfully to the final solution. Several of those are not as free as they might seem, either, for they are constrained by the others. For example, the lurch can't normally be positive or weakly negative because that would have to be caused by potential energies (of breaking the skull on both sides) that are much higher than anything reasonable or by a cloud that is excessively broad. The rearward lurch can't normally be faster than about 3 or 4 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 the skull). All these factors combine to give sizably negative lurches in most cases.