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 lurch—just 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.
Ahead to Introduction to a Frontal Hit
Back to Constraints on mcloud
and vcloud from mcloud and vcloud
Back to Intro to Constraints
Back to Physics of the Head Shot