23--Summary of solutions

The solutions
    We first summarize the solutions of all the previous sections, then summarize the major variables that have been revealed by the progression of solutions and the sensitivity analyses. Then (in the next section) we focus on the combinations of the most important variables in order to search for the most characteristic solutions.
    The progression of solutions for the snap and the lurch are listed in the tables below.
    The snap is a simple case, for there is only a single scenario each for linear and angular calculations. With the angular calculations, the speed of the snap decreases from its 5.0 ft s^-1 (linear) to 4.56 ft s^-1. The important point here is that both versions of the calculation explain the snap.

    The progression of calculations for the lurch is more complicated, since the linear and angular scenarios each have seven versions. The first five of each solve for the speed of the lurch and the speed of the could, while the last two substitute the mass of the could form the speed of the cloud. Examining the changes in the solutions as variables are added offers an initial way to assess the importance of the new variables.
    Consider first the linear calculations. The speed of the lurch in scenario 1 (body, bullet, and cloud) is 2.36 ft s^-1, substantially faster than is shown in the Zapruder film. Adding the large fragments (scenario 2) increases this speed only slightly, to 2.44 ft s^-1. This indicates that the large fragments do not materially affect the answer, although the offer important clues about the nature of the explosion and the forward movement of the cloud. By contrast, adding 3-D motion to the cloud (scenario 3) has a huge effect on the lurch, decreasing it speed to 0.88 ft s^-1 (i.e., by nearly a factor of three). This shows that the three-dimensional properties of the cloud must be estimated as accurately as possible. The reason for the decrease is easy to understand—motions along the Y- and Z-axes (in/out of the plane of the film and up/down) do not contribute to the rearward lurch (X-axis). Allowing a given mass of could to have Y- and Z-components of motions decreases the mass available to affect the motion in the X-direction). Adding 3-D motion to the large fragments (scenario 4) does not change the lurch perceptibly (0.88 ft s^-1 rearward becomes 0.87 ft s^-1 rearward). The same thing holds for adding 3-D motion to the body (scenario 5), where the lurch remains unchanged. Solving for m_cloud instead of v_cloud (scenario 6) hardly changes the solution, either; the lurch increases in speed to 0.90 ft s^-1.
    The final big change comes in scenario 7, which adds the four time intervals of frame 312 into frame 313. This step reduces the speed of the lurch from 0.90 ft s^-1 to 0.55 ft s^-1, and in so doing appears to bring it below the observed speed of 0.82 ft s^-1 to the rear. The disagreement may be more apparent than real, however, for the same reason discussed earlier for the snap—the initial lurch lasted less than a frame and so would have been faster than the full-frame average calculated from Thompson's data. Thus the apparent disagreement here does not mean much.
    We can conclude from the progression of lurches calculated linearly that the most important variables are the original ones (body, bullet, and cloud) plus the 3-D aspects of the cloud and the time intervals (which mostly hinge on the time for the snap).
    Similar conclusions on important variables can be derived from the angular scenarios. Although the rearward speeds are generally up to 50% greater than for the linear cases, they break with the same changes in variables and in the same directions. The time intervals seem to be more important for the angular calculation, however, for they take lurches that were greater than the linear cases (scenarios 1–6) and make them smaller than the linear case (scenario 7). But they do not change the overall list of important variables.

Scenario	Description	v_bodyafter, ft s^-1	v_cloud, ft s^-1
Lurch 1 Linear	Calculates from body, bullet, and cloud	-2.36	450
Lurch 2 Linear	Adds large fragments	-2.44	437
Lurch 3 Linear	Adds 3-D motion to cloud	-0.88	309
Lurch 4 Linear	Adds 3-D motion to large fragments	-0.87	307
Lurch 5 Linear	Adds 3-D motion to body	-0.87	307
			m_cloud, lb
Lurch 6 Linear	Sets v_cloud = v_frags and solves for m_cloud	-0.90	0.32
Lurch 7 Linear	Adds four time intervals	-0.55	0.18
			v_cloud, ft s^-1
Lurch 1 Angular	Rotational analog of Lurch 1 Linear	-3.53	449
Lurch 2 Angular	Adds large fragments	-3.29	436
Lurch 3 Angular	Adds conical 3-D motion of cloud	-1.30	282
Lurch 4 Angular	Adds 3-D motion of large fragments	-1.28	279
Lurch 5 Angular	Adds 3-D motion of body	-1.28	279
			m_cloud, lb
Lurch 6 Angular	Sets v_cloud = v_frags and solves for m_cloud	-1.14	0.26
Lurch 7 Angular	Adds four time intervals	-0.33	0.083

The most important variables
A better way to summarize the importance of the variables is to list them in the order of their range of values (of the lurch) created by reasonable variations (in the sensitivity analyses). The results are shown in the two large tables below. The first summarizes the variables and the range of values given to each, in roughly the order they were introduced into the calculations. The second shows the range of effect each variable had on each calculation.

Variable	Symbol	Default value	Range of values
Mass of bullet	m_bullet	161 gr	156–166 gr
Entrance velocity of bullet	v_bullet	1800 ft s^-1	1750–1850 ft s^-1
Exit velocity of bullet	v_bulletafter	200 ft s^-1	0–400 ft s^-1
Angle of bullet above horizontal	Θ	17°	12°–22°
Mass of head	m_head	7 lb	5–9 lb
Mass of upper torso	m_body	85 lb	65–105 lb
Vertical length of upper torso	L	3 ft	None—on both sides of equation
Mass of diffuse cloud of brain matter	m_cloud	0.3 lb	0.1–0.8 lb
Exit speed of cloud of brain matter	v_cloud	300 ft s^-1	100–500 ft s^-1
Potential energy created by bullet's breaking and transiting skull	PE	200 ft-lb	100–500 ft-lb
Lever arm of rotation of head about top of neck	R_head	4.5 in	3.5–5.5 in
Lever arm of bullet with respect to top of neck	R_bullet	5.75 in	4.75–6.75 in
Mass of large fragment 1	m_frag1	0.027 lb	0.022–0.032 lb
Exit velocity of large fragment 1	v_frag1	300 ft s^-1	100–500 ft s^-1
Upward angle of large fragment 1	Θ_frag1	40°	20°–60°
Mass of large fragments 2,3	m_frags23	0.01 lb	0.005–0.015 lb
Exit velocity of large fragments 2,3	v_frags23	300 ft s^-1	100–500 ft s^-1
Upward angle of large fragments 2,3	Θ_frags23	70°	50°–90°
Final speed of forward snap of head	v_snap	Calculated	Calculated
Final speed of rearward mechanical recoil (lurch) of head and upper torso	v_bodyafter, v_lurch	Calculated	Calculated
3-D term for speed of cloud (to reduce mean X-velocity)	f_xcl	0.7	0.4–1.0
3-D term for kinetic energy of cloud (to add Y, Z components of KE)	f_kecl	2	1–3
3-D terms for kinetic energy of large fragment 1 (to add Y, Z components of KE)	f_kefrag1	1.25	1.0–1.5
3-D term for kinetic energy of large fragments 2,3 (to add Y, Z components of KE)	f_kefrags23	1.25	1.0–1.5
3-D term for kinetic energy of lurching body (to add Y, Z components of KE)	f_kebody	1.2	1.0–1.4
Distance of bullet's transit through head	d_transit	4 in	0–9 in
Distance of forward snap of head	d_snap	2.2 in	1.4–3.0 in
Distance moved by large fragments in Z313	d_frags	6 ft	2–10 ft
Time delay to begin snap after Z312 closes	t_delay	0.002 s	0.000–0.006 s
Time duration of forward snap within 312,313	t_snap	Calculated	Calculated
Time for bullet to transit the head	t_transit	Calculated	Calculated
Time of lurch within open period of Z313	t_lurch	Calculated	Calculated
Half-angle of conical cloud of brain matter	Θ_cl	45°	25°–65°
Factor for improved moment of inertia of body	f_I	1.11	1.07–1.15

The most important variables are shown near the top of the table below, and are generally grouped in order of decreasing range of effect on the lurch in Lurch 7 Angular. Although the range of effects in L7A form something of a continuum, most of the largest ones come from variables associated with the head, the cloud, and the timing. The variable d_snap, which might seem to be an exception, is not because it is largely set by the speed of the forward snap, which in turn depends on the mass of the head. Even d_frags is not a real exception, because it has to do with the violence of the explosion and with the time intervals.
Equally instructive is to examine the variables with small effects. They have mostly to do with the bullet, the large fragments, and the body, and surprisingly include the mass of the body.

	Range of effect, ft s^-1, on lurch
Variable	L1L	L2L	L3L	L4L	L5L	L6L	L7L	L1A	L2A	L3A	L4A	L5A	L6A	L7A
v_cloud						4.20
d_snap							1.52							3.88
d_frags							1.68							2.97
m_cloud	3.29	3.28	1.63	1.62	1.62			4.52	4.80	2.32	2.30	2.30
m_head							1.44							1.78
R_head														1.49
R_bullet														1.20
Θ_cl										1.30	1.29	1.29	1.69	0.89
f_xcl			1.32	1.31	1.30	1.34	0.99
f_kecl			0.92	0.91	0.91	2.08	1.54
v_bulletafter	0.12	0.12	0.17	0.17	0.17	0.12	0.61	0.16	0.16	0.22	0.22	0.22	0.16	0.75
PE	0.67	0.69	0.34	0.35	0.35	0.71	0.55	1.34	0.93	0.49	0.49	0.49	0.91	0.55
t_delay							0.54							0.56
v_bullet	0.16	0.17	0.06	0.05	0.06	0.18	0.31	0.22	0.23	0.09	0.09	0.09	0.23	0.36
m_bullet	0.07	0.48	0.01	0.01	0.01	0.08	0.23	0.10	0.10	0.02	0.02	0.02	0.10	0.27
Θ_frag1		0.08	0.08	0.08	0.08	0.08	0.11		0.12	0.11	0.11	0.11	0.11	0.19
Θ	0.05	0.05	0.05	0.04	0.05	0.05	0.13	0.07	0.07	0.07	0.07	0.07	0.07	0.16
m_body	1.18	1.22	0.44	0.43	0.44	0.45	0.28	1.58	1.64	0.65	0.64	0.64	0.57	0.16
v_frag1		0.02	0.11	0.08	0.08				0.02	0.14	0.11	0.11
Θ_frags23		0.069	0.05	0.04	0.04	0.04	0.06		0.06	0.06	0.06	0.06	0.06	0.10
f_kefrag1				0.02	0.02	0.05	0.29				0.02	0.02	0.06	0.10
d_transit							0.05							0.07
m_frag1		0.03	0.04	0.04	0.04	0.02	0.03		0.04	0.06	0.05	0.05	0.04	0.06
f_I										0.12	0.12	0.12	0.10	0.03
v_frags23		0.05	0.00	0.01	0.01				0.04	0.01	0.01	0.01
m_frags23		0.00	0.01	0.01	0.01	0.01	0.01		0.00	0.02	0.01	0.01	0.01	0.01
f_kefrags23				0.01	0.01						0.01	0.01
f_kebody					0.00	0.00	0.00					0.00	0.00	0.00

The major questions to which the answers about the major variables are to be applied are threefold, at least at the beginning. We can express them in stepwise order as: (1) Can the mechanical interaction of the bullet and the head (the forward snap followed by the explosion of the head) ever produce a rearward lurch strictly as a result of physics? (2) If so, how large can it be? (3) How close does the answer with the optimum combination of variables come to the observed lurch?

We begin our examination of the three questions by ranking the most important variables and seeing whether external constraints on their values produce or guarantee a rearward lurch. Lurch 7 Angular (and very similarly Lurch 8 Angular) has revealed six variables that have the greatest effect on the answer, where "effect" means range of the answer coming from a reasonable range of the variable. Those variables are R_head, m_head, Q_cl, d_frag1, d_snap, and R_bullet. They and their effects are shown in the table below for Lurch 7 Angular.

The variables can be divided into three groups of two based on their effects: R_bullet and R_head (effects of 1.20 ft s^-1 and 1.439 ft s^-1), m_head and Q_cl (effects of 1.78 and 1.89 ft s^-1), and d_frag1 and d_snap (effects of 2.97 and 3.88 ft s^-1). Note that although none of these variables are constrained by "forbidden" values (other than d_frag1 and d_snap having to be positive), the three calculated variables v_lurch, m_cloud, and v_frags are constrained, as noted above. Thus we can eliminate any solution that violates these constraints, which are listed at the beginning of this section and again below. One effective way to do this is to prepare separate tables for each of these variables, in which their values are displayed vs. all combinations of that represent all combinations of d_frag1 and d_snap, the two variables with the biggest effects. Then the constraints can be applied to each set of results to establish the ranges of results for the variables, and ultimately for v_lurch. That makes three tables for each combination of the other variables (including the other four in the table above). Because the other four variables in the table exert significant effects on the results, they need to be allowed to vary, too. We begin by arbitrarily choosing m_head, giving it values of 6, 7, and 8 lb, and constructing the set of three tables for each value.
We need to distinguish external constraints from internal constraints. External constraints are those imposed by one variable on the validity of the solution for another variable (v_lurch in our case). In practice, this refers to forbidden values of m_cloud and v_cloud nullifying the corresponding solutions for v_lurch. These constraints make a powerful way of limiting the values of the lurch. Internal constraints are those imposed by a variable upon itself. The variable whose internal constraints we are most concerned about is, of course, v_lurch. Here those constraints are formed mainly by the calculated initial rearward lurch's not being allowed to exceed the magnitude of the observed initial lurch, which was a little less than 1 ft s^-1. Any solution that significantly exceeds this value (that is faster in the rearward direction than 1 ft s^-1), must be eliminated as wrong. But this is the second step—the first is to examine the effect of the external constraints.

Solution	Velocity of snap, ft s^-1
Snap linear	5.00
Snap angular	4.56

Variable	Sensitivity of v_lurch	*Range of v_lurch, ft s^-1*
R_bullet	-11.78	1.20
R_head	11.78	1.49
m_head	11.73	1.78
Q_cl	2.88	1.89
d_frag1	3.67	2.97
d_snap	11.68	3.88