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 allowed combinations of the most important variables in order to search for the most characteristic range of solutions.
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 5.1 ft s^-1 (linear) to 4.7 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 six of each solve for the velocity of the lurch and the velocity of the cloud, while the last one substitute potential energy for the cloud. Examining the changes in the solutions as variables are added offers an initial way to assess the importance of the variables.
    Consider first the linear calculations. The speed of the lurch in scenario 1 (body, bullet, and cloud) is 2.17 ft s^-1, substantially faster than the 0.8 ft s^-1 of the Zapruder film. Adding the large fragments (scenario 2) increases this speed only slightly, to 2.18 ft s^-1. This indicates that the large fragments do not materially affect the answer, although they 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 its speed to 1.06 ft s^-1 (i.e., by a factor of two). 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 cloud 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 (1.06 ft s^-1 rearward becomes 1.02 ft s^-1 rearward). The same thing holds for adding 3-D motion to the body (scenario 5), where the lurch remains unchanged. Adding the four time intervals (scenario 6) hardly changes the solution, either; the lurch increases in speed to 1.04 ft s^-1.
    The final big change comes in scenario 7, which solves for PE rather than v_cloud. This step reduces the speed of the lurch from 1.04 ft s^-1 to 0.62 ft s^-1, and in so doing appears to bring it below the observed speed of 0.8 ft s^-1 to the rear. The disagreement may be more apparent than real, however, for the solution does not represent a good combination of default values for the variables.
    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 35% greater than for the linear cases, they follow the same pattern as the scenarios change, and the important variables remain the same.

Scenario	Description	Linear		Angular
Scenario	Description	v_bodyafter, ft s^-1	v_cloud, ft s^-1	v_bodyafter, ft s^-1	v_cloud, ft s^-1
Lurch 1	Calculates from body, bullet, and cloud	-2.17	425	-2.92	425
Lurch 2	Adds large fragments	-2.18	388	-2.94	387
Lurch 3	Adds 3-D motion to cloud	-1.06	388	-1.44	388
Lurch 4	Adds 3-D motion to large fragments	-1.02	378	-1.38	378
Lurch 5	Adds 3-D motion to body	-1.02	378	-1.38	378
Snap/Lurch 6	Adds four time intervals	-1.04	401	-1.39	386
Snap/Lurch 7	Solves for PE instead of v_cloud	-0.62	633 (PE)	-0.91	576 (PE)

The most important variables
A better way to rank 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 tables below. The first shows the range of effect for each variable in each calculation. The second lists the variables in order of decreasing effect.

	Range of effect, ft s^-1, on lurch
Variable	L1L	L2L	L3L	L4L	L5L	SL6L	SL7L	L1A	L2A	L3A	L4A	L5A	SL6A	SL7A
v_cloud							2.52							3.40
d_snap						0.34	0.38						0.86	3.22
d_frags						0.02	0.28						0.21	0.47
m_cloud	3.20	2.91	1.73	1.68	1.68	1.79	2.95	4.28	3.90	2.33	2.27	2.23	2.31	3.99
m_head						0.47	2.76						0.28	1.40
R_head													0.54	0.84
R_bullet													0.45	0.71
Θ_cl			2.17	2.11	2.11	2.24	1.68			2.92	2.85	2.84	2.90	2.27
v_bulletafter	0.11	0.10	0.15	0.15	0.15	0.12	0.35	0.15	0.15	0.20	0.20	0.20	0.16	0.66
PE	1.08	1.20	0.71	0.73	0.73	0.68		1.46	1.61	0.96	0.98	0.98	0.96
t_delay						0.03	0.11						0.22	0.26
v_bullet	0.18	0.20	0.10	0.10	0.10	0.09	0.08	0.24	0.27	0.13	0.14	0.14	0.16	0.14
m_bullet	0.08	0.08	0.03	0.03	0.03	0.03	0.08	0.10	0.12	0.04	0.04	0.04	0.07	0.14
Θ_frag1		0.14	0.14	0.14	0.14	0.10	0.10		0.19	0.19	0.19	0.19	0.17	0.17
Θ	0.04	0.04	0.04	0.04	0.04	0.03	0.05	0.05	0.05	0.05	0.05	0.05	0.03	0.09
m_body	1.08	1.08	0.53	0.51	0.51	0.52	0.31	1.46	1.47	0.72	0.69	0.69	0.70	0.45
v_frag1		0.18	0.00	0.08	0.08				0.24	0.01	0.10	0.10
Θ_frags23		0.08	0.08	0.08	0.08	0.06	0.06		0.10	0.10	0.10	0.10	0.09	0.09
f_{kefrag1(frags)}				0.06	0.06	0.04	0.00				0.08	0.08	0.10	0.00
d_transit						0.00	0.01						0.01
m_frag1		0.02	0.04	0.03	0.03	0.04	0.07		0.02	0.06	0.04	0.04	0.05	0.11
f_I								0.26	0.26	0.13	0.12	0.12	0.12	0.08
v_frags23		0.11	0.05	0.08	0.08				0.15	0.07	0.10	0.10
m_frags23		0.04	0.00	0.02	0.02	0.00	0.03		0.05	0.01	0.02	0.02	0.01	0.05
f_kefrags23				0.02	0.02						0.03	0.03
f_kebody					0.00	0.00	0.00					0.00	0.00	0.00

The table above shows distinct patterns. Within the linear and angular sections, the effects generally start high and decrease as variables are added to the simulations (the new variables taking up some of the slack, so to speak). The numbers settle down at Scenario 3 (where the first 3-D effect is added) and remain stable through 6. They change again (in both directions) at 7, for reasons that are complex. In keeping with the higher lurches of the angular scenarios, the effects tend to be greater there, too.
The most representative group of effects seemed to be that zone between L3A and SL6A, because of its stability and because the angular calculations should represent the motions better than the linear calculations. Thus the average effects are based on L3A–SL6A.

Variable	Average effect (ft s^-1) L3A–SL6A
Θ_cl	2.88
m_cloud	2.28
PE	0.97
d_snap	0.86
m_body	0.70
R_head	0.54
R_bullet	0.45
m_head	0.28
t_delay	0.22
d_frags	0.21
v_bulletafter	0.19
Θ_frag1	0.18
v_bullet	0.14
f_I	0.12
Θ_frags23	0.10
v_frags23	0.09
f_{kefrag1(frags)}	0.09
v_frag1	0.07
m_frag1	0.05
m_bullet	0.05
Θ	0.04
f_kefrags23	0.03
m_frags23	0.02
d_transit	0.01
f_kebody	0.00

As seen earlier for the individual cases, the two most important variables are the half-angle and mass of the cloud (Θ_cl and m_cloud), with effects >2 ft s^-1. After this comes a big gap, with no variables between 1 and 2 ft s^-1. Then come five variables between about 0.5 and 1 ft s^-1 (PE, d_snap, m_body, R_head, and R_bullet), which are related to the head and the body). After this comes another gap. followed by a broad continuum of 18 variables with effects between 0.28 and 0.00 ft s^-1. They include properties of the fragments and, surprisingly, of the bullet.

Reviewing the first two questions
Recall that this monograph began by posing four basic questions. We are now prepared to answer the first two of them. They are:

    The answer to question 1 is clearly yes—the observed snap of >3.3 ft s^-1 is completely compatible with our linear and angular final speeds of 5.1 and 4.7 ft s^-1, respectively.
    The answer to question 2 is also a clear yes. Our 14 answers of 0.6–2.9 ft s^-1 rearward, for the 14 sets of scenarios and default conditions, are completely compatible with the observed lurch of 0.5–1.0 ft s^-1 rearward.
    But we must look further at the lurch, for these 14 answers represent single combinations of default conditions for 30+ variables. Could there be combinations of these variables that give significantly different answers, perhaps even with a majority being having the wrong direction or the wrong magnitude? In other words, could the default answers be only giving part of the story, and maybe a misleading part at that? We can address this possibility by taking those seven most-important variables (Θ_cl, m_cloud, PE, d_snap, m_body, R_head, R_bullet) and examining all sorts of combinations of values for them. It is feasible to do this for seven variables, but not for 30+. That is the task of the next section, where we examine constraints on the various combinations and see that the allowable answers represent a minority of the possible answers, and that the acceptable answers are constrained to a narrower range of lurches centered generally on the observations. That next section address the third question:

It answers the question in the affirmative, provided only that a forward-moving cloud of fragments is present.

Solution	Velocity of snap, ft s^-1
Snap linear	5.13
Snap angular	4.68