Math modeling of JFK’s motions in the Zapruder film is used to provide additional perspective on “back and to the left”
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[SUMMARY]
Background:
JFK’s unusual motion seen in the Zapruder film has been an issue debated for quite some time. This is especially true of his reactions seen over the film range from about z226 to z324 which includes his observed freezing up like a statue, followed by the famous “back and to the left” segment before collapsing over.
To explore the idea that some of these unusual motions from ~z226 to z324 could be interrelated in some way, further inspection of the Z-film was conducted and a simple anatomical math model was incorporated as a way to possibly help better explain these movement dynamics, especially the observed “back and to the left”.
The math modeling conducted to further understand JFK’s motions in the Z-film used a simple anatomical analog consisting of two (mass on a cantilever) systems that are coupled. A (head/neck) mass/cantilever system connected to a (upper-torso/lower-back) mass/cantilever system comprises the dual coupled cantilever used to model body motion and provide additional perspective on JFK's “back and to the left” motion. The output of this approach suggested that the overall observed motion on film could result solely from a rear shot. No shot strike from the front was indicated when looking at the overall integrated motion of the coupled (head/neck) and (upper-torso/lower-back) systems in this timeframe.
To help set up the math model, some additional JFK position measurements from the Z-film were used, and three unusual but key film questions required additional film examination to determine the likely dynamics involved: (1) Why it appeared that JFK acted like a statue, almost frozen in place, starting around z226 through z312?, (2) Could flaccid paralysis have set in the neck after the start of rearward motion of the head and torso after the strike, manifested by the head and neck apparently starting to go limp after ~z316?, and 3) Why it appeared there may not have been any significant back of upper torso collision/rebounding off the rear seat behind JFK, to create a force to smoothly stop and reverse the rearward torso motion in order to go forward again, at the upper torso's point of maximum rearward position around ~z320?
Key points summary:
The additional film inspection and film measurements, along with applying information from neurological resources, resulted in a final model adjusted for three time segments covering z312 to z324 which appeared to model/reproduce the actual observed horizontal data position measurements on the Z-film very well. The final model design that incorporated aspects from the additional Z-film review and neurology references relied on the following key points:
Stiffness of the cantilever segments of the model, i.e., the effective neck stiffness and effective lower-back stiffness (via the back brace), were important parameters with regard to the associated modeled motions.
The lower-back brace may have played a significant role in JFK’s motion after z313, and further appears to be the contributor to the upper torso motion going backwards and then forwards again (as opposed to JFK crashing into and bouncing off the back seat), before his lower body loses its anchoring support and tips further over to the left falling on Jackie. The head could also interact with the upper torso via its coupling through the neck.
A muscle stiffening reflex reaction (see this link spastic paralysis for a possible mechanism) in the neck and upper torso happened in the area affected by the shot through the lower neck (circa z222). Later, after the head shot, a removal of the neck stiffness looks like it may have occurred causing it to go limp (see flaccid paralysis vs spastic paralysis).
Conclusions:
- Overall, the model's solution for back of head and back of upper torso horizontal positions provided good agreement with the measured horizontal position data points seen on the Z-film.
- It appeared JFK’s motion observed over the time z312-z324 (forward->backward->forward) could be accounted for by a significant rear strike impulse with rearward motion influenced by a jet effect and the physical properties of JFK (which included his back brace and changing neck stiffness) over that timeframe.
- The significant jet effect appeared to be involved in the initial rearward head motion around z313.
Possibly some rearward neuromuscular reactions were also involved early on, but if so, it is believed they were likely shut down around z316 where it appears total body flaccid paralysis begins, setting in shortly after the major brain blowout that happened between z312/z313.
- This model indicated a scenario where Potential Energy, garnished and stored from the Kinetic Energy of the z312/z313 rear bullet strike, comes into play and interacts with the stiffened “system” components. In principle a perfect oscillating system (before dampening sets in) in theory wouldn’t need any additional forces (like a jet effect, or a neuromuscular reaction or its removal, or someone tugging backwards or a second bullet impulse coming from the front, etc., etc.) as long as there was some mechanism to store potential energy derived from the initial kinetic energy impulse, and which is reversible to facilitate oscillation. This dynamic is analogous to what is seen in ballistic pendulums.
- This modeling and Zapruder film analysis suggested a scenario where JFK’s extended motions on the Z-film (from z313 thru z324), after a rear head strike that occurred at inter-frame z312/z313, were oscillatory in nature and could be associated with a macro interaction involving two semi-independent principal mass centers, his head and upper torso, at a point in time when both of the structures supporting those components, the (neck) and (lower-back) respectively, happened to be under the influence of extraneous stiffening effects (with the neck stiffness changing shortly after the z312/z313 head strike).
- There was no indication of a frontal shot striking the President in this timeframe based on this analysis.
[ANALYSIS DETAILS]
The analysis details can be found in a video on the left below which goes over the complete study within a PowerPoint presentation. The raw PPt files are found to the right.
The video was accompanied by a rough voice over, primarily to add additional background material for those not as familiar with some of the aspects of the case. The video became quite lengthy in the process of documenting with the additional voice comments on all the procedures leading to the final model design, so an index list found further below may be useful to identify, and help advance to, a specific time location in the video for a particular topic of interest.
PowerPoint File Recorded on Video
Back and to the Left.mp4
The PowerPoint slides used in the video are added below here as one large .ppt file
Note: Google may try to display the .ppt file above when the link is clicked on, but because of size it will likely be necessary to download the file, and run separately as a Powerpoint presentation, to get it to fully display properly.
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The video presentation on the left however includes everything and should play straight away with one or two clicks, but may take some time to load.
The complete analysis using a dual coupled mass-cantilever system: PowerPoint slides & related video time references
PPt Slide # Video Time Topic discussed
1 0:00:00 Introduction
2 0:01:17 Study Approach & Design
3 0:02:50 Overall Findings
4 0:04:50 Model design basis
5 0:08:19 Background on what initiated this study/review
6 0:11:01 Slide 5 continued
7 0:13:39 Review on related background material (Jet effect/neuromuscular reaction)
8 0:21:48 4 Steps involved in the analysis
9 0:23:53 Review & comparison to previous Z-film movement studies
10 0:27:10 Z-Film measurement technique for this study
11 0:28:58 Directional head motions that were not quantified in this study
12 0:31:53 Josiah Thompson’s film measurement data on the head
13 0:34:20 ITEK’s Head and Shoulder change in position measurements
14 0:37:42 Comparison of ITEK & Thompson’s head measurement data (using the same format)
15 0:39:02 Comparison of this study’s head data vs. ITEK & Thompson
16 0:40:11 Comparison of this study’s upper torso data to ITEK shoulder data when using position
17 0:42:16 Comparison of this study’s head displacement data to Thompson’s
18 0:43:08 Summary of this study’s head and upper torso data (relative to a 0 at z312)
19 0:45:11 Review of this study’s head data using either the limo, or back of upper torso, as a frame of reference
20 0:54:12 Review on some dynamics of the bullet energy and a rear head strike
21 0:56:48 Identify any discontinuous motion points appearing on film to account for in modeling
22 0:58:08 Graph showing points of discontinuity that indicate a model adjustment there is necessary
23 1:01:01 Summary of points where making a model adjustment would be indicated
24 1:03:14 Visualizing on a graph where three model segments for head motion are indicated
25 1:05:36 Physical and anatomical basis for the model design setup
26 1:09:02 Modeling Approach using a Force Balance
27 1:10:35 Introducing a Cantilever analogy
28 1:12:06 Example of simple cantilever in operation, and when falling over
29 1:14:45 Force Balance and solution equation for a simple cantilever
30 1:16:22 Introducing a Dual Coupled Cantilever system for the system model
31 1:18:27 How Z-film measurements relate to the model’s output for the principal mass positions vs time
32 1:19:55 Summary of the model approach & limitations and the format of its analytical solutions
33 1:23:19 A coupled cantilever mechanical example
34 1:24:13 Video of the Hyperflex dual cantilever mechanical example
35 1:26:26 Hyperflex video measurements and how well the model can fit that
36 1:29:18 Insurance Institute crash test dummy testing and similarities to Z-film dynamics
37 1:32:38 Video example of a crash test dummy in Industry testing
38 1:33:53 Head motion similarities in mechanical crash test dummy vs Z-film head motions
39 1:35:08 Looking for the oscillation effects seen in the Z-film in other areas like sports impacts
40 1:35:50 Neck stiffening in soccer and boxing and its capability on transmitting forces
41 1:36:57 Oscillation nature of the head from impulses like those seen from strikes in boxing
42 1:39:40 Another boxing KO video, and body motion similarities to reactions seen in the Zapruder film
43 1:43:24 Summary of mechanical and human head reactions after receiving an impulse strike
44 1:45:31 Conclusion on the applicability of using the oscillatory cantilever model and its segments
45 1:46:23 Back of head and back of upper torso raw film data template used to plot the model results
46 1:47:05 Modeled first segment from z312 to z313, and the need for the jet effect here to make the model segment work
47 1:52:14 Modeled second segment from z313 to z315.6; head initial conditions and mass adjusted
48 1:53:11 Modeled third segment from z315.6 to z324; if no neck stiffness adjustments are made
49 1:55:20 Modeled third segment from z315.6 to z324; with neck losing ~90% of its stiffness (flaccid paralysis occurred)
50 1:57:48 Key points indicated from this review
51 2:00:50 Overall Conclusions
52 2:02:58 Some final observations & thoughts
53 2:05:39 Appendix 1 – References on stiffness of the lower-back
54 2:07:00 Appendix 1a – References on stiffness diagnosis of the neck/upper torso area
55 2:07:44 Appendix 2 – Film measurements for the study
56 2:08:05 Appendix 3 – Estimates for input variables and calculated constants for model segments
57 2:08:53 Appendix 4 – Model variables and estimates for modeling of the Hyperflex equipment
58 2:09:48 Appendix 4 (continued) – details on model variable estimates
59 2:12:37 Appendix 4 (continued) – details on model segments initial conditions estimates
60 2:13:17 Hyperlinks to additional references used in various slides
B.J. Roselle 9/2021
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