Alvarez [] considered this problem based upon simple idealized energy and momentum conservation arguments, along with simple experiments involving tape-wrapped melons as proxies for live human heads; subsequent experiments have repeated Alvarez's results using more realistic proxies []. In this paper, the backward movement is reexamined theoretically in more detail. Like the current author, Prof. Alvarez did not take pleasure in having to delve into this subject matter, but was “convinced that the conclusions…are important” and thus strove to “make the text as free from emotional content as possible.” With that said, Prof. Alvarez realized that “the President's head did not fall back, but was driven back by some real force” [emphasis his]. Alvarez estimated that the recoil would be about twice the initial velocity brought about by the collision. He was cautious about the magnitude, but more emphatic about the direction, and modestly deferred the details of the KE transfer mechanism to “someone more knowledgeable in the theory of fluid mechanics.” However, he had already intuited the basic idea, namely that the “conical shape of the interaction zone is the key to the non-negligible efficiency of energy transfer.”

The temporary cavitation effect leads to an expanding “interaction zone” and can deliver devastating damage to the tissues of a human target, especially less-elastic organs such as the brain []. Tragically this is observed in the Zapruder Film. Here the large wound inflicted on the President's head was not a bullet exit wound, but rather the region of maximum temporary cavitation associated with KE transfer []. This KE deposit propagated radially outward in the form of an expanding pressure wave [] resulting in a rupture and explosion of the skull. Restoring force undulations are also (gruesomely) apparent in the Zapruder Film as brain tissue pulsing and dripping out of the wound in frames Z323–Z330 (not shown here), and Mr. Zapruder himself observed this while filming, as he would later testify to the WC “then I saw his head opened up and the blood and everything came out…” However, all this said, note well that because such explosions are not necessarily the bullet outshoots, the momentum directly carried forward by a given bullet during passage may not be the primary player in a recoil effect.

As pointed out by Alvarez [], the collision between a high-speed projectile and a human body target is inelastic, with much of the incoming KE going toward “heating” the target. However, the “heating” in question consists primarily in perturbing and deforming the target. In the case of a Newtonian fluid, the stresses imposed by such perturbations under high Reynolds number conditions (i.e.,) leads to the development of a separated flow field and rotational flow features in the form of random eddies (i.e., turbulence) that carry energy and momentum away from their origin. In the case of soft tissue (as opposed to a purely Newtonian fluid such as water), an analogous phenomenon is manifested in what is called temporary cavitation [], namely the temporary development of a near-vacuum in the wake of the bullet that is rapidly closed via the pressure gradient and elasticity in the tissue [], resulting in restoring forces that can lead to additional violent undulations before the material fully returns to equilibrium [].

Although the phenomenon of a high-energy projectile passing through a heterogenous body (e.g., a human head composed of hard and soft tissue) falls squarely within the realm of classical Newtonian mechanics, it is nevertheless something not readily observed in everyday experience, and thus not immediately intuitive to the average lay person. Indeed, it is often the case that scientific descriptions of nature are counter-intuitive and can run contrary to common sense [e.g.,, pp. xi–xii, 1–2]. However, the development of high-speed cameras has gone a long way toward facilitating observation and physical understanding of gunshot wound ballistics.

Model

Thus, in this paper a different method is sought. Rather than attempting to demonstrate or prove the general hypothetical question of whether or not high-speed projectile impacts on head cavities can lead to recoil effects (which Prof. Alvarez and others have demonstrated in the affirmative), one only needs to consider the physics of this particular special case. The objective here is simply to explain the observed behavior in the Zapruder Film, treating it as a case study.

3, p. 540 Warren Commission Report Report of the President's Commission on the Assassination of President Kennedy. 12 Lattimer J.K.

Lattimer J.K.

Lattimer G.

Haubner E.

Laidlaw A.

Forgett V. Differences in the wounding behavior of the two bullets that struck President Kennedy; an experimental study. 8 16, pp. 163–164 Sturdivan L.M. The JFK Myths: A Scientific Investigation of the Kennedy Assassination. One may witness this in high-speed camera footage, for example those presented in the 2008 Discovery Channel program “JFK: Inside the Target Car,” as well in other documentaries. High-speed camera frames from a Carcano bullet are also shown in [], where a complete near-isotropic rupture of the skull is evident. 52 Itek Corporation John Kennedy Assassination Film Analysis. 12 Lattimer J.K.

Lattimer J.K.

Lattimer G.

Haubner E.

Laidlaw A.

Forgett V. Differences in the wounding behavior of the two bullets that struck President Kennedy; an experimental study. 7 Thomas D.B. Hear No Evil: Politics, Science & the Forensic Evidence in the Kennedy Assassination. In Z313–Z316 ( Figure 5 ) an expulsion of mass (i.e., the “jet”) is observed resulting from an explosion caused in the wake of a high-speed projectile passage. Although the explosion emanates over a range of angles within a roughly conical cloud, the explosion of mass nevertheless is observed to escape from the single large wound on the right front of the President's head (described in the Autopsy Report [] and in Lattimer et al. []). Note that this is not a universal occurrence—depending on the firearm, bullet, target, entry and exit locations, etc., different “explosions” can result.But in this case a directional expulsion of mass is observed in the Zapruder Film. It is this escape of the explosion from one end of the cavity, but not the other, that creates a directional component to the mass expulsion, and thus a “jet.” In the author's study of the high resolution digital frames, it was noticed that there were particles that maintained their size and shape over adjacent frames, unlike the rest of the material in the cloud. It was subsequently realized that these were in fact solid skull fragments within a cloud of non-solid tissue, and the author has since learned that previous investigators had already ascertained this []. But here it is noted that because these solid particles hold together in flight, they can effectively act as tracers, whereby one may estimate the velocity of the ejected mass within the explosion (assuming they travel at the same velocity as the rest of the bulk material).

52 Itek Corporation John Kennedy Assassination Film Analysis. 12 Lattimer J.K.

Lattimer J.K.

Lattimer G.

Haubner E.

Laidlaw A.

Forgett V. Differences in the wounding behavior of the two bullets that struck President Kennedy; an experimental study. 7 Thomas D.B. Hear No Evil: Politics, Science & the Forensic Evidence in the Kennedy Assassination. Shown in Figure 5 are annotations pointing out the locations of 3 such tracer fragments that were sufficiently large enough for the author to identify. Two of these three particles appear in at least two frames, and they appear as double or even quadruple images, this probably resulting from a rapid rotation of the particles [] with a frequency of 1–2 complete rotations during the course of the shutter exposure. As the skull fragments rotated along their longitudinal axes, they aligned such that the flat sides were perpendicular to the camera FOV. Because they are solid, they hold together during flight, which along with the rotation, facilitates tracking of these pieces as projectiles on the images.

V x e (typically propellant exhaust via combustion, but any mass ejection would also qualify, including the jettisoning of stages in a multistage rocket) in a direction opposite the direction of travel. This is the result of momentum conservation for the system, involving vehicle and exhaust, for which it can be shown that [e.g., 53 Marion J.B.

Thornton S.T. Classical Dynamics of Particles and Systems. Δ V x ≡ V x ( t ) − V x ( 0 ) = − V x e ln ⁡ ( M 0 M 0 − δ m ) , (27)

where V x e is the velocity of the exhaust in the moving vehicle frame, defined here as positive along +x (left to right relative to the camera FOV), M 0 is the starting total mass of the system (vehicle plus propellant) and δm is the propellant mass. Thus it is seen that the thrust is dependent on both the mass and velocity at which the propellant is expelled. It is thus from these tracer particles that one has sufficient information in hand whereby the classical equation for rocket motion may be applied. To create thrust, the engine of a rocket (or generically, a jet propulsion vehicle) jettisons mass δm with a velocity(typically propellant exhaust via combustion, but any mass ejection would also qualify, including the jettisoning of stages in a multistage rocket) in a direction opposite the direction of travel. This is the result of momentum conservation for the system, involving vehicle and exhaust, for which it can be shown that [e.g.,, pp. 88–89]whereis the velocity of the exhaust in the moving vehicle frame, defined here as positive along +x (left to right relative to the camera FOV),is the starting total mass of the system (vehicle plus propellant) and δm is the propellant mass. Thus it is seen that the thrust is dependent on both the mass and velocity at which the propellant is expelled.

20 ± 10 % of the total brain mass. 9 25 Breo D.L. JFK's death—the plain truth from the MDs who did the autopsy. 54 Henery C.C.

Mayhew T.M. The cerebrum and cerebellum of the fixed human brain: efficient and unbiased estimates of volumes and cortical surface areas. 10 Alvarez L.W. A physicist examines the Kennedy assassination film. These values are an estimate based upon autopsy pathologist testimony that “two thirds of the right cerebrum had been blown away” [] (i.e., about 1/3 of the total cerebrum, which is 90% of the total brain [], thus 30%), and recognizing that a fraction of this total mass lost occurred well after the “peak explosion” and does not factor into the jet, thus <30%. Alvarez [] assumed a jet mass of 10% the total weight of the head in his paper, but noted that “the assassination buffs” considered this value to be too high (which may in fact be the case). 52, p. 63 Itek Corporation John Kennedy Assassination Film Analysis. p = δ m v s , where v s ≡ r ˙ s (the velocity of the exhaust spray), and thus the x-component (defined as positive down Elm Street, counter to the bullet trajectory and left-to-right in the Zapruder frames) is given by p x = δ m r ˙ s ( θ , δ ϕ ) cos ⁡ ( θ ) cos ⁡ ( δ ϕ ) , (28)

where θ is the mean relative elevation angle and δϕ is the mean relative azimuth angle, measured clockwise from x around the local zenith, z, of the spray CM. Assuming δ ϕ ≈ 0 ∘ , one may then estimate the maximum velocity of the spray's CM by examining the tracers identified in V x e = r ˙ s cos ⁡ ( θ ) . (29)

Regarding the current application, while the exact mass of the exhaust “jet” is not known, one may estimate it to be on the order of% of the total brain mass.However, unlike a typical rocket vehicle, the jet under consideration was not constrained along a tube to exit mostly along a single vectorial direction, but rather erupted in a roughly conical cloud over a finite range of directions. Nevertheless, because it originated from a opening on one side of the cavity, the jet (or, perhaps more accurately, “spray”) exhibits a mean direction of motion in the forward-upward-right direction relative to Elm Street []. The directional momentum for the spray's CM is, where(the velocity of the exhaust spray), and thus the x-component (defined as positive down Elm Street, counter to the bullet trajectory and left-to-right in the Zapruder frames) is given bywhere θ is the mean relative elevation angle and δϕ is the mean relative azimuth angle, measured clockwise from x around the local zenith, z, of the spray CM. Assuming, one may then estimate the maximum velocity of the spray's CM by examining the tracers identified in Figure 5 . The x-component of this would then constitute the jet exhaust speed in ( 27 ), which from Eq. ( 28 ) one then has

v p = S / δ t = S f z , where S is the slant path of the particle trajectory computed as S = δ x 2 + δ y 2 . From this crude approach, speeds for Particles 2 and 3 identified in v p ≈ 3000 and 1400 cm / s (≈67 and 32 mph, respectively). ≈ 3600 cm / s ) [ 52, p. 62 Itek Corporation John Kennedy Assassination Film Analysis. 7, p. 333 Thomas D.B. Hear No Evil: Politics, Science & the Forensic Evidence in the Kennedy Assassination. Using the “head snap” estimate of 2.3 inches, and assuming that the particle positions fall roughly within a plane orthogonal to the camera's FOV, the author was to able perform a rough conversion of image pixels to centimeters and thus grid the approximate locations of the particles. Shown in Figure 6 are Cartesian plots charting the locations and movement of the particles—the lefthand plot shows the particles for each frame and the righthand plot shows the frame-to-frame motion of individual particles. Particle 1 is not readily visible over two frames and the time interval between impact and Z313 is not perfectly known (as discussed in Section 2.1.3 ); thus, this particle is not used for estimating the explosion speed, but will be returned to below. However, Particles 2 and 3 span multiple frames, Z313–Z314 and Z314–Z315, respectively, and thus the shutter frequency provides the time interval. From the multi-frame points plotted in Figure 6 b, the velocity of the particles may be estimated from two successive frames as, where S is the slant path of the particle trajectory computed as. From this crude approach, speeds for Particles 2 and 3 identified in Figure 6 b are estimated as3000 and(≈67 and 32 mph, respectively). Figure 6 b shows estimated ballistic trajectories neglecting air resistance based upon these estimated speeds and trajectory angles, which although not perfectly matched (due to neglecting drag), nevertheless illustrate that the particles roughly follow ballistic trajectories. The estimate for Particle 2 is smaller than, but reasonably close to, the estimate obtained by the Itek Corporation for the HSCA, which was ≈80 mph () [], []. The slower speed estimates obtained empirically by the author may be attributed to the fact that they were obtained from Z313–Z314 (Particle 2) and Z314–Z315 (Particle 3), whereby air resistance had already damped their speed, and in the case of Particle 3 the explosive KE from the wound had already partially dissipated.

Figure 6 Graphs of the approximate locations of solid particles observed in the Zapruder Film: (a) particle locations by frame indicated by different colored circles, and (b) multi-frame locations of 3 individual particles (distinguished by color), with the approximate “origins” indicated in red circles, and dashed gray lines indicating approximate ballistic trajectories (neglecting air resistance) for the estimated angles and particle speeds.

δ K B ≡ K B ( t 1 ) − K B ( t 2 ) = Q B , (30)

where δ K B is the KE loss of the bullet during passage through the target (the target here being the brain), Q B is the energy imparted to the target, and t 1 , t 2 are entry and exit times of the bullet as expressed in Eq. ( 55 Joos G. Theoretical Physics. Q B = 1 2 ∑ i = 1 N m i r ˙ i 2 + 1 2 M B R ˙ B 2 , (31)

where M B is the total brain mass, and R ˙ B and r ˙ i are velocities of the brain CM and individual particles relative to the CM, respectively. The second term on the right is equivalent to the KE associated with the soft-tissue impulse contributing to the observed forward head-snap (which shall be denoted K I ) and has been calculated in Section r ˙ 2 ‾ , Eq. ( r ˙ 2 ‾ ≡ ∑ i = 1 N m i r ˙ i 2 ∑ i = 1 N m i ⟹ 1 2 ∑ i = 1 N m i r ˙ i 2 = 1 2 M B r ˙ 2 ‾ ⟹ Q B = 1 2 M B r ˙ 2 ‾ + K I ,

which when substituted back into Eq. ( r ˙ = 2 M B ( δ K B − K I ) . (32)

One may thus use Eq. ( r ˙ ≈ r ˙ s ≡ v s given values of δ K B and K I (as calculated in Section M B . Although the autopsy reported the brain mass to be 1500 g, this number will be too small due to the loss of tissue, blood and fluid from the gunshot wound. Given the assumed brain loss for the jet spray of 20 ± 10 % (as discussed in Footnote 9 25 Breo D.L. JFK's death—the plain truth from the MDs who did the autopsy. 54 Henery C.C.

Mayhew T.M. The cerebrum and cerebellum of the fixed human brain: efficient and unbiased estimates of volumes and cortical surface areas. 10 Alvarez L.W. A physicist examines the Kennedy assassination film. These values are an estimate based upon autopsy pathologist testimony that “two thirds of the right cerebrum had been blown away” [] (i.e., about 1/3 of the total cerebrum, which is 90% of the total brain [], thus 30%), and recognizing that a fraction of this total mass lost occurred well after the “peak explosion” and does not factor into the jet, thus <30%. Alvarez [] assumed a jet mass of 10% the total weight of the head in his paper, but noted that “the assassination buffs” considered this value to be too high (which may in fact be the case). M B ≈ 2100 g. From these parameters, the theoretical spray speed v s is subsequently calculated to be in the range of ≈ 3200 – 3500 cm / s (depending on the assumed exit wound diameter), which are in general agreement with the observed values from the Zapruder Film, thus lending confidence to these estimates. It may also be noted that although these particle exit speeds appear “fast” by everyday experience, they are significantly slower than the projectile depositing the energy; thus the explosion lags the projectile impulse [ 50, p. 173 HSCA Investigation of the Assassination of President John F. Kennedy, Appendix to Hearings, vol. VII, Medical and Firearms Evidence. However, these empirical velocity estimates for the spray (or jet) may be theoretically arrived at by revisiting the conservation of energy for the systemwhereis the KE loss of the bullet during passage through the target (the target here being the brain),is the energy imparted to the target, andare entry and exit times of the bullet as expressed in Eq. ( 2 ). The resulting KE of the system of particles comprising the target can be separated into two parts: (1) the KE of individual particles relative to the CM frame, and (2) the KE of the CM relative to an inertial frame. This is expressed as [e.g.,, pp. 109–110]whereis the total brain mass, andandare velocities of the brain CM and individual particles relative to the CM, respectively. The second term on the right is equivalent to the KE associated with the soft-tissue impulse contributing to the observed forward head-snap (which shall be denoted) and has been calculated in Section 2.1 . The first term, on the other hand, is associated with “heating” (viz., inelastic disruption and deformation) of the target, a part of which goes toward the momentum of the spray that erupts from the cavity. If one defines a mean-square speed of the particles (relative to the CM),, Eq. ( 31 ) may be simplified as followswhich when substituted back into Eq. ( 30 ) leavesOne may thus use Eq. ( 32 ) to estimategiven values ofand(as calculated in Section 2.1.3 ), and an estimate of the brain mass,. Although the autopsy reported the brain mass to be 1500 g, this number will be too small due to the loss of tissue, blood and fluid from the gunshot wound. Given the assumed brain loss for the jet spray of% (as discussed in Footnote), the living brain mass before wounding is taken to be 40% more than the autopsy-measured value, thusg. From these parameters, the theoretical spray speedis subsequently calculated to be in the range of(depending on the assumed exit wound diameter), which are in general agreement with the observed values from the Zapruder Film, thus lending confidence to these estimates. It may also be noted that although these particle exit speeds appear “fast” by everyday experience, they are significantly slower than the projectile depositing the energy; thus the explosion lags the projectile impulse [].

δ m ≈ ( 0.2 ± 0.1 ) M B and θ ≈ 40 – 80 ∘ , the exact values are not known (the latter estimate is from visual inspection of Z313). As in Section Δ X r = Δ V x ⋅ Δ t ZF ; for convenience in interpretation of the results, they are given in the head-snap frame, thus the initial motion of the “vehicle” (i.e., the head) is taken to be zero, V x ( 0 ) = 0 cm / s . In reality, the head was initially moving at speed Δ V x ( t 3 ) from a pre-existing momentum imparted by the impulse from the bullet collision as per Eq. ( V x e , calculations are simply repeated for parameter ranges encompassing realistic values [ 48, pp. 78–79 Taylor J.R. An Introduction to Error Analysis: The Study of Uncertainties in Physical Measurements. Returning to the application of Eqs. ( 27 ) and ( 29 ), although one has reasonable ranges for the mass and mean angle of the jet spray, that isand, the exact values are not known (the latter estimate is from visual inspection of Z313). As in Section 2.1.3 , the recoil distances are obtained from the change in velocities calculated from Eq. ( 27 ) as; for convenience in interpretation of the results, they are given in the head-snap frame, thus the initial motion of the “vehicle” (i.e., the head) is taken to be zero,. In reality, the head was initially moving at speedfrom a pre-existing momentum imparted by the impulse from the bullet collision as per Eq. ( 2 ), but it is easier to observe the recoil effect without this initial velocity superimposed. To allow for the uncertainties in the parameters δm, θ and, calculations are simply repeated for parameter ranges encompassing realistic values [], with the subsequent results (in inches per Zapruder frame) conveniently summarized as contour plots in Figure 7