The effect produced by such high-energy projectiles [14] , [15] upon collision with a human head is catastrophic as has been characterized through ballistics experiments [12] , [16]. In examining this, three separate dynamical phenomena will be considered in Section 2 that explain behaviors observed in the Zapruder Film, namely 1 the initial effect of the forces directly imparted to the target head by the projectile discussed in Section 2. Kennedy assassination crime scene in Dealey Plaza, Dallas, Texas.
The map is oriented with true north pointing toward the top of the page; non-permanent geographic features e. In the Zapruder Film President Kennedy is seen to react to three separate gunshots, the first missing him and the limo [6] , [16] , [4] , [17] , the next two hitting him with increasing accuracy. Further discussion of the first two shots is beyond the scope of this paper other than to note that all three gunshots had associated 1—2 frame anomalous movements, and a outward impulse is observed on the jacket lapel of Texas Governor John Connally who had accompanied President and Mrs.
Kennedy in the motorcade and was collaterally wounded in chest at the same time the President begins showing signs of being injured [12] , [6] , [7] , but no other discernable impulses are otherwise seen on either of the two men prior to the third shot. Figure 2 shows the two Zapruder Film frames that captured the fatal shot, namely Z the moment just before impact and Z the moment just following impact. In Z the catastrophic effect of the energy deposit from a supersonic projectile passing through a human head is clearly evident.
However, while it is not immediately noticeable at this scale and not detectable while viewed in motion at normal speed , a careful comparison between the two frames also reveals that President Kennedy's head snaps forward from Z to Z [9, pp. There is nothing new in this observation—early researchers with access to the still frames first noticed this in the mid-to-late s.
Notable among these is author Josiah Thompson, who estimated the position of the President's head relative to two fixed objects on the rear of the limousine, the results of which are plotted in his book Six Seconds in Dallas [9, p.
It is also crucial to note that this anomalous forward impulse at Z is only observed on Kennedy's head—it is not observed on any of the other limo occupants with reference to dotted lines c , d and f , nor is it even observed on Kennedy's own torso line b , wherein lies his body's center-of-mass CM.
This implies that an isolated real force acted directly and solely upon the President's head just prior to Z; the only plausible source for this instantaneous, isolated forcing mechanism is manifestly and unequivocally the projectile impact. Therefore, what follows in Sections 2. Un-enhanced high-resolution digital copies of Zapruder Film Frames and Z and Z showing the high-powered rifle gunshot that fatally wounded President Kennedy.
Frame has been horizontally adjusted to correct for tracking error between the two frames. The annotated red line segments demarcate features relevant to the kinematical discussion in the text: dashed line O is flush with the back of Kennedy's head in Z; lines a and e are stable limo features viz. Kennedy's hat and hair; line d is Governor and Mrs. Connally; and line f is the Secret Service agents. The interaction of a firearm projectile passing through a human body target constitutes an inelastic collision , that is, one where the kinetic energy KE of the two-body bullet-target system is not conserved e.
This concept will be returned to in Section 2. Because such collisions are neither elastic nor perfectly inelastic, the physical description of the interaction can be more complicated. Nevertheless, assuming in the case under consideration that the projectile collision occurred near the center of the target thus not imparting torque , it is possible to consider the interaction in terms of a linear impulse, J x , which in one dimension for the CM frame is given as [e.
Equation 1 applies to the bullet passage through an entire head, but this may be broken down into three separate impulses brought on by 1 the skull entrance, 2 passage through the skull cavity i. For simplicity, any resistance presented by the hair and scalp is assumed to be of second order and thus negligible [e. Equation 1 is therefore rewritten as. From Eq. Attention is first given to the terms involving passage through bone, namely the first and last terms on the right side of Eq.
Sturdivan and Bexon [20] derived a model describing the probability of penetration of human skulls by high-speed projectiles viz. As explained by Sturdivan [20] , [16] , when a high-speed projectile impacts the head and penetrates the skull, it first perforates a hole in the outer table of approximately the same diameter of its own presented area.
However, as the bullet breaks through into the inner, less-dense trabecular layer, the area expands roughly in a cone shape until it finally breaks out a wider hole through the inner table. It is this cratering process that assists forensic pathologists in establishing skull entrance or exit wounds [25] , [22] , thus facilitating the determination of the direction of the bullet's passage relative to the body and the direction of origin of the shot.
Building upon the skull-cratering model described above [20] , the impulse force is modeled in the current paper by assuming a perfectly inelastic collision [26, pp. The interaction of the two bodies during collision according to Newton's Second and Third Laws is given by. Solution of ordinary differential equations ODEs 3 and 4 is easily obtained by first recasting them as. Combining Eqs. Finally, substituting Eq.
Equations 11 and 8 form the basis for calculating the first and third terms of Eq. In this case, the collision would then be a perfectly inelastic collision, and the total incoming momentum of the projectile p 1 would be completely transferred to the skull and head.
Using dimensional analysis, Sturdivan [33] derived a drag law for the passage of a spherical projectile through tissue as a combination of an inertial term and a viscous friction term. The tissue-strength term in Eq. Similarly, for Eq. However, it is known in practice that a high-speed projectile will undergo deformation from collisions with bone [36] , [37] , [16] , [14] , [38] , as well as during subsequent passage through soft tissue [16] , [38].
This was certainly the case for the Kennedy assassination, as evidenced by the badly deformed bullet fragments recovered from the limousine and autopsy, and will be addressed further in Section 2.
For the bullet velocity at impact, v 1 , the retardation of the projectile through the intervening atmosphere must be taken into account. However, to corroborate the measured value, and to facilitate application of the model to an arbitrary source sniper location and projectile, the theoretical modeling study in this paper is extended here to include calculations accounting for the projectile passage through the ambient atmosphere of the crime scene.
The bullet impact velocity at the target location may be estimated theoretically by accounting for aerodynamic drag. The one-dimensional aerodynamic drag force is given by [e. The equation of motion for the projectile may thus be written as. Values for the crime scene parameters may be found in the literature [16, p. Although the influence of departures of the ambient atmospheric temperature, humidity and wind from a mean state are small [41] , the projectile's KE is proportional to the square of its speed.
T v is calculated as [42, p. Hourly meteorological surface observations at Dallas Love Field from the U. From these values and Eq. Windspeed and direction were included in Table 1 because they factor into the aerodynamic drag calculations 19 and 21 by altering the bullet velocity relative to the fluid.
It can be seen that winds were out of the west, which was obliquely against the direction of the bullet trajectory, 5 thus increasing the relative speed and drag. Substituting into Eq. The known parameters from the assassination crime scene may now be applied to the equations derived above. Throughout this paper centimeter-gram-second CGS units are used unless otherwise noted. Table 2 summarizes the values used for the parameters along with their sources for performing impulse force calculations.
For the skull passage Eqs. Values for these parameters may be found in Table 2. The presented areas of the projectile may be estimated from its diameters at skull entrance and exit. Because the Carcano bullet is extremely stable during normal flight through air, the entrance diameter d 0 is simply the cross-sectional diameter of an unfired bullet, which is known to be 0.
Although the bullet was broken into at least three fragments [38] , it is assumed that the autopsy measurements nevertheless provide an objective best-estimate of the effective presented area of the projectile on exit [Sturdivan, L. To allow for uncertainties [48, pp. Finally, the stress-strain curve reported by Keaveny et al. Soft tissue force calculations may be performed based upon Eqs.
The total path of the projectile through the brain tissue is taken to be 11 cm, which corresponds to the U. As mentioned previously, the equations for velocity retardation assume a constant presented area A , and the force equations are dependent upon both A and v.
Given that most of the bullet deformation occurs initially before rapidly diminishing with decreasing velocity [16, p. Then given A L and v L , the drag force through the layers F d A L , v L may be calculated using either 15 or 16 ; Figure 3 shows the results for the range of assumed projectile effective exit diameters d e. Here it is seen that both models yield very comparable results for the case under consideration, both in terms of magnitude and in variation, thereby providing confidence in their application in this work.
Finite-difference layer drag force computations for deforming half-parabolic growth spherical projectile passage through visco-elastic soft tissue i. The different colored lines correspond to different effective exit wound diameters, d e , spanning the range described in the Autopsy Report.
The integrated drag-force impulse is calculated via the finite-difference approximation as. Equation 24 forms the basis for calculating the second term of Eq. Note again that the tissue drag force equations were derived assuming spheres [33] , [34] , so a spherical projectile is also implicitly assumed here.
This approximation conveniently eliminates the need to specify the 2-axis orientation of the bullet during passage e. It is recognized that the projectile is assumed to deform and fragment after the initial collision with the skull which breaches the copper jacket [12] , [38] to the extent that tumbling may be ignored [14] , and a primary effect of tumbling is simply to increase the presented area of the projectile [49] which is already accounted for in the finite-difference model Eq.
Although the visco-elastic soft tissue presents less resistance than rigid bone, the increased presented area caused by the initial bullet-bone collision and subsequent deformation over a much greater path length yields this greater integrated impulse. Figure 4 b shows the reduction of the bullet speed during passage, which translates to the power-of-two loss of KE to the surrounding environment Figure 4 c. Here it is seen that the KE transfer from the projectile to the soft tissue brain is maximized before the midpoint, which arises primarily from the increased presented area [16, p.
This large deposit of energy is propagated away from the projectile path via a separated flow field and pressure wave known as temporary cavitation [36] , [37] , [34] , [16] , [49] , [14] , [15] discussed more in Section 2.
Modeled effects of collision of a deforming, high-speed spherical projectile i. However, to the author's knowledge, there simply does not exist a precise measurement of the target mass, namely that of President Kennedy's head although it may be noted that Kennedy's hat size is reported to have been 7 3 8 , 6 which is an average size for an adult male.
Thus, in this work a best estimate for Kennedy's head mass is obtained from a tabulation of published anthropometric datasets compiled by Yoganandan et al. Zero relative motion was assumed before impact, so to obtain an estimate of the total movement of the head from the previous observed position viz. From the projectile velocity calculations Figure 4 b , the total time for the bullet passage through the head i. The uncertainty estimates correspond to the uncertainty in the estimated target mass.
The last row shows values derived from momentum conservation for a total inelastic collision of the projectile with the target i. Note that nearly identical results were obtained using the Sturdivan model [33] for the soft-tissue calculations, albeit slightly larger and thus in slightly better agreement, but are not shown in the interest of brevity.
Included for reference in Table 3 are calculations assuming a perfectly inelastic collision of the projectile with the target i. Note it may also be deduced that the bullet was well airborne at Z and the moment of impact probably occurred just after the shutter closed. In fact, the bullet may very well have been just outside or within the camera's field-of-view FOV at Z The explosion observed in the Zapruder Film and its dynamical effects are explored in more detail below in Section 2.
The calculations may have slightly underestimated the observed head snap given that they were based upon a linear impulse imparted to the back-center of the head [16, p. Additionally, given the downward trajectory of the projectile and initial inclination of the target , the earth's gravity which was implicitly neglected would have imposed a small additional acceleration to the impulse. Another consideration is that the growth of the projectile presented area, A , may have been larger than that estimated using the effective exit diameter—this is quite possible given that the projectile had fragmented fragments were found in skull cavity during the autopsy and it would have created secondary missiles [14] from the initial skull collision.
Other factors include the assumptions employed in the soft tissue drag formulas, whereby parameters such as the drag coefficient C d were assumed constant, but more likely varied as the projectile's speed slowed e. Finally, while uncertainties in parameters were accounted for as much as possible, this could not be done for a handful of them, especially bio-mechanical parameters e.
But all said, the computed magnitudes are found to be physically consistent with less than the limiting case of a perfect inelastic collision i.
The calculated changes in momentum as defined by Eq. Additionally, the small deficits in the target momentum gains are consistent with the consideration that the impulse calculations may have slightly underestimated the observed head snap. Conservation of momentum for bullet and target interaction cf. Clearly there is a backward movement, but the movement is delayed and slower than the forward impulse discussed above.
The impact occurred just following the shutter closure of Z; this means that the bullet what was left of it was long gone by the time of the shutter opening at Z [50] , [16]. Frames Z—Z have been adjusted to remove jiggle caused by camera tracking errors of the cameraman.
Solid particles i. Note that the bullet was already long gone by the time the shutter opened at Z [50, p. Alvarez [10] 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 [11] , [12] , [13].
In this paper, the backward movement is reexamined theoretically in more detail. Like the current author, Prof. Alvarez estimated that the recoil would be about twice the initial velocity brought about by the collision.
Although the phenomenon of a high-energy projectile passing through a heterogenous body e. Indeed, it is often the case that scientific descriptions of nature are counter-intuitive and can run contrary to common sense [e.
However, the development of high-speed cameras has gone a long way toward facilitating observation and physical understanding of gunshot wound ballistics. In the case of a Newtonian fluid, the stresses imposed by such perturbations under high Reynolds number conditions i. 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 [36] , [37] , [34] , [16] , [49] , [14] , [15] , 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 [14] , [49] , [15] , [18] , resulting in restoring forces that can lead to additional violent undulations before the material fully returns to equilibrium [14] , [15].
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 [49] , [16].
This KE deposit propagated radially outward in the form of an expanding pressure wave [14] , [15] 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 Z—Z not shown here , and Mr.
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.
In Z—Z Figure 5 an expulsion of mass i. 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 [3, p. Note that this is not a universal occurrence—depending on the firearm, bullet, target, entry and exit locations, etc. 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 [52] , [12] , [7].
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. 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 [52] , [12] , [7] 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.
It is thus from these tracer particles that one has sufficient information in hand whereby the classical equation for rocket motion may be applied. This is the result of momentum conservation for the system, involving vehicle and exhaust, for which it can be shown that [e.
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