9. Lurch 1 Linear—simplest analysis, with bullet, head cloud, and PE

The model and its justification
    We begin the linear calculations of the lurch by using the simplest representation (model), in which the speed of the rearward lurch is calculated from only the the bullet, the body, and the diffuse cloud of fragments. The idea is that the impact of the bullet supplies momentum and energy that afterwards appear as a movement of the cloud forward and, in response, a movement of the body rearward. The variables involved here are mass of bullet, initial and final speeds of bullet, mass of body (torso), mass and velocity of cloud of fragments, angle of incoming bullet, and potential energy (kinetic energy used up in penetrating the skull on both sides). For simplicity, we assume that the cloud moves horizontally forward. The two variables that we solve for are the speed of the lurch (v_bodyafter) and the speed of the cloud (v_cloud). All the rest are either known or guessed at.

The potential energy term
    The potential energy term represents kinetic energy that is lost in ways that cannot later be transformed back into KE. The two prime ways are degradation into heat energy inside the head and loss by breaking the skull as the bullet enters and exits. For simplicity, we neglect the heat term.
    To help me find a reasonable value of the PE for breaking bone, I called the Armed Forces Institute of Pathology (AFIP) and spoke with Dr. Steve Cogswell. He told me that most of the kinetic energy of bullets passing through heads remains as kinetic energy. About 200 ft-lb (the units of kinetic energy used by the American ballistic community) is needed to stretch skin to the breaking point. By contrast, 10–20 ft-lb is needed to cut skin with a knife, with a second cut being easier than a first cut. About 30–70 ft-lb is needed to create a linear fracture in a cranial vault. Breaking the skull and the skin would require 200–300 ft-lb of kinetic energy, with 200 being "reasonable" and 300 "conservative." This I have taken 200 ft-lb for the starting point in my calculations, and use a range of 0–450 ft-lb to be sure.

Mass of brain missing
    The mass of brain blown out in the diffuse cloud is a very important variable, and must be estimated carefully. The calculations below represent a better estimate than given in the section on plausibility, which were first estimates.
    After fixing in Formalin, the remains of the brain weighed 1500 g (WCR p. 544). Dr. John Lattimer reported that 70% of the right cerebral hemisphere was missing from the brain (Kennedy and Lincoln). If each cerebral hemisphere made up 1/3 of the brain, the fraction missing was 0.7(1/3), or 0.233. The total weight of the original brain would have then been 1500/0.767, or 1956 g. The mass lost would have been 0.233(1956 g), or 456 g. Thus we can say that the maximum brain blown out would be 456 g, or about 1 lb (454 g). That 1 lb would represent 1/8 of the total mass of the head (8 lb or so).

Solving the simultaneous equations
    The conservation equations for momentum and energy that are to be solved simultaneously are shown below. Note that even though they are called masses, the "m" terms are now actually weights, and are divided by 32 to make the proper masses. Starting values for the seven variables are shown below them. The solutions for v_bodyafter and v_cloud, -2.36 ft s^-1 and 450 ft s^-1, respectively are shown below the variables. Note that again we have an embarrassment of riches, with the basic solution for the speed of the rearward lurch well exceeding the observed value. This must mean that this simple representation is not good enough.

Conservation of X-momentum

Conservation of total energy

Default values of the variables

Default solutions to the simultaneous equations

v_bodyafter = -2.36 ft s^-1  v_cloud = 450 ft s^-1

Distributions of momentum and energy

    It is interesting to use the table above to see where the bullet's momentum and kinetic energy go. The original momentum, 1.24 lb ft s^-1, is divided into positive and negative components (forward and backward) of 4.22 for the cloud. -3.12 for the body, and 0.14 for the bullet, assuming the default exited speed of 200 ft s^-1. The energy divides itself very differently. The original 1164 ft-lb goes overwhelmingly to the cloud (948 ft-lb), with 200 going into potential energy (by default) and meager 14 and 2 to the exiting bullet and the body, respectively. To put it another way, the lurching body gets nearly half the momentum but less than 1% of the energy. The cloud get 90% of the kinetic energy.

Sensitivity analysis
    The sensitivity analysis is shown below. It revealed that, as with the snap, the two most sensitive variables were m_body and v_bullet. But the greatest importance (range of lurch) went to m_cloud, with its range was 3.3 ft s^-1 in the lurch, far greater than from any other variable. This puts these calculations in a difficult light, for the variable with the biggest effect is again the one we know the least about (hard to reliably estimate the mass of the cloud).

Sensitivity tests, Lurch 1 linear
(Standard conditions in boldface)

Ordered summary of sensitivities

    We must be very careful to note that positive sensitivities for the (rearward) lurch mean that increases in a driving variable increase the velocity of the lurch, which means making it less negative (because all the answers are negative, or rearward). This is counterintuitive, and must not be forgotten. Negative sensitivities mean that increases in the driving variable make the velocity of the lurch more negative, that is, increase its rearward component. In brief, positive sensitivities decrease the observed rearward lurch; negative sensitivities increase it. For simplicity, we will write "increase the lurch" when we mean strengthen it, or make it faster in the rearward direction, and "decrease the lurch" when we mean weaken the observed rearward motion.
    That said, we can examine the table immediately above. Note first that five of the seven variables increase the lurch, and only two decrease it. Of the five strengtheners, only the mass of the cloud matters much; the mass of the bullet, its entrance and exit velocities, and its angle of impact have negligible effects on the lurch. Of the two weakeners, the potential energy has a small effect and the mass of the body a moderate effect. The two variables that matter most to the final solution are the mass of the cloud and the mass of the body, and the mass of the cloud outweighs in importance the mass of the body. (This is the reason that Lurch 6 rearranges some of the variables in order to solve explicitly for the mass of the cloud.)

Summary
    This first model for the lurch produces a value that significantly exceeds the observed value. A more sophisticated approach must be tried. The two most important variables are the mass of the cloud and the mass of the body, in decreasing order.

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