11. Lurch 3 Linear—add 3-D motion of the diffuse cloud

The model and its justification
    Like Lurch 2 Linear, Lurch 3 Linear calculates the speed of the rearward lurch of the upper body from the bullet, the body, the cloud, and the large fragments. To these it adds three-dimensional motion of the diffuse cloud. This three-dimensionality decreases the intensity of the lurch dramatically, from -2.44 ft s^-1 to -0.88 ft s^-1 for default values of the variables. This result shows that the 3-D motion of the diffuse cloud is an extremely important property to consider.

Solving the simultaneous equations
    As before, the equations for conservation of X-momentum and total energy are solved simultaneously. To the version in Lurch 2 are added two terms that express the three-dimensionality of the motion of the diffuse cloud—f_xcl and f_kecl. The f_xcl expresses something like the fraction of a horizontal two-dimensional velocity that is in the forward direction. Specifically, f_xcl is the average of cos Q between the leftmost forward angle considered reasonable and the rightmost. If all the cloud is considered to move straight ahead, f_xcl = 1. If the cloud is 30° wide, f_xcl = 0.94. If the cloud extends from -90° to +90°, which means from straight right to straight left, f_xcl is 0.4. For the starting value, I used the intermediate 0.7. For the range, I used 0.4–1. Of course, the real cloud was three-dimensional rather than two-dimensional. The third dimension is added in Lurch 3 Angular.
    The f_kecl term expresses the kinetic energy of the cloud beyond what is contained in the X-component of the velocity of the exiting fragments (i.e., for pure forward motion). If the cloud is shot out equally in all three directions (like an expanding sphere), then KE_x = KE_y = KE_z and f_kecl = 3. If it is shot out as an octant, a quadrant, or a hemisphere, the same relations hold. This will make the maximum value of f_kecl equal to 3 and the minimum equal to 1. For this reason, I used a starting value of 2 and a range of 1–3.
    The solutions for v_bodyafter and v_cloud, -0.88 ft s^-1 and 309 ft s^-1, respectively, are shown below the variables. Both are drastically reduced from Lurch 2 Linear.

Conservation of X-momentum

Conservation of total energy

Default values of the variables

Default solutions to the simultaneous equations

v_bodyafter = -0.88 ft s^-1  v_cloud = 309 ft s^-1

Distributions of momentum and energy

Sensitivity analysis
    

Sensitivity analysis, Lurch 3 Linear
(Standard conditions in boldface)

Ordered summary of sensitivities

    Making the cloud three-dimensional not only changed the basic solution (see above), but also the hierarchy of sensitivities. Into second and third place for effect (range of lurch) moved the two spatial variables f_xcl and f_kecl. Only the effect of m_cloud remained larger. All the other 12 variables had small effects on the solution. Thus the three important variables all had to do with the cloud—its mass and the three-dimensional distributions of its momentum and kinetic energy. In short, the properties of the diffuse cloud became all-important to calculating the lurch.

Summary
    Adding three-dimensionality to the diffuse cloud decreased the magnitude of the lurch by a factor of three, from -2.44 ft s^-1 to -0.88 ft s^-1. It also made the mass and dimensionality of the cloud the most important variables (greatest range of lurch predicted from the range of each of the variables). As much as anything, this result expressed the difficulty is estimating the true attributes of the cloud.

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