12. Lurch 4 Linearadd 3-D motion of the large fragments

The model and its justification
Like Lurch 3 Linear, Lurch 4 Linear calculates the speed of the rearward lurch of the upper body from the bullet, the body, the cloud, the large fragments, and three-dimensional motion of the cloud. To these it adds three-dimensional motion of the large fragments. This addition hardly changes the answer (from -0.88 ft s-1 to -0.87 ft s-1 for default values of the variables). This shows that the 3-D motion of the large fragments is not an important consideration.

Solving the simultaneous equations
As before, the equations for conservation of X-momentum and total energy are solved simultaneously. To the version in Lurch 3 are added two terms that express the three-dimensionality of the motion of the two large fragmentsfkefrag1 and fkefrags23. These terms are roughly analogous to the earlier fkecl, and range upward from 1 because the the kinetic energy of a fragment will always be greater than the value calculated from the X- and Z-components of its velocity that are derived from the Zapruder film. To this must be added a component perpendicular to the film, which we can call the Y-component. A fkefrag of 2 would mean that the fragment has twice the kinetic energy of that calculated from its X- and Z-velocities.
There is no reliable way to tell how much more kinetic energy the two large fragments had than is shown in the film's XZ plane, because we can't determine their velocities perpendicular to the film. Thus we can only estimate values for fkefrag1 and fkefrags23. To be conservative, we use default values of 1.25 for both and ranges of 1 to 1.5 It turns out not to matter much, for the speed of the lurch is very insensitive to both terms.

# Preset variables

 mbullet = 161 gr vbullet = 1800 ft s-1 mfrag1 = 0.027 gr mbody = 85 lb vbulletafter = 200 ft s-1 mfrags23 = 0.01 gr mcloud = 0.3 lb Qfrag1 = 40° vfrag1 = 500 ft s-1 Q = 12° Qfrags23 = 70° vfrags23 = 500 ft s-1 PE = 300 ft-lb Qcl = 70° fkefrag1 = 1.25 fxcl = 0.59 (calc.) fkefrags23 = 1.25

### Default solutions to the simultaneous equations

vbodyafter = -1.02 ft s-1  vcloud = 378 ft s-1

## Distributions of momentum and energy

 Momentum Energy, ft-lb Before After Before After Pbullet = 1.26 Pbulletafter = 0.14 KEbullet = 1164 KEbulletafter = 14 Pbodyafter = -1.35 KEbodyafter = 0.3 Pcloud = 2.10 KEcloud = 669 Pfrag1 = 0.32 KEfrag1 = 132 Pfrags23 = 0.05 KEfrags23 = 49 PE = 300

Sensitivity analysis
This sensitivity analysis is practically identical to the one for Lurch 3 Linear. The two new terms, fkefrag1 and fkefrags23, have minimal sensitivities and minimal effects on the lurch. Thus they are unimportant variables.

Sensitivity analysis, Lurch 4 linear
(Standard conditions in boldface)

 mbullet vlurch mbody vlurch mcloud vlurch PE vlurch vbullet vlurch 0 -1.343 156 -1.004 50 -1.293 1750 -0.970 157 -1.007 65 -1.336 100 -1.242 1760 -0.980 158 -1.011 70 -1.240 150 -1.189 1770 -0.990 159 -1.014 75 -1.157 0.1 -0.351 200 -1.135 1780 -1.000 160 -1.017 80 -1.085 0.2 -0.729 250 -1.079 1790 -1.011 161 -1.021 85 -1.021 0.3 -1.021 300 -1.021 1800 -1.021 162 -1.024 90 -0.964 0.4 -1.267 350 -0.960 1810 -1.031 163 -1.027 9 -0.913 0.5 -1.484 400 -0.898 1820 -1.040 164 -1.030 100 -0.867 0.6 -1.682 450 -0.832 1830 -1.050 165 -1.033 105 -0.826 0.7 -1.863 500 -0.763 1840 -1.060 166 -1.036 0.8 -2.033 550 -0.690 1850 -1.070 600 -0.613 Sensitivity =  [(-1.027 + 1.014)/4]/ [1.021/161] = -0.51 Sensitivity =  [(-0.964 + 1.085)/10]/ [1.021/85] = 1.01 Sensitivity =  [(-1.267 + 0.729)/0.2]/ [1.021/0.3] = -0.79 Sensitivity =  [(-0.960 + 1.079)/100]/ [1.021/300] = 0.35 Sensitivity =  [(-1.031 + 1.011)/20]/ [1.021/1800] = -1.76 Range = 0.03 Range = 0.51 Range = 1.68 Range = 0.73 Range = 0.10

 vbulletafter vlurch Q vlurch Qcl vlurch mfrag1 vlurch mfrags23 vlurch 7 -1.007 20 -2.005 0.022 -1.004 0.005 -1.029 0 -0.929 8 -1.009 30 -1.877 0.023 -1.007 0.006 -1.027 50 -0.955 9 -1.011 40 -1.705 0.024 -1.011 0.007 -1.026 100 -0.979 10 -1.014 50 -1.499 0.025 -1.014 0.008 -1.024 150 -1.001 11 -1.017 60 -1.268 0.026 -1.017 0.009 -1.022 200 -1.021 12 -1.021 70 -1.021 0.027 -1.021 0.010 -1.021 250 -1.038 13 -1.024 80 -0.769 0.028 -1.024 0.011 -1.019 300 -1.053 14 -1.028 90 -0.522 0.029 -1.027 0.012 -1.017 350 -1.067 15 -1.033 100 -0.289 0.030 -1.030 0.013 -1.015 400 -1.077 16 -1.037 110 -0.078 0.031 -1.034 0.015 -1.014 17 -1.042 120 +0.015 0.032 -1.037 0.015 -1.012 Sensitivity =  [(-1.038 + 1.001)/100]/ [1.021/200] = -0.07 Sensitivity =  [(-1.028 + 1.014)/4]/ [1.021/12] = -0.04 Sensitivity =  [(-0.769 + 1.268)/20]/ [1.021/70] = 1.71 Sensitivity =  [-1.024 + 1.017)/0.002]/ [1.021/0.027] = -0.09 Sensitivity =  [(-1.017 + 1.024)/0.004]/ [1.021/0.010] = 0.02 Range = 0.15 Range = 0.04 Range = 2.11 Range = 0.03 Range = 0.02

 vfrag1 vlurch vfrags23 vlurch Qfrag1 vlurch Qfrags23 vlurch 250 -1.011 250 -1.043 300 -1.020 300 -1.041 20 -1.076 50 -1.056 350 -1.025 350 -1.038 25 -1.065 55 -1.048 400 -1.027 400 -1.033 30 -1.052 60 -1.039 450 -1.026 450 -1.028 35 -1.038 65 -1.030 500 -1.021 500 -1.021 40 -1.021 70 -1.021 550 -1.012 550 -1.012 45 -1.002 75 -1.011 600 -0.999 600 -1.003 50 -0.981 80 -1.001 650 -0.982 650 -0.992 55 -0.959 85 -0.991 700 -0.961 700 -0.980 60 -0.936 90 -0.980 700 -0.934 750 -0.967 Sensitivity =  [(-1.012 + 1.026)/100]/ [1.021/500] = 0.07 Sensitivity =  [(-1.012 + 1.028)/100]/ [1.021/500] = 0.08 Sensitivity =  [(-1.002 + 1.038)/10]/ [1.021/40] = 0.14 Sensitivity =  [(-1.011 + 1.030)/10]/ [1.021/70] = 0.13 Range = 0.08 Range = 0.08 Range = 0.14 Range = 0.08

 fkefrag1 vlurch fkefrags23 vlurch 1.00 -1.051 1.00 -1.032 1.05 -1.045 1.05 -1.030 1.10 -1.039 1.10 -1.028 1.15 -1.033 1.15 -1.025 1.20 -1.027 1.20 -1.023 1.25 -1.021 1.25 -1.021 1.30 -1.014 1.30 -1.018 1.35 -1.008 1.35 -1.016 1.40 -1.002 1.40 -1.014 1.45 -0.995 1.45 -1.011 1.50 -0.989 1.50 -1.009 Sensitivity =  [(-1.014 + 1.027)/0.10]/ [1.021/1.25] = 0.16 Sensitivity =  [(-1.018 + 1.023)/0.1]/ [1.021/1.25] = 0.06 Range = 0.06 Range = 0.02

Ordered summary of sensitivities

 Variable Sensitivity of vlurch Range of vlurch, ft s-1 Magnitude Positive effect on lurch (reduces rearward velocity) mfrags23 0.02 0.02 Small fkefrags23 0.06 0.02 Small fkefrag1 0.16 0.06 Small vfrag1 0.07 0.08 Small vfrags23 0.08 0.08 Small Qfrags23 0.13 0.08 Small Qfrag1 0.14 0.14 Small mbody 1.01 0.51 Medium PE 0.35 0.73 Medium Qcl 1.71 2.11 Large Negative effect on lurch (increases rearward velocity) mbullet -0.51 0.03 Small mfrag1 -0.09 0.03 Small Q -0.04 0.04 Small vbullet -1.76 0.10 Small vbulletafter -0.07 0.15 Small mcloud -0.79 1.68 Large

This list of sensitivities is virtually the same as for Lurch 3 Linear. The two additional variables from the three-dimensionality of the fragments hardly changed the solutio, and had two of the three smallest effects of any of the ten intensifying variables.

Summary
Adding three-dimensional effects for the two large fragments hardly changed anything about the solution, presumably because the fragments were minor parts of the solution to begin with. The dominant contributors to the solution remained the mass and half-angle of the cloud.