13. Lurch 5 Linearadd 3-D motion of body

The model and its justification
Having added 3-D motion to the large fragments in Lurch 4 Linear, we now try the same for the body. We can neglect the component of the momentum into or out of the plane of the Z-film because are dealing only with X-momentum. But we cannot neglect that component of the kinetic energy, because the equation for conservation of total energy uses total kinetic energy. As happened when adding the 3-D component of the fragments, the 3-D component of the body's KE hardly changes the answer at all. In fact, the previous -1.02 ft s-1 for the default values of variables remains the same here. This the 3-D motion of the body is an insignificant term in the overall equation.
The equations calculate the speed of the rearward lurch of the upper body from the bullet, the body (in 3-D), the cloud (in 3-D), and the large fragments (also in 3-D).

Solving the simultaneous equations
The three-dimensionality of the body's kinetic energy is expressed by the term fkebody, which is similar in form and function to fkefrag. We assume a range of 11.4 and a starting value of 1.2. ##### Conservation of X-momentum ##### Conservation of total energy Default values of 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 fkebody = 1.2 fxcl = 0.59 (calc.) fkefrags23 = 1.25

### Default solutions to 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 = 0.4 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 4 Linear. The new term, fkebody, has undetectable sensitivity and range of effect.

Sensitivity analysis, Lurch 5 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 -1.030 163 -1.027 95 -0.913 0.5 -1.484 400 -0.897 1820 -1.040 164 -1.030 100 -0.867 0.6 -1.681 450 -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 =  [(-1.024 + 1.017)/2]/ [1.021/161] = -0.55 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 =  [(-1.030 + 1.011)/20]/ [1.021/1800] = -1.68 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 -1.006 20 -2.005 0.022 -1.004 0.005 -1.029 0 -0.929 8 -1.009 30 -1.876 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 -1.267 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 -1.066 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 -1.033 0.015 -1.013 17 -1.042 120 +0.105 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 + 1.267)/20]/ [1.021/70] = 1.71 Sensitivity =  [-1.027 + 1.014)/0.004]/ [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 -1.025 450 -1.027 35 -1.037 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 -0.990 700 -0.961 700 -0.980 60 -0.936 90 -0.980 700 -0.934 750 -0.967 Sensitivity =  [(-1.012 + 1.025)/100]/ [1.021/500] = 0.06 Sensitivity =  [(-1.012 + 1.027)/100]/ [1.021/500] = 0.07 Sensitivity =  [(-1.002 + 1.037)/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 fkebody vlurch 1.00 -1.051 1.00 -1.032 1.05 -1.045 1.05 -1.030 1.00 -1.021 1.10 -1.039 1.10 -1.027 1.05 -1.021 1.15 -1.033 1.15 -1.025 1.10 -1.021 1.20 -1.027 1.20 -1.023 1.15 -1.021 1.25 -1.021 1.25 -1.021 1.20 -1.021 1.30 -1.014 1.30 -1.018 1.25 -1.021 1.35 -1.008 1.35 -1.016 1.30 -1.020 1.40 -1.002 1.40 -1.014 1.35 -1.020 1.45 -0.995 1.45 -1.011 1.40 -1.020 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)/0.10]/ [1.021/1.25] = 0.06 Sensitivity =  [(-1.020 + 1.021)/0.4]/ [1.021/1.20] = 0.00 Range = 0.06 Range = 0.02 Range = 0.00

Ordered summary of sensitivities

 Variable Sensitivity of vlurch Range of vlurch, ft s-1 Magnitude Positive effect on lurch (reduces rearward velocity) fkebody 0.00 0.00 Small mfrags23 0.02 0.02 Small fkefrags23 0.06 0.02 Small fkefrag1 0.16 0.06 Small vfrag1 0.06 0.08 Small vfrags23 0.07 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 -0.55 0.03 Small mfrag1 -0.09 0.03 Small Q -0.04 0.04 Small vbullet -1.68 0.10 Small vbulletafter -0.07 0.15 Small mcloud -0.79 1.68 Large

These sensitivities and effects are virtually the same as for Lurch 4 Linear. The new term, fkebody, has undetectable sensitivity and range of effect, i.e., counts for nothing. This is a consequence of the fact that the body gets virtually none of the final kinetic energy.

Summary
Adding three-dimensionality to the body's KE does not change the solution perceptibly, presumably because the body gets almost none of the KE after the explosion (<0.1%). This is confirmed by the undetectable sensitivity and range of effect for fkebody. The two variables with the largest effects remain Qcl (2.11 ft s-1) and mcloud (1.68 ft s-1). Only two variables have medium effects, PE (0.73 ft s-1) and mbody (0.51 ft s-1). Thus the cloud remains all-important to the solution.