17. Lurch 2 Angular—add large fragments

The model and its justification
Lurch 2 Angular is the rotational analog of Lurch 2 Linear. It calculates the speed of the rearward lurch from the bullet, the body, the diffuse cloud of fragments, and the large fragments. Its result for the lurch, -2.94 ft s-1, is within 1% of Lurch 1 Angular (-2.92 ft s-1) and 35% greater than for Lurch 2 Linear (-2.18 ft s-1).

Solving the simultaneous equations
As with Lurch 2 Linear, the large fragments are represented by six terms, two for masses, two for exit speeds, and two for exit angles.

# 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 fI = 1.11

### Solutions to simultaneous equations

vbodyafter = -2.94 ft s-1  vcloud = 387 ft s-1

## Distributions of momentum and energy

 Angular momentum Energy, ft-lb Before After Before After Ωbullet = 3.80 Ωbulletafter = 0.43 KEbullet = 1164 KEbulletafter = 14 Ωbodyafter = -8.64 KEbodyafter = 4 Ωcloud = 10.88 KEcloud = 701 Ωfrag1 = 0.97 KEfrag1 = 105 Ωfrags23 = 0.16 KEfrags23 = 39 PE = 300

The angular momentum and energy are distributed similarly to the linear momentum and energy. The momentum is partitioned roughly equally between the forward movement of the cloud and the rearward movement of the body. By contrast, nearly all the kinetic energy goes to the cloud, except for the 300 units consumed in the potential energy of breaking the skull and scalp.

Sensitivity analysis
The sensitivity analysis for Lurch 2 Angular is very similar to that for Lurch 2 Linear. Only three terms matter much, the mass of the body, the mass of the cloud, and the potential energy. The first and last of these are negatively related to the strength of the lurch, whereas the second is positively related. The latter (the mass of the cloud), is also the strongest.

Sensitivity analysis, Lurch 2 Angular
(Standard conditions in boldface)

 mbullet vlurch mbody vlurch mcloud vlurch PE vlurch vbullet vlurch 0 -3.660 156 -2.883 50 -3.549 1750 -2.807 157 -2.896 65 -3.849 100 -3.435 1760 -2.835 158 -2.908 70 -3.574 150 -3.317 1770 -2.862 159 -2.919 75 -3.335 0.1 -1.379 100 -3.196 1780 -2.889 160 -2.931 80 -3.127 0.2 -2.264 250 -3.072 1790 -2.916 161 -2.943 85 -2.943 0.3 -2.943 300 -2.943 1800 -2.943 162 -2.955 90 -2.779 0.4 -3.515 350 -2.809 1810 -2.970 163 -2.966 95 -2.633 0.5 -4.018 400 -2.671 1820 -2.996 164 -2.978 100 -2.501 0.6 -4.472 450 -2.526 1830 -3.023 165 -2.989 105 -2.382 0.7 -4.890 500 -2.375 1840 -3.049 166 -3.000 0.8 -5.278 550 -2.216 1850 -3.075 600 -2.048 Sensitivity =  [(-2.955 + 2.931)/2]/ [2.943/161] = -0.66 Sensitivity =  [(-2.779 + 3.127)/10]/ [2.943/85] = 1.00 Sensitivity =  [(-3.515 + 2.264)/0.2]/ [2.943/0.3] = -0.64 Sensitivity =  [(-2.809 + 3.072)/100]/ [2.943/300] = 0.27 Sensitivity =  [(-2.970 + 2.916)/20]/ [2.943/1800] = -1.74 Range = 0.12 Range = 1.47 Range = 3.90 Range = 1.61 Range = 0.27

 vbulletafter vlurch Q vlurch mfrag1 vlurch mfrags23 vlurch vfrag1 vlurch 7 -2.924 0.022 -2.933 0.005 -2.967 250 -2.981 0 -2.835 8 -2.927 0.023 -2.935 0.006 -2.962 300 -2.985 50 -2.869 9 -2.930 0.024 -2.937 0.007 -2.957 350 -2.983 100 -2.898 10 -2.934 0.025 -2.939 0.008 -2.952 400 -2.976 150 -2.923 11 -2.938 0.026 -2.941 0.009 -2.948 450 -2.962 200 -2.943 12 -2.943 0.027 -2.943 0.010 -2.943 500 -2.943 250 -2.958 13 -2.948 0.028 -2.945 0.011 -2.938 550 -2.917 300 -2.968 14 -2.953 0.029 -2.947 0.012 -2.933 600 -2.885 350 -2.974 15 -2.959 0.030 -2.949 0.013 -2.928 650 -2.845 400 -2.974 16 -2.965 0.031 -2.950 0.014 -2.924 700 -2.798 17 -2.972 0.032 -2.952 0.015 -2.919 750 -2.743 Sensitivity =  [(-2.958 + 2.923)/100]/ [2.943/200] = -0.02 Sensitivity =  [(-2.953 + 2.934)/4]/ [2.943/12] = -0.02 Sensitivity =  [(-2.945 + 2.941)/0.002]/ [2.943/0.027] = -0.02 Sensitivity =  [(-2.938 + 2.948)/0.002]/ [2.943/0.010] = 0.02 Sensitivity =  [(-2.917 + 2.962)/100]/ [2.943/500] = 0.08 Range = 0.14 Range = 0.05 Range = 0.02 Range = 0.05 Range = 0.24

 vfrags23 vlurch Qfrag1 vlurch Qfrags23 vlurch fI vlurch 250 -2.992 1.06 -3.081 300 -2.986 20 -3.017 50 -2.991 1.07 -3.053 350 -2.979 25 -3.003 55 -2.980 1.08 -3.024 400 -2.969 30 -2.986 60 -2.968 1.09 -2.997 450 -2.957 35 -2.966 65 -2.956 1.10 -2.970 500 -2.943 40 -2.943 70 -2.943 1.11 -2.943 550 -2.927 45 -2.918 75 -2.930 1.12 -2.917 600 -2.908 50 -2.890 80 -2.916 1.13 -2.891 650 -2.888 55 -2.861 85 -2.903 1.14 -2.866 700 -2.865 60 -2.829 90 -2.889 1.15 -2.841 750 -2.840 1.16 -2.817 Sensitivity =  [(-2.927 + 2.957)/100]/ [2.943/500] = 0.05 Sensitivity =  [(-2.918 + 2.966)/10]/ [2.943/40] = 0.06 Sensitivity =  [(-2.930 + 2.956)/10]/ [2.943/70] = 0.06 Sensitivity =  [(-2.917 + 2.970)/0.02]/ [2.943/1.11] = 1.00 Range = 0.15 Range = 0.19 Range = 0.10 Range = 0.26

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 0.05 Small vfrags23 0.05 0.15 Small Qfrags23 0.06 0.10 Small Qfrag1 0.06 0.19 Small vfrag1 0.08 0.24 Small fI 1.00 0.26 Small mbody 1.00 1.47 Large PE 0.27 1.61 Large Negative effect on lurch (increases rearward velocity) mfrag1 -0.02 0.02 Small Q -0.02 0.05 Small mbullet -0.66 0.12 Small vbulletafter -0.02 0.14 Small vbullet -1.74 0.27 Small mcloud -0.64 3.90 Large

Adding the large fragments and their six variables does not affect the sensitivity analysis materially, for all six terms have small sensitivities (<0.1) and small effects (<0.25 ft s-1). All the other terms remain as before, with the body, the cloud, and the potential energy still being the only large terms.

Summary
Adding the large fragments hardly changes the lurch or any of the basic sensitivities. The six new variables all have small sensitivities and effects.