10. Lurch 2 linearadd large fragments

The model and its justification
The next step in making the simulation more realistic is to add the two large fragments to it. These are the while lines seen in Z313 and to a small extent in Z314. Their masses can be estimated from the pictures in Lattimer's book and the density of bone. They come out to be 0.027 gr for fragment 1 (with the upward angle of 40° and 0.01 gr for the one with an angle of 70° (here called frags23 because it may include a smaller third fragment as well). We use a default velocity for each fragment of 500 ft s-1, as measured on Z313 (keeping in mind that the path represents less than a full frame's exposure). These velocities turn out to be unimportant.

Solving the simultaneous equations
The equations to be solved simultaneously are shown below. Note that the mass of the lurching body has been reduced by the masses of the fragments. The starting values of the variables are shown below the equations.
The solutions for vbodyafter and vcloud, -2.18 ft s-1 and 388 ft s-1, respectively, are shown below the variables. Both are very similar to the values from the first linear simulation, -2.17 and 425 ft s-1.

##### 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

### Default solutions to simultaneous equations

vbodyafter = -2.18 ft s-1  vcloud = 388 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 = -2.89 KEbodyafter = 2 Pcloud = 3.63 KEcloud = 704 Pfrag1 = 0.32 KEfrag1 = 105 Pfrags23 = 0.05 KEfrags23 = 39 PE = 300

The table of distributions is very similar to the one without the large fragments, particularly for the momentum. The fragments removed 144 ft-lb of KE from the cloud, however (17% of its value). But the lurch changed by only 1%. Thus adding the fragments didn't change the basic solution very much.

Sensitivity analysis
Although the sensitivity analysis looks different from the previous one, its essentials are the same. None of the six new variables (two masses, two speeds, two angles) have important sensitivities or ranges.

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

 mbullet vlurch mbody vlurch mcloud vlurch PE vlurch vbullet vlurch 0 -2.715 156 -2.139 50 -2.632 1750 -2.082 157 -2.148 65 -2.857 100 -2.548 1760 -2.103 158 -2.157 70 -2.652 150 -2.461 1770 -2.123 159 -2.166 75 -2.475 0.1 -1.021 100 -2.371 1780 -2.143 160 -2.174 80 -2.320 0.2 -1.678 250 -2.279 1790 -2.163 161 -2.183 85 -2.183 0.3 -2.183 300 -2.183 1800 -2.183 162 -2.192 90 -2.061 0.4 -2.609 350 -2.084 1810 -2.203 163 -2.200 95 -1.953 0.5 -2.986 400 -1.981 1820 -2.223 164 -2.209 100 -1.855 0.6 -3.327 450 -1.874 1830 -2.242 165 -2.217 105 -1.766 0.7 -3.641 500 -1.762 1840 -2.262 166 -2.225 0.8 -3.933 550 -1.644 1850 -2.281 600 -1.519 Sensitivity =  [(-2.192 + 2.174)/2]/ [2.183/161] = -0.66 Sensitivity =  [(-2.061 + 2.320)/10]/ [2.183/85] = 1.01 Sensitivity =  [(-2.609 + 1.678)/0.2]/ [2.183/0.3] = -0.64 Sensitivity =  [(-2.084 + 2.279)/100]/ [2.183/300] = 0.27 Sensitivity =  [(-2.203 + 2.163)/20]/ [2.183/1800] = -1.65 Range = 0.09 Range = 1.09 Range = 2.91 Range = 1.20 Range = 0.20

 vbulletafter vlurch Q vlurch mfrag1 vlurch mfrags23 vlurch vfrag1 vlurch 7 -2.169 0.022 -2.175 0.005 -2.200 250 -2.211 0 -2.102 8 -2.171 0.023 -2.177 0.006 -2.197 300 -2.214 50 -2.128 9 -2.174 0.024 -2.179 0.007 -2.194 350 -2.213 100 -2.150 10 -2.176 0.025 -2.180 0.008 -2.190 400 -2.207 150 -2.168 11 -2.180 0.026 -2.182 0.009 -2.187 450 -2.197 200 -2.183 12 -2.183 0.027 -2.183 0.010 -2.183 500 -2.183 250 -2.194 13 -2.187 0.028 -2.184 0.011 -2.179 550 -2.164 300 -2.202 14 -2.191 0.029 -2.186 0.012 -2.176 600 -2.140 350 -2.206 15 -2.195 0.030 -2.187 0.013 -2.172 650 -2.111 400 -2.206 16 -2.199 0.031 -2.189 0.014 -2.169 700 -2.076 17 -2.204 0.032 -2.190 0.015 -2.165 750 -2.035 Sensitivity = [(-2.194 + 2.168)/100]/ [2.183/200] = -0.02 Sensitivity = [(-2.191 + 2.176)/4]/ [2.183/12] = -0.02 Sensitivity = [(-2.186 + 2.180)/0.004]/ [2.183/0.027] = -0.02 Sensitivity = [(-2.179 + 2.187)/0.002]/ [2.183/0.01] = 0.02 Sensitivity = [(-2.164 + 2.197)/100]/ [2.183/500] = 0.08 Range = 0.10 Range = 0.04 Range = 0.02 Range = 0.04 Range = 0.18

 vfrags23 vlurch Qfrag1 vlurch Qfrags23 vlurch 250 -2.219 300 -2.215 20 -2.238 50 -2.218 350 -2.209 25 -2.228 55 -2.210 400 -2.202 30 -2.215 60 -2.202 450 -2.193 35 -2.200 65 -2.192 500 -2.183 40 -2.183 70 -2.183 550 -2.171 45 -2.164 75 -2.173 600 -2.157 50 -2.144 80 -2.163 650 -2.142 55 -2.122 85 -2.153 700 -2.125 60 -2.098 90 -2.143 750 -2.106 Sensitivity =  [(-2.171 + 2.193)/100]/ [2.183/500] = 0.05 Sensitivity =  [(-2.164 + 2.200)/10]/ [2.183/40] = 0.07 Sensitivity =  [(-2.173 + 2.192)/10]/ [2.183/70] = 0.06 Range = 0.11 Range = 0.14 Range = 0.08

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 0.04 Small Qfrags23 0.06 0.08 Small vfrags23 0.05 0.11 Small Qfrag1 0.07 0.14 Small vfrag1 0.08 0.18 Small mbody 1.01 1.09 Medium PE 0.27 1.20 Medium Negative effect on lurch (increases rearward velocity) mfrag1 -0.02 0.02 Small Q -0.02 0.04 Small mbullet -0.66 0.09 Small vbulletafter -0.02 0.10 Small vbullet -1.65 0.20 Small mcloud -0.64 2.91 Large

None of the six new variables (associated with the mass, velocity, and angle of inclination of the two large fragments) affect the speed of the lurch very much, as seen above. They all have low sensitivities and small ranges of effects. Five of them have positive effects, one negative. The effects of the previous variables remain much the same. In other words, the large fragments are not important to the solution, at least for these early calculations.

Summary
The answer with the two large fragments is fundamentally the same as without them. They are thus minor variables. This is confirmed by the six sensitivities and effects associated with them, which are all very small. The next step is to add three-dimensional motions, beginning with the diffuse cloud of fragments.