10--L2L

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 fragments as well). We use a default velocity for each fragment of 300 ft s^-1, as measured on Z313. (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 v_bodyafter and v_cloud, -2.44 ft s^-1 and 437 ft s^-1, respectively, are shown below the variables. Both are very similar to the values from the first simulation, -2.36 and 450 ft s^-1.

Conservation of X-momentum

Conservation of total energy

Default solutions to simultaneous equations

Distributions of momentum and energy

The table of distributions is very similar to the one without the large fragments. That means that adding the fragments didn't change the basic solution very much.

Sensitivity analysis
The sensitivity analysis is very similar to the previous one. None of the six new variables (two masses, two speeds, two angles) have important sensitivities or ranges.

m_bullet	v_lurch	m_body	v_lurch	m_cloud	v_lurch	PE	v_lurch	v_bullet	v_lurch
156	-2.406							1750	-2.357
157	-2.413	65	-3.197			0	-2.77	1760	-2.374
158	-2.421	70	-2.968			50	-2.691	1770	-2.391
159	-2.428	75	-2.769	0.1	-1.132	100	-2.611	1780	-2.408
160	-2.435	80	-2.596	0.2	-1.873	150	-2.528	1790	-2.425
161	-2.443	85	-2.443	0.3	-2.443	200	-2.443	1800	-2.443
162	-2.45	90	-2.307	0.4	-2.924	250	-2.355	1810	-2.46
163	-2.457	95	-2.185	0.5	-3.348	300	-2.265	1820	-2.477
164	-2.464	100	-2.075	0.6	-3.733	350	-2.172	1830	-2.494
165	-2.471	105	-1.976	0.7	-4.087	400	-2.076	1840	-2.511
166	-2.478			0.8	-4.417			1850	-2.528
Sensitivity = [(-2.45 + 2.435)/2]/ [2.443/161] = -0.49		Sensitivity = [(-2.307 + 2.596)/10]/ [2.443/85] = 1.01		Sensitivity = [(-2.924 + 1.873)/0.2]/ [2.443/0.3] = -0.64		Sensitivity = [(-2.355 + 2.528)/100]/ [2.443/200] = 0.14		Sensitivity = [(-2.46 + 2.425)/20]/ [2.443/1800] = -1.29
Range = 0.48		Range = 1.22		Range = 3.28		Range = 0.69		Range = 0.17

v_bulletafter	v_lurch	Q	v_lurch	m_frag1	v_lurch	m_frags23	v_lurch	v_frag1	v_lurch
		12	-2.421	0.022	-2.428	0.005	-2.443
0	-2.359	13	-2.425	0.023	-2.431	0.006	-2.443	100	-2.403
50	-2.385	14	-2.429	0.024	-2.434	0.007	-2.443	150	-2.419
100	-2.407	15	-2.433	0.025	-2.437	0.008	-2.443	200	-2.43
150	-2.426	16	-2.438	0.026	-2.44	0.009	-2.443	250	-2.438
200	-2.443	17	-2.443	0.027	-2.443	0.010	-2.443	300	-2.443
250	-2.456	18	-2.448	0.028	-2.446	0.011	-2.443	350	-2.443
300	-2.466	19	-2.453	0.029	-2.449	0.012	-2.443	400	-2.44
350	-2.472	20	-2.459	0.030	-2.452	0.013	-2.443	450	-2.433
400	-2.476	21	-2.465	0.031	-2.455	0.014	-2.443	500	-2.421
		22	-2.471	0.032	-2.458	0.015	-2.443
Sensitivity = [(-2.456 + 2.426)/100]/ [2.443/200] = -0.025		Sensitivity = [(-2.448 + 2.438)/2]/ [2.443/17] = -0.035		Sensitivity = [(-2.446 + 2.44)/0.002]/ [2.443/0.027] = -0.033		Sensitivity = [(-2.443 + 2.443)/0.002]/ [2.443/0.01] = -0.000		Sensitivity = [(-2.443 + 2.438)/100]/ [2.443/300] = -0.006
Range = 0.12		Range = 0.050		Range = 0.03		Range = 0.000		Range = 0.018

v_frags23	v_lurch	Q_frag1	v_lurch	Q_frags23	v_lurch

100	-2.451	20	-2.476	50	-2.475
150	-2.452	25	-2.469	55	-2.467
200	-2.451	30	-2.462	60	-2.459
250	-2.448	35	-2.453	65	-2.451
300	-2.443	40	-2.443	70	-2.443
350	-2.436	45	-2.431	75	-2.434
400	-2.426	50	-2.419	80	-2.425
450	-2.415	55	-2.406	85	-2.416
500	-2.401	60	-2.392	90	-2.406
Sensitivity = [(-2.436 + 2.448)/100]/ [2.443/300] = 0.015		Sensitivity = [(-2.431 + 2.453)/10]/ [2.443/40] = 0.061		Sensitivity = [(-2.434 + 2.451)/10]/ [2.443/70] = 0.049
Sensitivity = [(-2.436 + 2.448)/100]/ [2.443/300] = 0.015		Sensitivity = [(-2.431 + 2.453)/10]/ [2.443/40] = 0.061		Sensitivity = [(-2.434 + 2.451)/10]/ [2.443/70] = 0.049
Range = 0.050		Range = 0.084		Range = 0.069

The six additional variables associated with the mass, velocity, and angle of inclination of the two large fragments change the speed of the lurch very little, as seen above. The reason is that they all have low sensitivities and small ranges of effects. In other words, they are not essential to the solution, at least at this early stage in the calculations.

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

m_bullet = 161 gr	v_bullet = 1800 ft s^-1	m_frag1 = 0.027 gr
m_body = 85 lb	v_bulletafter = 200 ft s^-1	m_frags23 = 0.01 gr
m_cloud = 0.3 lb	Q_frag1 = 40°	v_frag1 = 500 ft s-1
Q = 12°	Q_frags23 = 70°	v_frags23 = 500 ft s-1
PE = 300 ft-lb

Momentum		Energy, ft-lb
Before	After	Before	After
P_bullet = 1.24	P_bulletafter = 0.14	KE_bullet = 1164	KE_bulletafter = 14
	P_bodyafter = -3.23		KE_bodyafter = 2
	P_cloud = 4.10		KE_cloud = 896
	P_frag1 = 0.19		KE_frag1 = 38
	P_frags23 = 0.03		KE_frags23 = 14
			PE = 200

Variable	Sensitivity of v_lurch	*Range of v_lurch, ft s^-1*	Magnitude
Positive effect on lurch (reduces rearward velocity)
v_frags23	0.015	0.05	Small
Q_frags23	0.049	0.07	Small
Q_frag1	0.061	0.08	Small
PE	0.14	0.69	Small
m_body	1.01	1.22	Medium
Negative effect on lurch (increases rearward velocity)
m_frags23	-0.000	0.00	Small
v_frag1	-0.0006	0.02	Small
m_frag1	-0.033	0.03	Small
Q	-0.035	0.05	Small
v_bulletafter	-0.025	0.12	Small
v_bullet	-1.29	0.17	Small
m_bullet	-0.49	0.48	Small
m_cloud	-0.64	3.28	Large