13--L5L

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 -0.87 ft s-1 for the default values of variables remain 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 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 f_kebody, which is similar in form and function to f_kefrag. We assume a range of 1–1.5 and a starting value of 1.2.

Conservation of X-momentum

Conservation of total energy

Default solutions to simultaneous equations

Distributions of momentum and energy

Sensitivity analysis
This sensitivity analysis is practically identical to the one for Lurch 4 Linear. The new term, f_kebody, has undetectable sensitivity and range of effect.

m_bullet	v_lurch	m_body	v_lurch	m_cloud	v_lurch	PE	v_lurch	v_bullet	v_lurch
156	-0.863							1750	-0.838
157	-0.864	65	-1.137			0	-1.032	1760	-0.844
158	-0.865	70	-1.055			50	-0.992	1770	-0.85
159	-0.866	75	-0.984	0.1	-0.224	100	-0.952	1780	-0.856
160	-0.867	80	-0.923	0.2	-0.588	150	-0.911	1790	-0.862
161	-0.868	85	-0.868	0.3	-0.868	200	-0.868	1800	-0.868
162	-0.869	90	-0.82	0.4	-1.105	250	-0.824	1810	-0.874
163	-0.87	95	-0.776	0.5	-1.314	300	-0.779	1820	-0.88
164	-0.871	100	-0.738	0.6	-1.504	350	-0.733	1830	-0.886
165	-0.872	105	-0.702	0.7	-1.679	400	-0.685	1840	-0.892
166	-0.873			0.8	-1.842			1850	-0.898
Sensitivity = [(-0.869 + 0.867)/2]/ [0.868/161] = -0.186		Sensitivity = [(-0.82 + 0.923)/10]/ [0.868/85] = 1.01		Sensitivity = [(-1.105 + 0.588)/0.2]/ [0.868/0.3] = -0.89		Sensitivity = [(-0.824 + 0.911)/100]/ [0.868/200] = 0.20		Sensitivity = [(-0.874 + 0.862)/20]/ [0.868/1800] = -1.24
Range = 0.01		Range = 0.44		Range = 1.62		Range = 0.35		Range = 0.06

v_bulletafter	v_lurch	Q	v_lurch	m_frag1	v_lurch	m_frags23	v_lurch	v_frag1	v_lurch
		12	-0.847	0.022	-0.848	0.005	-0.864
0	-0.772	13	-0.85	0.023	-0.852	0.006	-0.864	100	-0.806
50	-0.798	14	-0.854	0.024	-0.856	0.007	-0.865	150	-0.825
100	-0.823	15	-0.859	0.025	-0.86	0.008	-0.866	200	-0.842
150	-0.846	16	-0.863	0.026	-0.864	0.009	-0.867	250	-0.856
200	-0.868	17	-0.868	0.027	-0.868	0.01	-0.868	300	-0.868
250	-0.888	18	-0.873	0.028	-0.872	0.011	-0.869	350	-0.878
300	-0.907	19	-0.879	0.029	-0.876	0.012	-0.87	400	-0.885
350	-0.924	20	-0.884	0.03	-0.88	0.013	-0.871	450	-0.889
400	-0.939	21	-0.89	0.031	-0.884	0.014	-0.872	500	-0.891
		22	-0.897	0.032	-0.888	0.015	-0.873
Sensitivity = [(-0.888 + 0.846)/100]/ [0.868/200] = -0.097		Sensitivity = [(-0.873 + 0.863)/2]/ [0.868/17] = -0.098		Sensitivity = [(-0.872 + 0.864)/0.002]/ [0.868/0.027] = -0.124		Sensitivity = [(-0.869 + 0.867)/0.002]/ [0.868/0.01] = -0.012		Sensitivity = [-0.878 + 0.856)/100]/ [0.868/300] = -0.076
Range = 0.17		Range = 0.05		Range = 0.04		Range = 0.01		Range = 0.08

v_frags23	v_lurch	Q_frag1	v_lurch	Q_frags23	v_lurch	f_xcl	v_lurch	f_kecl	v_lurch
100	-0.865	20	-0.901	50	-0.889			1	-1.499
150	-0.867	25	-0.895	55	-0.885	0.4	-0.215	1.25	-1.271
200	-0.868	30	-0.887	60	-0.879	0.5	-0.433	1.5	-1.104
250	-0.869	35	-0.878	65	-0.874	0.6	-0.651	1.75	-0.973
300	-0.868	40	-0.868	70	-0.868	0.7	-0.868	2	-0.868
350	-0.867	45	-0.857	75	-0.865	0.8	-1.086	2.25	-0.781
400	-0.864	50	-0.845	80	-0.856	0.9	-1.303	2.5	-0.707
450	-0.861	55	-0.831	85	-0.85	1.0	-1.52	2.75	-0.644
500	-0.857	60	-0.817	90	-0.844			3	-0.589
Sensitivity = [(-0.869 + 0.867)/100]/ [0.868/300] = -0.0069		Sensitivity = [(-0.857 + 0.878)/10]/ [0.868/40] = 0.097		Sensitivity = [(-0.865 + 0.874)/10]/ [0.868/70] = 0.073		Sensitivity = [(-1.086 + 0.651)/0.2]/ [0.868/0.7] = -1.75		Sensitivity = [(-0.781 + 0.973)/0.5]/ [0.868/2] = 0.88
Range = 0.008		Range = 0.084		Range = 0.045		Range = 1.30		Range = 0.91

f_kefrag1	v_lurch	f_kefrags23	v_lurch	f_kebody	v_lurch
1.0	-0.876	1.0	-0.874	1.0	-0.868
1.1	-0.873	1.1	-0.87	1.1	-0.868
1.25	-0.868	1.25	-0.868	1.2	-0.868
1.4	-0.863	1.4	-0.866	1.3	-0.868
1.5	-0.86	1.5	-0.865	1.4	-0.868
Sensitivity = [(-0.863 + 0.873)/0.3]/ [0.868/1.25] = 0.048		Sensitivity = [(-0.866 + 0.87)/0.3]/ [0.868/1.25] = 0.019		Sensitivity = [(-0.868 + 0.868)/0.2]/ [0.868/1.2] = 0.000
Range = 0.016		Range = 0.009		Range = 0.000

These sensitivities and effects are virtually the same as for Lurch 4 Linear. The new term, f_kebody, 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 movement (its KE) does not change the solution perceptibly, presumably because the body gets almost none of the KE after the explosion (<<1%). This is confirmed by the undetectable sensitivity and range of effect for f_kebody.

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 = 300 ft s^-1
Q = 17°	Q_frags23 = 70°	v_frags23 = 300 ft s^-1
PE = 200 ft-lb	f_xcl = 0.7	f_kefrag1 = 1.25
	f_kecl = 2	f_kefrags23 = 1.25
		f_kebody = 1.2

Momentum		Energy, ft-lb
Before	After	Before	After
P_bullet = 1.24	P_bulletafter = 0.14	KE_bullet = 1164	KE_bulletafter = 14
	P_bodyafter = -1.15		KE_bodyafter = 0.3
	P_cloud = 2.02		KE_cloud = 885
	P_frag1 = 0.19		KE_frag1 = 47
	P_frags23 = 0.03		KE_frags23 = 18
			PE = 200

Variable	Sensitivity of v_lurch	*Range of v_lurch, ft s^-1*	Magnitude
Positive effect on lurch (reduces rearward velocity)
f_kebody	0.000	0.00	Small
f_kefrags23	0.019	0.01	Small
f_kefrag1	0.048	0.02	Small
Q_frags23	0.073	0.04	Small
Q_frag1	0.097	0.08	Small
PE	0.20	0.35	Small
m_body	1.01	0.44	Small
f_kecl	0.88	0.91	Medium
Negative effect on lurch (increases rearward velocity)
v_frags23	-0.007	0.01	Small
m_frags23	-0.012	0.01	Small
m_bullet	-0.19	0.01	Small
m_frag1	-0.124	0.04	Small
Q	-0.098	0.05	Small
v_bullet	-1.24	0.06	Small
v_frag1	-0.076	0.08	Small
v_bulletafter	-0.097	0.17	Small
f_xcl	-1.75	1.30	Large
m_cloud	-0.89	1.62	Large