12--L4L

The model and its justification
Like Lurch 3 Linear, Lurch 4 Linear calculates the speed of the rearward lurch of the upper body from the bullet, the body, the cloud, the large fragments, and three-dimensional motion of the cloud. To these it adds three-dimensional motion of the large fragments. This addition hardly changes the answer (from -0.88 ft s^-1 to -0.87 ft s^-1 for default values of the variables). This shows that the 3-D motion of the large fragments is not an important consideration.

Solving the simultaneous equations
As before, the equations for conservation of X-momentum and total energy are solved simultaneously. To the version in Lurch 3 are added two terms that express the three-dimensionality of the motion of the two large fragments—f_kefrag1 and f_kefrags23. These terms are roughly analogous to the earlier f_kecl, and range upward from 1 because the the kinetic energy of a fragment will always be greater than the value calculated from the X- and Z-components of its velocity that are derived from the Zapruder film. To this must be added a component perpendicular to the film, which we can call the Y-component. A f_kefrag of 2 would mean that the fragment has twice the kinetic energy of that calculated from its X- and Z-velocities.
There is no reliable way to tell how much more kinetic energy the two large fragments had than is shown in the film's XZ plane, because we can't determine their velocities perpendicular to the film. Thus we can only estimate values for f_kefrag1 and f_kefrags23. To be conservative, we use default values of 1.25 for both and ranges of 1 to 1.5 It turns out not to matter much, for the speed of the lurch is very insensitive to both terms.

Conservation of X-momentum

Conservation of total energy

Preset variables

Default solutions to the simultaneous equations

Distributions of momentum and energy

Sensitivity analysis
This sensitivity analysis is practically identical to the one for Lurch 3 Linear. The two new terms, f_kefrag1 and f_kefrags23, have minimal sensitivities and minimal effects on the lurch. Thus they are unimportant variables.

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	-0.825	1810	-0.874
163	-0.87	95	-0.776	0.5	-1.314	300	-0.78	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] = -0.19		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 = [(-0.825 + 0.911)/100]/ [0.868/200] = 0.20		Sensitivity = [(-0.874 + 0.862)/20]/ [0.868/1800] = -1.24
Range = 0.01		Range = 0.43		Range = 1.62		Range = 0.35		Range = 0.05

v_bulletafter	v_lurch	Q	v_lurch	m_frag1	v_lurch	m_frags23	v_lurch	v_frag1	v_lurch
		12	-0.847	0.022	-0.849	0.005	-0.864
0	-0.772	13	-0.851	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	0.010	-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 = 0.04		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	-1.272
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	-0.862	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	-1.521	2.75	-0.644
500	-0.857	60	-0.817	90	-0.844			3	-0.589
Sensitivity = [(-0.867 + 0.869)/100]/ [0.868/300] = 0.007		Sensitivity = [(-0.857 + 0.868)/10]/ [0.868/40] = 0.097		Sensitivity = [(-0.862 + 0.874)/10]/ [0.868/70] = 0.097		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 = 0.01		Range = 0.08		Range = 0.04		Range = 1.31		Range = 0.91

f_kefrag1	v_lurch	f_kefrags23	v_lurch
1	-0.876	1	-0.871
1.1	-0.873	1.1	-0.87
1.25	-0.868	1.25	-0.868
1.4	-0.863	1.4	-0.866
1.5	-0.86	1.5	-0.865
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 = [(-0.863 + 0.873)/0.3]/ [0.868/1.25] = 0.048		Sensitivity = [(-0.866 + 0.87)/0.3]/ [0.868/1.25] = 0.019
Range = 0.02		Range = 0.01

The two additional variables from the three-dimensionality of the fragments changed the solution imperceptibly and had the smallest effects of any of the seven intensifying variables.

Summary
Adding three-dimensional effects for the two large fragments hardly changed anything about the solution, presumably because these variables were minor to begin with.

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

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 = 0.2
	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)
v_frags23	0.007	0.01	Small
f_kefrags23	0.019	0.01	Small
f_kefrag1	0.048	0.02	Small
Q_frags23	0.097	0.04	Small
Q_frag1	0.097	0.08	Small
PE	0.20	0.35	Small
m_body	1.01	0.43	Small
f_kecl	0.88	0.91	Medium
Negative effect on lurch (increases rearward velocity)
m_frags23	-0.012	0.01	Small
m_bullet	-0.19	0.01	Small
m_frag1	-0.124	0.04	Small
Q	-0.098	0.04	Small
v_bullet	-1.24	0.05	Small
v_frag1	-0.076	0.08	Small
v_bulletafter	-0.097	0.17	Small
f_xcl	-1.75	1.31	Large
m_cloud	-0.89	1.62	Large