12. Lurch 4 linear

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
						0	-1.343
156	-1.004					50	-1.293	1750	-0.970
157	-1.007	65	-1.336			100	-1.242	1760	-0.980
158	-1.011	70	-1.240			150	-1.189	1770	-0.990
159	-1.014	75	-1.157	0.1	-0.351	200	-1.135	1780	-1.000
160	-1.017	80	-1.085	0.2	-0.729	250	-1.079	1790	-1.011
161	-1.021	85	-1.021	0.3	-1.021	300	-1.021	1800	-1.021
162	-1.024	90	-0.964	0.4	-1.267	350	-0.960	1810	-1.031
163	-1.027	9	-0.913	0.5	-1.484	400	-0.898	1820	-1.040
164	-1.030	100	-0.867	0.6	-1.682	450	-0.832	1830	-1.050
165	-1.033	105	-0.826	0.7	-1.863	500	-0.763	1840	-1.060
166	-1.036			0.8	-2.033	550	-0.690	1850	-1.070
						600	-0.613
Sensitivity = [(-1.027 + 1.014)/4]/ [1.021/161] = -0.51		Sensitivity = [(-0.964 + 1.085)/10]/ [1.021/85] = 1.01		Sensitivity = [(-1.267 + 0.729)/0.2]/ [1.021/0.3] = -0.79		Sensitivity = [(-0.960 + 1.079)/100]/ [1.021/300] = 0.35		Sensitivity = [(-1.031 + 1.011)/20]/ [1.021/1800] = -1.76
Range = 0.03		Range = 0.51		Range = 1.68		Range = 0.73		Range = 0.10

v_bulletafter	v_lurch	Q	v_lurch	Q_cl	v_lurch	m_frag1	v_lurch	m_frags23	v_lurch
		7	-1.007	20	-2.005	0.022	-1.004	0.005	-1.029
0	-0.929	8	-1.009	30	-1.877	0.023	-1.007	0.006	-1.027
50	-0.955	9	-1.011	40	-1.705	0.024	-1.011	0.007	-1.026
100	-0.979	10	-1.014	50	-1.499	0.025	-1.014	0.008	-1.024
150	-1.001	11	-1.017	60	-1.268	0.026	-1.017	0.009	-1.022
200	-1.021	12	-1.021	70	-1.021	0.027	-1.021	0.010	-1.021
250	-1.038	13	-1.024	80	-0.769	0.028	-1.024	0.011	-1.019
300	-1.053	14	-1.028	90	-0.522	0.029	-1.027	0.012	-1.017
350	-1.067	15	-1.033	100	-0.289	0.030	-1.030	0.013	-1.015
400	-1.077	16	-1.037	110	-0.078	0.031	-1.034	0.015	-1.014
		17	-1.042	120	+0.015	0.032	-1.037	0.015	-1.012
Sensitivity = [(-1.038 + 1.001)/100]/ [1.021/200] = -0.07		Sensitivity = [(-1.028 + 1.014)/4]/ [1.021/12] = -0.04		Sensitivity = [(-0.769 + 1.268)/20]/ [1.021/70] = 1.71		Sensitivity = [-1.024 + 1.017)/0.002]/ [1.021/0.027] = -0.09		Sensitivity = [(-1.017 + 1.024)/0.004]/ [1.021/0.010] = 0.02
Range = 0.15		Range = 0.04		Range = 2.11		Range = 0.03		Range = 0.02

v_frag1	v_lurch	v_frags23	v_lurch	Q_frag1	v_lurch	Q_frags23	v_lurch
250	-1.011	250	-1.043
300	-1.020	300	-1.041	20	-1.076	50	-1.056
350	-1.025	350	-1.038	25	-1.065	55	-1.048
400	-1.027	400	-1.033	30	-1.052	60	-1.039
450	-1.026	450	-1.028	35	-1.038	65	-1.030
500	-1.021	500	-1.021	40	-1.021	70	-1.021
550	-1.012	550	-1.012	45	-1.002	75	-1.011
600	-0.999	600	-1.003	50	-0.981	80	-1.001
650	-0.982	650	-0.992	55	-0.959	85	-0.991
700	-0.961	700	-0.980	60	-0.936	90	-0.980
700	-0.934	750	-0.967
Sensitivity = [(-1.012 + 1.026)/100]/ [1.021/500] = 0.07		Sensitivity = [(-1.012 + 1.028)/100]/ [1.021/500] = 0.08		Sensitivity = [(-1.002 + 1.038)/10]/ [1.021/40] = 0.14		Sensitivity = [(-1.011 + 1.030)/10]/ [1.021/70] = 0.13
Range = 0.08		Range = 0.08		Range = 0.14		Range = 0.08

f_kefrag1	v_lurch	f_kefrags23	v_lurch
1.00	-1.051	1.00	-1.032
1.05	-1.045	1.05	-1.030
1.10	-1.039	1.10	-1.028
1.15	-1.033	1.15	-1.025
1.20	-1.027	1.20	-1.023
1.25	-1.021	1.25	-1.021
1.30	-1.014	1.30	-1.018
1.35	-1.008	1.35	-1.016
1.40	-1.002	1.40	-1.014
1.45	-0.995	1.45	-1.011
1.50	-0.989	1.50	-1.009
Sensitivity = [(-1.014 + 1.027)/0.10]/ [1.021/1.25] = 0.16		Sensitivity = [(-1.018 + 1.023)/0.1]/ [1.021/1.25] = 0.06
Range = 0.06		Range = 0.02

This list of sensitivities is virtually the same as for Lurch 3 Linear. The two additional variables from the three-dimensionality of the fragments hardly changed the solutio, and had two of the three smallest effects of any of the ten intensifying variables.

Summary
Adding three-dimensional effects for the two large fragments hardly changed anything about the solution, presumably because the fragments were minor parts of the solution to begin with. The dominant contributors to the solution remained the mass and half-angle of the cloud.

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	Q_cl = 70°	f_kefrag1 = 1.25
	f_xcl = 0.59 (calc.)	f_kefrags23 = 1.25

Momentum		Energy, ft-lb
Before	After	Before	After
P_bullet = 1.26	P_bulletafter = 0.14	KE_bullet = 1164	KE_bulletafter = 14
	P_bodyafter = -1.35		KE_bodyafter = 0.3
	P_cloud = 2.10		KE_cloud = 669
	P_frag1 = 0.32		KE_frag1 = 132
	P_frags23 = 0.05		KE_frags23 = 49
			PE = 300

Variable	Sensitivity of v_lurch	*Range of v_lurch, ft s^-1*	Magnitude
Positive effect on lurch (reduces rearward velocity)
m_frags23	0.02	0.02	Small
f_kefrags23	0.06	0.02	Small
f_kefrag1	0.16	0.06	Small
v_frag1	0.07	0.08	Small
v_frags23	0.08	0.08	Small
Q_frags23	0.13	0.08	Small
Q_frag1	0.14	0.14	Small
m_body	1.01	0.51	Medium
PE	0.35	0.73	Medium
Q_cl	1.71	2.11	Large
Negative effect on lurch (increases rearward velocity)
m_bullet	-0.51	0.03	Small
m_frag1	-0.09	0.03	Small
Q	-0.04	0.04	Small
v_bullet	-1.76	0.10	Small
v_bulletafter	-0.07	0.15	Small
m_cloud	-0.79	1.68	Large