22. Lurch 7 angular

The model and its justification
    This simulation differs from Snap/Lurch 6 Angular only in that it solves for PE rather than v_cloud. It focuses on PE because it is so important in setting constraints for other variables.
    As before, this procedure calculates the speed of the rearward lurch of JFK's upper body from the bullet, the body, the cloud, and the large fragments, and incorporates 3-D motions of the cloud, the large fragments, and the body: Before doing that, however, it interposes three steps having to do with the times. (1) It first calculates v_snap and uses it to calculate t_snap. (2) Then it calculates t_transit and combines it with t_snap with t_delay to calculate t_lurch. (3) Then it uses t_lurch and d_frags to calculate v_frags. At this point it returns to the procedure of Lurch 5 Linear and (4) solves for v_bodyafter and v_cloud. To simplify matters a bit, it sets v_frags23 = v_frag1 and f_kefrags23 = f_kefrag1.
    This minor change in variables changes the default solution considerably, however. It gives -0.91 ft s^-1 for the lurch (a 35% contraction) and 576 ft-lb for the PE (double the previous default value). The sensitivities and effects also change considerably. As with Snap/Lurch 7 Linear, it appears that the default values of the variables are an improper combination, and consequently that this default solution should not be regarded seriously. That does not represent a problem, however, for the main purpose of Snap/Lurch 7 is for use in setting the constraints that are discussed in later sections.

Conservation of X-momentum

Conservation of total energy

Solutions to simultaneous equations

v_bodyafter = -0.91 ft s^-1 PE = 576 ft-lb
v_frag1 = v_frags23 = 458 ft s^-1
v_snap = 4.7 ft s^-1
t_transit = 0.33 ms
t_snap = 39.2 ms
t_lurch = 13.1 ms

Distributions of momentum and energy

Angular momentum		Energy, ft-lb
Before	After	Before	After
Ω_bullet = 3.80	Ω_bulletafter = 0.43	KE_bullet = 1164	KE_bulletafter = 14
	Ω_bodyafter = -2.66		KE_bodyafter = 0.5
	Ω_cloud = 4.99		KE_cloud = 422
	Ω_frag1 = 0.89		KE_frag1 = 111
	Ω_frags23 = 0.15		KE_frags23 = 41
			PE = 300

m_bullet	v_lurch	m_body	v_lurch	m_cloud	v_lurch	m_head	v_lurch	v_bullet	v_lurch
						4.0	-0.708
156	-0.979					4.5	-0.724	1750	-0.980
157	-0.964	65	-1.187			5.0	-0.744	1760	-0.965
158	-0.949	70	-1.102			5.5	-0.769	1770	-0.949
159	-0.934	75	-1.028	0.1	+0.226	6.0	-0.801	1780	-0.935
160	-0.920	80	-0.963	0.2	-0.339	6.5	-0.844	1790	-0.920
161	-0.906	85	-0.906	0.3	-0.906	7.0	-0.906	1800	-0.906
162	-0.893	90	-0.856	0.4	-1.475	7.5	-1.002	1810	-0.893
163	-0.880	95	-0.811	0.5	-2.044	8.0	-1.169	1820	-0.879
164	-0.867	100	-0.770	0.6	-2.615	8.5	-1.536	1830	-0.866
165	-0.854	105	-0.733	0.7	-3.188	9.0	-2.980	1840	-0.853
166	-0.842			0.8	-3.761	9.5	+4.604	1850	-0.841
						10.0	+0.696
Sensitivity = [(-0.893 + 0.920)/2]/ [0.906/161] = 0.00		Sensitivity = [(-0.856 + 0.963)/10]/ [0.906/85] = 1.00		Sensitivity = [(-1.475 + 0.339)/0.2]/ [0.906/0.3] = -1.88		Sensitivity = [(-1.002 + 0.844)/1.0]/ [0.906/7.0] = -1.22		Sensitivity = [(-0.893 + 0.920)/20]/ [0.906/1800] = 2.68
Range = 0.14		Range = 0.45		Range = 3.99		Range = 1.40		Range = 0.14

v_bulletafter	v_lurch	v_cloud	v_lurch	Q	v_lurch	Q_cl	v_lurch	m_frag1	v_lurch
		0	+0.793
		50	+0.510	7	-0.871	20	-1.965	0.022	-0.850
0	-0.671	100	+0.227	8	-0.876	30	-1.827	0.023	-0.861
50	-0.723	150	-0.057	9	-0.883	40	-1.642	0.024	-0.873
100	-0.779	200	-0.340	10	-0.890	50	-1.421	0.025	-0.884
150	-0.840	250	-0.623	11	-0.898	60	-1.172	0.026	-0.895
200	-0.906	300	-0.906	12	-0.906	70	-0.906	0.027	-0.906
250	-0.982	350	-1.190	13	-0.916	80	-0.636	0.028	-0.918
300	-1.070	400	-1.473	14	-0.927	90	-0.371	0.029	-0.929
350	-1.180	450	-1.756	15	-0.938	100	-0.121	0.030	-0.940
400	-1.327	500	-2.039	16	-0.951	110	+0.105	0.031	-0.951
		550	-2.323	17	-0.965	120	+0.302	0.032	-0.962
		600	-2.606
Sensitivity = [(-0.982 + 0.840)/100]/ [0.906/200] = -0.31		Sensitivity = [(-1.190 + 0.623)/100]/ [0.906/300] = -1.88		Sensitivity = [(-0.927 + 0.890)/4]/ [0.906/12] = -0.12		Sensitivity = [(-0.636 + 1.172)/20]/ [0.906/70] = 2.07		Sensitivity = [-0.918 + 0.895)/0.002]/ [0.906/0.027] = -0.34
Range = 0.66		Range = 3.40		Range = 0.09		Range = 2.27		Range = 0.11

m_frags23	v_lurch	Q_frag1	v_lurch	Q_frags23	v_lurch	f_kefrags	v_lurch	f_kebody	v_lurch
0.005	-0.881					1.00	-0.906
0.006	-0.886	20	-0.975	50	-0.950	1.05	-0.906	1.00	-0.906
0.007	-0.891	25	-0.962	55	-0.940	1.10	-0.906	1.05	-0.906
0.008	-0.896	30	-0.946	60	-0.929	1.15	-0.906	1.10	-0.906
0.009	-0.901	35	-0.927	65	-0.918	1.20	-0.906	1.15	-0.906
0.010	-0.906	40	-0.906	70	-0.906	1.25	-0.906	1.20	-0.906
0.011	-0.911	45	-0.883	75	-0.894	1.30	-0.906	1.25	-0.906
0.012	-0.916	50	-0.858	80	-0.882	1.35	-0.906	1.30	-0.906
0.013	-0.921	55	-0.830	85	-0.869	1.40	-0.906	1.35	-0.906
0.015	-0.926	60	-0.801	90	-0.856	1.45	-0.906	1.40	-0.906
0.015	-0.931					1.50	-0.906
Sensitivity = [(-0.911 + 0.901)/0.002]/ [0.906/0.010] = -0.06		Sensitivity = [(-0.883 + 0.927)/10]/ [0.906/40] = 0.19		Sensitivity = [(-0.894 + 0.918)/10]/ [0.906/70] = 0.18		Sensitivity = [(-0.906 + 0.906)/0.50]/ \|[0.906/1.25] = 0.00		Sensitivity = [(-0.906 + 0.906)/0.40]/ [0.906/1.20] = 0.00
Range = 0.05		Range = 0.17		Range = 0.09		Range = 0.00		Range = 0.00

f_I	v_lurch	d_transit	v_lurch	d_snap	v_lurch	d_frag1	v_lurch	t_delay	v_lurch
1.06	-0.949
1.07	-0.940	0	-0.898			2	-0.671
1.08	-0.932	1	-0.900	1.6	-0.748	3	-0.730
1.09	-0.923	2	-0.902	1.8	-0.782	4	-0.789	0.000	-0.860
1.10	-0.915	3	-0.904	2.0	-0.831	5	-0.848	0.001	-0.881
1.11	-0.906	4	-0.906	2.2	-0.906	6	-0.906	0.002	-0.906
1.12	-0.898	5	-0.909	2.4	-1.038	7	-0.965	0.003	-0.936
1.13	-0.890	6	-0.911	2.6	-1.327	8	-1.024	0.004	-0.970
1.14	-0.883	7	-0.913	2.8	-2.468	9	-1.083	0.005	-1.011
1.15	-0.875	8	-0.916			10	-1.141	0.006	-1.061
1.16	-0.867							0.007	-1.124
Sensitivity = [(-0.898 + 0.915)/0.02]/ [0.906/1.11] = 1.04		Sensitivity = [(-0.916 + 0.898)/8]/ [0.906/4] = -0.01		Sensitivity = [(-1.038 + 0.831)/0.4]/ [0.906/2.2] = -1.26		Sensitivity = [(-0.965 + 0.848)/2]/ [0.906/6] = -0.39		Sensitivity = [(-0.936 + 0.881)/0.002]/ [0.906/0.002] = -0.06
Range = 0.08		Range = 0.02		Range = 3.22		Range = 0.47		Range = 0.26

R_bullet	v_lurch	R_head	v_lurch
4.75	-1.506	3.50	-0.766
5.00	-1.193	3.75	-0.789
5.25	-1.047	4.00	-0.818
5.50	-0.962	4.25	-0.856
5.75	-0.906	4.50	-0.906
6.00	-0.867	4.75	-0.977
6.25	-0.838	5.00	-1.082
6.50	-0.816	5.25	-1.257
6.75	-0.798	5.50	-1.605
Sensitivity = [(-0.867 + 0.962)/0.50]/ [0.906/5.75] = 1.21		Sensitivity = [(-0.977 + 0.856)/0.50]/ [0.906/4.50] = -1.20
Range = 0.71		Range = 0.84

These sensitivities and effects are very similar to those of the linear analog Snap/Lurch 7 Linear. They differ from Snap/Lurch 6 in that they are systematically shifted toward negative effects. As opposed to Snap/Lurch 6, which had 15 positive effects and 7 negative sensitivities, Snap/Lurch 7 has 10 positive and 12 negative effects, for a net shift of five effects from positive to negative. One might then expect the rearward lurch to become stronger, but the opposite happened—it decreased in intensity from -1.39 ft s^-1 to -0.91 ft s^-1, a loss of 35%. As before, however, there was a great dichotomy between the few strong terms (five, with effects between 1.40 ft s^-1 and 3.99 ft s^-1) and the many small and medium terms (17, with effects between 0.00 ft s^-1 and 0.84 ft s^-1). Of the five strong terms, three had to do with the cloud (m_cloud, v_cloud, and Q_cl) and two with the snap (m_head and d_snap). This reinforces the importance of the cloud to the final solution.

Summary
Solving for PE instead of v_cloud created five more negative effects but reduced the strength of the negative lurch from -1.39 ft s^-1 to -0.91 ft s^-1. Of the five large effects, three were properties of the cloud, two of the head and snap.

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°	f_kefrag1 = 1.25
m_head = 7 lb	Q_frags23 = 70°	f_kefrags23 = f_kefrag1
Q = 12°	Q_cl = 70°	f_kebody = 1.2
v_cloud = 300 ft s^-1	t_delay = 0.002 s	f_xcl = 0.59 (calc.)
d_transit = 4 in	d_snap = 2.2 in	d_frag1 = 6 ft

Variable	Sensitivity of v_lurch	*Range of v_lurch, ft s^-1*	Magnitude
Positive effect on lurch (reduces rearward velocity)
f_kebody	0.00	0.00	Small
f_kefrag1	0.00	0.00	Small
f_I	1.04	0.08	Small
Q_frags23	0.18	0.09	Small
m_bullet	0.00	0.14	Small
v_bullet	2.16	0.14	Small
Q_frag1	0.19	0.17	Small
m_body	1.00	0.45	Small
R_bullet	1.21	0.71	Medium
Q_cl	2.07	2.27	Large
Negative effect on lurch (increases rearward velocity)
d_transit	-0.01	0.02	Small
m_frags23	-0.06	0.05	Small
Q	-0.12	0.09	Small
m_frag1	-0.34	0.11	Small
t_delay	-0.06	0.26	Small
d_frag1	-0.39	0.47	Small
v_bulletafter	-0.31	0.66	Medium
R_head	-1.20	0.84	Medium
m_head	-1.22	1.40	Large
d_snap	-1.26	3.22	Large
v_cloud	-1.88	3.40	Large
m_cloud	-1.88	3.99	Large