15--L7L

15. Lurch 7 Linear—add four time intervals

The model and its justification
    Adding the four time intervals to the model represents a significant complication to it. It represents the fact that we know only what happened from one frame to the next, not anything within a frame. But the details within a frame can strongly change the answer, and we must take account of their effect on the answer and on its uncertainties.
    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: It also sets v_cloud = v_frags and calculates m_cloud. This far, it is the same as Lurch 6 Linear. 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 6 Linear and (4) sets v_cloud = v_frags and solves for m_cloud and v_bodyafter.
    The result of including these time intervals is to reduce the default lurch to -0.55 ft s^-1, a significant contraction from the previous -0.90 ft s^-1 (Lurch 6 Linear). [Reason?]

(1) Equations for t_snap

(2) Equations for t_lurch

(3) Equations for v_frags

v_frag1 = v_frags23 = v_cloud = 385 ft s^-1

(4) Simultaneous equations for m_cloud and v_bodyafter

Conservation of X-momentum

Conservation of total energy

Default values of variables

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_head = 7 lb	Q_frag1 = 40°
Q = 17°	Q_frags23 = 70°	f_kefrag1 = 1.25
PE = 200 ft-lb	f_xcl = 0.7	f_kefrags23 = f_kefrag1
d_transit = 4 in	f_kecl = 2	f_kebody = 1.2
d_snap = 2.2 in	d_frag1 = 6 ft	t_delay = 0.002 s

Solutions to simultaneous equations

v_bodyafter = -0.55 ft s^-1 m_cloud = 0.18 lb

Distributions of momentum and energy

Momentum		Energy
Before	After	Before	After
P_bullet = 1.24	P_bulletafter = 0.14	KE_bullet = 1164	KE_bulletafter = 14
	P_bodyafter = -0.73		KE_bodyafter = 0.1
	P_cloud = 1.53		KE_cloud = 843
	P_frag1 = 0.25		KE_frag1 = 78
	P_frags23 = 0.04		KE_frags23 = 3
			PE = 200

Sensitivity analysis

Sensitivity analysis, Lurch 7 Linear
(Standard conditions in boldface)

m_bullet	v_lurch	m_body	v_lurch	m_head	v_lurch	Q	v_lurch	PE	v_lurch
156	-0.438					12	-0.603
157	-0.461	65	-0.721	5	-1.403	13	-0.594	0	-0.825
158	-0.483	70	-0.669	5.5	-1.187	14	-0.584	50	-0.757
159	-0.505	75	-0.624	6	-0.972	15	-0.574	100	-0.688
160	-0.528	80	-0.585	6.5	-0.76	16	-0.563	150	-0.619
161	-0.551	85	-0.551	7	-0.551	17	-0.551	200	-0.551
162	-0.573	90	-0.52	7.5	-0.346	18	-0.538	250	-0.482
163	-0.596	95	-0.493	8	-0.149	19	-0.524	300	-0.413
164	-0.619	100	-0.468	8.5	+0.034	20	-0.509	350	-0.345
165	-0.642	105	-0.446	9		21	-0.493	400	-0.276
166	-0.665					22	-0.476
Sensitivity = [(-0.573 + 0.528)/2]/ [0.551/161] = -6.57		Sensitivity = [(-0.52 + 0.585)/10]/ [0.551/85] = 1.00		Sensitivity = [(-0.346 + 0.76)/1]/ [0.551/7] = 5.26		Sensitivity = [(-0.538 + 0.563)/2]/ [0.551/17] = 0.39		Sensitivity = [(-0.482 + 0.619)/100]/ [0.551/200] = 0.50
Range = 0.23		Range = 0.28		Range = 1.44		Range = 0.13		Range = 0.55

v_bullet	v_lurch	v_bulletafter	v_lurch	m_frag1	v_lurch	m_frags23	v_lurch	Q_frag1	v_lurch
1750	-0.401			0.022	-0.536	0.005	-0.555
1760	-0.43	0	-0.806	0.023	-0.539	0.006	-0.554	20	-0.593
1770	-0.46	50	-0.753	0.024	-0.542	0.007	-0.553	25	-0.585
1780	-0.49	100	-0.693	0.025	-0.545	0.008	-0.552	30	-0.575
1790	-0.52	150	-0.625	0.026	-0.548	0.009	-0.552	35	-0.564
1800	-0.551	200	-0.551	0.027	-0.551	0.01	-0.551	40	-0.551
1810	-0.582	250	-0.47	0.028	-0.554	0.011	-0.55	45	-0.536
1820	-0.613	300	-0.384	0.029	-0.557	0.012	-0.549	50	-0.52
1830	-0.646	350	-0.293	0.03	-0.56	0.013	-0.548	55	-0.504
1840	-0.678	400	-0.198	0.031	-0.563	0.014	-0.547	60	-0.485
1850	-0.711			0.032	-0.566	0.015	-0.546
Sensitivity = [(-0.582 + 0.52)/20]/ [0.551/1800] = -10.13		Sensitivity = [(-0.47 + 0.625)/100]/ [0.551/200] = 0.56		Sensitivity = [(-0.554 + 0.548)/0.002]/ [0.551/0.027] = -0.147		Sensitivity = [(-0.55 + 0.552)/0.002]/ [0.551/0.01] = 0.018		Sensitivity = [(-0.536 + 0.564)/10]/ [0.551/40] = 0.20
Range = 0.31		Range = 0.61		Range = 0.03		Range = 0.009		Range = 0.11

Q_frags23	v_lurch	f_xcl	v_lurch	f_kecl	v_lurch	f_kefrag1	v_lurch	f_kebody	v_lurch
						0	-0.698
50	-0.578			1	-1.709	0.25	-0.668
55	-0.572	0.4	-0.055	1.25	-1.246	0.5	-0.639
60	-0.565	0.5	-0.22	1.5	-0.937	0.75	-0.609	1.0	-0.551
65	-0.558	0.6	-0.385	1.75	-0.716	1	-0.58	1.1	-0.551
70	-0.551	0.7	-0.551	2	-0.551	1.25	-0.551	1.2	-0.551
75	-0.543	0.8	-0.716	2.25	-0.422	1.5	-0.521	1.3	-0.551
80	-0.535	0.9	-0.881	2.5	-0.319	1.75	-0.492	1.4	-0.551
85	-0.528	1.0	-1.046	2.75	-0.235	2	-0.462
90	-0.52			3	-0.165	2.25	-0.433
						2.5	-0.404
Sensitivity = [(-0.543 + 0.558)/10]/ [0.551/70] = 0.19		Sensitivity = [(-0.716 + 0.385)/0.2]/ [0.551/0.7] = -2.1		Sensitivity = [(-0.422 + 0.716)/0.5]/ [0.551/2] = 2.13		Sensitivity = [(-0.521 + 0.58)/0.5]/ [0.551/1.25] = 0.27		Sensitivity = [(-0.551 + 0.551)/0.2]/ [0.551/1.2] = 0.000
Range = 0.058		Range = 0.99		Range = 1.54		Range = 0.29		Range = 0.000

d_transit	v_lurch	d_snap	v_lurch	d_frag1	v_lurch	t_delay	v_lurch
0	-0.577
1	-0.57	1.6	-1.364	3	-1.829
2	-0.564	1.8	-1.089	4	-1.182	0.000	-0.71
3	-0.557	2	-0.818	5	-0.8	0.001	-0.63
4	-0.551	2.2	-0.551	6	-0.551	0.002	-0.551
5	-0.544	2.4	-0.291	7	-0.376	0.003	-0.472
6	-0.537	2.6	-0.047	8	-0.248	0.004	-0.394
7	-0.531	2.8	+0.157	9	-0.152	0.005	-0.317
8	-0.524					0.006	-0.241
						0.007	-0.166
Sensitivity = [(-0.544 + 0.557)/2]/ [0.551/4] = 0.047		Sensitivity = [(-0.291 + 0.818)/4]/ [0.551/2.2] = 5.26		Sensitivity = [(-0.376 + 0.8)/2]/ [0.551/6] = 2.31		Sensitivity = [(-0.472 + 0.63)/0.002]/ [0.551/0.002] = 0.029
Range = 0.053		Range = 1.52		Range = 1.68		Range = 0.54

Ordered summary of sensitivities

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
m_frags23	0.018	0.01	Small
d_transit	0.047	0.05	Small
Q_frags23	0.19	0.06	Small
Q_frag1	0.20	0.11	Small
Q	0.39	0.13	Small
m_body	1.010	0.28	Small
f_kefrag1	0.27	0.29	Small
t_delay	0.029	0.54	Medium
PE	0.50	0.55	Medium
m_head	5.26	1.44	Large
d_snap	5.26	1.52	Large
f_kecl	2.13	1.54	Large
d_frag1	2.31	1.68	Large
Negative effect on lurch (increases rearward velocity)
m_frag1	-0.147	0.03	Small
m_bullet	-6.57	0.23	Small
v_bullet	-10.13	0.31	Small
v_bulletafter	-0.56	0.61	Medium
f_xcl	-2.10	0.99	Large

The most striking thing about this ordered summary of sensitivities is that they are so dominated by terms that reduce the lurch. Only five of the 19 terms act to intensify the lurch, and only one of those (f_xcl) is has a large effect. By contrast, the 14 terms that reduce the lurch contain four with large effects, and they represent properties of the head, the snap, the cloud, and the large fragments. So it is no wonder that the resultant speed of the lurch in Lurch 7 Linear is reduced from the -0.90 ft s^-1 of Lurch 6 Linear to the -0.55 here.

Summary
Adding the four time intervals to Lurch 6 Linear complicates the simulation and shows how details within a frame can strongly change the answer. The preponderance of reducing terms decreases the magnitude of the lurch from the -0.90 ft s^-1 of Lurch 6 Linear to -0.55 ft s^-1. The rearrangement among the variables changes several of the sensitivities and creates a situation where most of the new variables act to reduce the speed of the lurch.

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