15. Lurch 7 linear

The model and its justification
    This simulation is very similar to Snap/Lurch 6. It differs only in solving 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. It gives -0.62 ft s^-1 for the lurch (a 40% contraction) and 633 ft s^-1 for the PE (double the previous default value). The sensitivities and effects also change considerably. It appears that the default 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 = -0.62 ft s^-1 PE = 633 ft-lb
v_frag1 = v_frags23 = 363 ft s^-1
v_snap = 5.1 ft s^-1
t_transit = 0.33 ms
t_snap = 35.8 ms
t_lurch = 16.5 ms

Distributions of momentum and energy

m_bullet	v_lurch	m_body	v_lurch	m_cloud	v_lurch	m_head	v_lurch	v_bullet	v_lurch
						4.0	-0.517
156	-0.658					4.5	-0.526	1750	-0.659
157	-0.650	65	-0.807			5.0	-0.538	1760	-0.650
158	-0.641	70	-0.749			5.5	-0.551	1770	-0.642
159	-0.633	75	-0.699	0.1	+0.221	6.0	-0.568	1780	-0.633
160	-0.625	80	-0.655	0.2	-0.197	6.5	-0.589	1790	-0.625
161	-0.617	85	-0.617	0.3	-0.617	7.0	-0.617	1800	-0.617
162	-0.609	90	-0.582	0.4	-1.037	7.5	-0.655	1810	-0.609
163	-0.601	95	-0.552	0.5	-1.459	8.0	-0.709	1820	-0.601
164	-0.593	100	-0.524	0.6	-1.881	8.5	-0.796	1830	-0.593
165	-0.585	105	-0.499	0.7	-2.305	9.0	-0.952	1840	-0.585
166	-0.578			0.8	-2.729	9.5	-1.322	1850	-0.577
						10.0	-3.275
Sensitivity = [(-0.609 + 0.625)/2]/ [0.617/161] = 2.09		Sensitivity = [(-0.582 + 0.655)/10]/ [0.617/85] = 1.01		Sensitivity = [(-1.037 + 0.197)/0.2]/ [0.617/0.3] = -2.04		Sensitivity = [(-0.655 + 0.589)/1.0]/ [0.617/7.0] = -0.75		Sensitivity = [(-0.609 + 0.625)/20]/ [0.617/1800] = 2.33
Range = 0.08		Range = 0.31		Range = 2.95		Range = 2.76		Range = 0.08

v_bulletafter	v_lurch	v_cloud	v_lurch	Q	v_lurch	Q_cl	v_lurch	m_frag1	v_lurch
		0	+0.641
		50	+0.431	7	-0.596	20	-1.400	0.022	-0.584
0	-0.467	100	+0.222	8	-0.599	30	-1.298	0.023	-0.590
50	-0.502	150	+0.012	9	-0.603	40	-1.161	0.024	-0.597
100	-0.539	200	-0.197	10	-0.607	50	-0.997	0.025	-0.604
150	-0.577	250	-0.407	11	-0.612	60	-0.813	0.026	-0.610
200	-0.617	300	-0.617	12	-0.617	70	-0.617	0.027	-0.617
250	-0.660	350	-0.826	13	-0.622	80	-0.417	0.028	-0.623
300	-0.707	400	-1.036	14	-0.629	90	-0.221	0.029	-0.630
350	-0.760	450	-1.246	15	-0.635	100	-0.036	0.030	-0.636
400	-0.821	500	-1.455	16	-0.642	110	+0.132	0.031	-0.643
		550	-1.665	17	-0.650	120	+0.278	0.032	-0.650
		600	-1.874
Sensitivity = [(-0.660 + 0.577)/100]/ [0.617/200] = -0.27		Sensitivity = [(-0.826 + 0.407)/100]/ [0.617/300] = -2.04		Sensitivity = [(-0.629 + 0.607)/4]/ [0.617/12] = -0.11		Sensitivity = [(-0.417 + 0.813)/20]/ [0.617/70] = 2.25		Sensitivity = [-0.623 + 0.610)/0.002]/ [0.617/0.027] = -0.28
Range = 0.35		Range = 2.52		Range = 0.05		Range = 1.68		Range = 0.07

m_frags23	v_lurch	Q_frag1	v_lurch	Q_frags23	v_lurch	f_kefrags	v_lurch	f_kebody	v_lurch
0.005	-0.602					1.00	-0.617
0.006	-0.605	20	-0.657	50	-0.642	1.05	-0.617	1.00	-0.617
0.007	-0.608	25	-0.649	55	-0.637	1.10	-0.617	1.05	-0.617
0.008	-0.611	30	-0.640	60	-0.630	1.15	-0.617	1.10	-0.617
0.009	-0.614	35	-0.629	65	-0.624	1.20	-0.617	1.15	-0.617
0.010	-0.617	40	-0.617	70	-0.617	1.25	-0.617	1.20	-0.617
0.011	-0.620	45	-0.603	75	-0.610	1.30	-0.617	1.25	-0.617
0.012	-0.623	50	-0.588	80	-0.602	1.35	-0.617	1.30	-0.617
0.013	-0.626	55	-0.572	85	-0.595	1.40	-0.617	1.35	-0.617
0.015	-0.628	60	-0.555	90	-0.587	1.45	-0.617	1.40	-0.617
0.015	-0.631					1.50	-0.617
Sensitivity = [(-0.620 + 0.614)/0.002]/ [0.617/0.010] = -0.05		Sensitivity = [(-0.603 + 0.629)/10]/ [0.617/40] = 0.17		Sensitivity = [(-0.610 + 0.624)/10]/ [0.617/70] = 0.16		Sensitivity = [(-0.617 + 0.617)/0.50]/ [0.617/1.25] = 0.00		Sensitivity = [(-0.617 + 0.617)/0.40]/ [0.617/1.20] = 0.00
Range = 0.03		Range = 0.10		Range = 0.06		Range = 0.00		Range = 0.00

d_transit	v_lurch	d_snap	v_lurch	d_frag1	v_lurch	t_delay	v_lurch
0	-0.613			2	-0.479
1	-0.614	1.6	-0.540	3	-0.513
2	-0.615	1.8	-0.558	4	-0.548	0.000	-0.594
3	-0.616	2.0	-0.583	5	-0.582	0.001	-0.605
4	-0.617	2.2	-0.617	6	-0.617	0.002	-0.617
5	-0.618	2.4	-0.667	7	-0.651	0.003	-0.630
6	-0.619	2.6	-0.751	8	-0.686	0.004	-0.645
7	-0.620	2.8	-0.915	9	-0.720	0.005	-0.663
8	-0.621			10	-0.755	0.006	-0.683
						0.007	-0.707
Sensitivity = [(-0.621 + 0.613)/8]/ [0.617/4] = -0.01		Sensitivity = [(-0.667 + 0.583)/0.4]/ [0.617/2.2] = -0.75		Sensitivity = [(-0.651 + 0.582)/2]/ [0.617/6] = -0.34		Sensitivity = [(-0.630 + 0.605)/0.002]/ [0.617/0.002] = -0.04
Range = 0.01		Range = 0.38		Range = 0.28		Range = 0.11

The biggest change in this ordered summary of sensitivities is the shift toward negative effects. As opposed to Snap/Lurch 6, which had 12 positive effects and 7 negative, Snap/Lurch 7 has 8 positive and 11 negative effects, for a net shift of four effects from positive to negative. One might then expect the rearward lurch to become stronger, but the opposite happened—it decreased in intensity from -1.04 ft s^-1 to -0.62 ft s^-1, a loss of 40%. As before, however, there was a great dichotomy between the few strong terms (four, with effects between 1.68 ft s^-1 and 2.95 ft s^-1) and the many small terms (14, with effects between 0.00 ft s^-1 and 0.38 ft s^-1). Of the four strong terms, three had to do with the cloud (m_cloud, v_cloud, and Q_cl) and one with the head (m_head). This reinforces the all-importance of the cloud to the final solution.

Summary
Solving for PE instead of v_cloud created four more negative effects but reduced the strength of the negative lurch from -1.04 ft s^-1 to -0.62 ft s^-1. Of the four large effects, three were properties of the cloud, one the mass of the head.

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

Momentum		Energy
Before	After	Before	After
P_bullet = 1.26	P_bulletafter = 0.14	KE_bullet = 1164	KE_bulletafter = 14
	P_bodyafter = -0.82		KE_bodyafter = 0.2
	P_cloud = 1.66		KE_cloud = 422
	P_frag1 = 0.24		KE_frag1 = 70
	P_frags23 = 0.04		KE_frags23 = 26
			PE = 633

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_kefrags	0.00	0.00	Small
Q_frags23	0.16	0.06	Small
m_bullet	2.09	0.08	Small
v_bullet	2.33	0.08	Small
Q_frag1	0.17	0.10	Small
m_body	1.01	0.31	Small
Q_cl	2.25	1.68	Large
Negative effect on lurch (increases rearward velocity)
d_transit	-0.01	0.01	Small
m_frags23	-0.05	0.03	Small
Q	-0.11	0.05	Small
m_frag1	-0.28	0.07	Small
t_delay	-0.04	0.11	Small
d_frag1	-0.34	0.28	Small
v_bulletafter	-0.27	0.35	Small
d_snap	-0.75	0.38	Small
m_head	-0.75	2.76	Large
v_cloud	-2.04	2.52	Large
m_cloud	-2.04	2.95	Large