14. Lurch 6 linear

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: 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 former procedure 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.
    Including these time intervals leaves the default solution about the same: -1.04 and 401 ft s^-1 for lurch and cloud, respectively, versus -1.02 and 378 ft s^-1 in Lurch 5 Linear. The sensitivities and effects change, however, presumably to better reflect the true situation.

Conservation of X-momentum

Conservation of total energy

Solutions to simultaneous equations

v_bodyafter = -1.04 ft s^-1 v_cloud = 401 ft s^-1
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	-1.017
156	-1.025					4.5	-1.021	1750	-0.993
157	-1.029	65	-1.362			5.0	-1.026	1760	-1.003
158	-1.032	70	-1.265			5.5	-1.031	1770	-1.012
159	-1.035	75	-1.180	0.1	-0.329	6.0	-1.035	1780	-1.022
160	-1.038	80	-1.106	0.2	-0.731	6.5	-1.039	1790	-1.031
161	-1.041	85	-1.041	0.3	-1.041	7.0	-1.041	1800	-1.041
162	-1.043	90	-0.983	0.4	-1.302	7.5	-1.035	1810	-1.050
163	-1.046	95	-0.931	0.5	-1.533	8.0	-1.012	1820	-1.059
164	-1.048	100	-0.884	0.6	-1.742	8.5	-0.931	1830	-1.068
165	-1.051	105	-0.842	0.7	-1.935	9.0	-0.548	1840	-1.076
166	-1.053			0.8	-2.115	9.5	—	1850	-1.085
						10.0	—
Sensitivity = [(-1.043 + 1.038)/2]/ [1.041/161] = -0.39		Sensitivity = [(-0.983 + 1.106)/10]/ [1.041/85] = 1.00		Sensitivity = [(-1.302 + 0.731)/0.2]/ [1.041/0.3] = -0.82		Sensitivity = [(-1.012 + 1.035)/2]/ [1.041/7.0] = 0.08		Sensitivity = [(-1.050 + 1.031)/20]/ [1.041/1800] = -1.64
Range = 0.03		Range = 0.52		Range = 1.79		Range = 0.47		Range = 0.09

v_bulletafter	v_lurch	PE	v_lurch	Q	v_lurch	Q_cl	v_lurch	m_frag1	v_lurch
		0	-1.347
		50	-1.299	7	-1.027	20	-2.086	0.022	-1.022
0	-0.944	100	-1.250	8	-1.029	30	-1.949	0.023	-1.026
50	-0.972	150	-1.200	9	-1.031	40	-1.767	0.024	-1.029
100	-0.997	200	-1.149	10	-1.034	50	-1.549	0.025	-1.033
150	-1.020	250	-1.095	11	-1.037	60	-1.303	0.026	-1.037
200	-1.041	300	-1.041	12	-1.041	70	-1.041	0.027	-1.041
250	-1.057	350	-0.984	13	-1.044	80	-0.773	0.028	-1.044
300	-1.070	400	-0.925	14	-1.048	90	-0.511	0.029	-1.048
350	-1.074	450	-0.864	15	-1.052	100	-0.264	0.030	-1.052
400	-1.066	500	-0.801	16	-1.057	110	-0.040	0.031	-1.055
		550	-0.734	17	-1.061	120	+0.155	0.032	-1.059
		600	-0.664
Sensitivity = [(-1.057 + 1.020)/100]/ [1.041/200] = -0.07		Sensitivity = [(-0.984 + 1.095)/100]/ [1.041/300] = 0.32		Sensitivity = [(-1.048 + 1.034)/4]/ [1.041/12] = -0.04		Sensitivity = [(-0.773 + 1.303)/20]/ [1.041/70] = 1.78		Sensitivity = [-1.048 + 1.033)/0.004]/ [1.041/0.027] = -0.10
Range = 0.12		Range = 0.68		Range = 0.03		Range = 2.24		Range = 0.04

m_frags23	v_lurch	Q_frag1	v_lurch	Q_frags23	v_lurch	f_kefrags	v_lurch	f_kebody	v_lurch
0.005	-1.040					1.00	-1.062
0.006	-1.040	20	-1.081	50	-1.066	1.05	-1.058	1.00	-1.041
0.007	-1.040	25	-1.073	55	-1.060	1.10	-1.053	1.05	-1.041
0.008	-1.040	30	-1.064	60	-1.054	1.15	-1.049	1.10	-1.041
0.009	-1.041	35	-1.053	65	-1.048	1.20	-1.045	1.15	-1.041
0.010	-1.041	40	-1.041	70	-1.041	1.25	-1.041	1.20	-1.041
0.011	-1.041	45	-1.027	75	-1.033	1.30	-1.036	1.25	-1.041
0.012	-1.041	50	-1.012	80	-1.026	1.35	-1.032	1.30	-1.041
0.013	-1.041	55	-0.996	85	-1.019	1.40	-1.028	1.35	-1.041
0.015	-1.041	60	-0.979	90	-1.011	1.45	-1.024	1.40	-1.041
0.015	-1.041					1.50	-1.019
Sensitivity = [(-1.041 + 1.040)/0.010]/ [1.041/0.010] = -0.00		Sensitivity = [(-1.027 + 1.053)/10]/ [1.041/40] = 0.10		Sensitivity = [(-1.033 + 1.048)/10]/ [1.041/70] = 0.10		Sensitivity = [(-1.036 + 1.045)/0.10]/ [1.041/1.25] = 0.11		Sensitivity = [(-1.041 + 1.041)/0.40]/ [1.041/1.20] = 0.00
Range = 0.00		Range = 0.10		Range = 0.06		Range = 0.04		Range = 0.00

d_transit	v_lurch	d_snap	v_lurch	d_frag1	v_lurch	t_delay	v_lurch
0	-1.041			2	-0.995
1	-1.041	1.6	-1.027	3	-1.015
2	-1.041	1.8	-1.033	4	-1.030	0.000	-1.040
3	-1.041	2.0	-1.038	5	-1.038	0.001	-1.041
4	-1.041	2.2	-1.041	6	-1.041	0.002	-1.041
5	-1.041	2.4	-1.032	7	-1.036	0.003	-1.040
6	-1.041	2.6	-0.981	8	-1.025	0.004	-1.037
7	-1.040	2.8	-0.684	9	-1.005	0.005	-1.033
8	-1.040			10	-0.977	0.006	-1.026
						0.007	-1.014
Sensitivity = [(-1.040 + 1.041)/8]/ [1.041/4] = 0.00		Sensitivity = [(-1.032 + 1.038)/0.4]/ [1.041/2.2] = 0.03		Sensitivity = [(-1.025 + 1.030)/4]/ [1.041/6] = 0.01		Sensitivity = [(-1.037 + 1.040)/0.004]/ [1.041/0.002] = 0.00
Range = 0.00		Range = 0.34		Range = 0.02		Range = 0.03

This ordered summary of sensitivities is strikingly similar to the previous one, given the differences in calculation introduced here. All the previous variables maintained very similar sensitivities and ranges, with Q_cl and m_cloud continuing to dominate. The five new variables related to the timing and the snap all had positive sensitivities, that is, they worked to increase the velocity of the lurch (to reduce its rearward velocity). In spite of this, the lurch hardly changed, and if anything, increased in rearward speed (from -1.02 ft s^-1 to -1.04 ft s^-1). This tiny effect is probably because the sensitivities and effects of the new variables were so small—three were negligible (0.00–0.03 ft s^-1), and two were small to medium (0.34 and 0.47 ft s^-1). Thus, adding the four time intervals did not change the basic result.

Summary
The basic solution remained virtually unchanged when the four time intervals were added: The lurch of -1.02 ft s^-1 in Lurch 5 Linear increased (in magnitude) only to -1.04 ft s^-1 in Snap/Lurch 6 Linear. This important result was probably a consequence of the small sensitivities and effects of the five new 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_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
PE = 300 ft-lb	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 = -1.38		KE_bodyafter = 0.4
	P_cloud = 2.22		KE_cloud = 754
	P_frag1 = 0.24		KE_frag1 = 70
	P_frags23 = 0.04		KE_frags23 = 26
			PE = 300

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
d_transit	0.00	0.00	Small
d_frag1	0.01	0.02	Small
t_delay	0.00	0.03	Small
f_kefrag1	0.11	0.04	Small
Q_frags23	0.10	0.06	Small
Q_frag1	0.10	0.10	Small
d_snap	0.03	0.34	Small
m_head	0.08	0.47	Medium
m_body	1.00	0.52	Medium
PE	0.32	0.68	Medium
Q_cl	1.78	2.24	Large
Negative effect on lurch (increases rearward velocity)
m_frags23	-0.00	0.00	Small
m_bullet	-0.39	0.03	Small
Q	-0.04	0.03	Small
m_frag1	-0.10	0.04	Small
v_bullet	-1.64	0.09	Small
v_bulletafter	-0.07	0.12	Small
m_cloud	-0.82	1.79	Large