14--L6L

The model and its justification
This version of the simulation tries to directly get values for m_cloud, one of the hardest variables to estimate, by solving for it directly. In order to do that, some other variable must take on an assumed value, because only two variables can be solved for by the two equations. Since v_cloud was coming out to values very similar to those assumed for v_frags, I decided to set v_frags23 = v_frag1 and v_frags (really v_frag1) = v_cloud in order to free up a degree of freedom that could then be used to solve for m_cloud. I was gratified to see that the value of m_cloud, at least for the default conditions, came out very similar to the default value I had previously taken (0.32 lb vs. 0.30 lb previously used). Of course, the other variable solved for remained v_bodyafter. Just to be clear, this version of the equations calculated the speed of the rearward lurch of JFK's upper body from the bullet, the body, the cloud, and the large fragments, with 3-D motion of cloud, large fragments, and body.

Solving the simultaneous equations
Since no new variables were introduced here, the form of the two equations remained the same as in Lurch 5 Linear. Since the assumed value for v_cloud was very close to the value solved for there, the default solution for v_bodyafter (-0.90 ft s^-1) was very similar to the previous value of -0.87 ft s^-1.

Conservation of X-momentum

Conservation of total energy

Default solutions to simultaneous equations

Default distributions of momentum and energy

Sensitivity analysis
Linking the speed of the cloud to the speed of the fragments made the speed of the lurch considerably more sensitive to the mass of the bullet (7 times) and to the potential energy (2 times) and the initial speed of the bullet (3 times). Sensitivities to the other variables remained about the same.

m_bullet	v_lurch	m_body	v_lurch	PE	v_lurch	v_bullet	v_lurch	v_bulletafter	v_lurch
156	-0.867					1750	-0.818
157	-0.875	65	-1.184	0	-1.258	1760	-0.835	0	-0.822
158	-0.882	70	-1.099	50	-1.17	1770	-0.852	50	-0.847
159	-0.89	75	-1.026	100	-1.081	1780	-0.87	100	-0.87
160	-0.897	80	-0.962	150	-0.993	1790	-0.887	150	-0.889
161	-0.905	85	-0.905	200	-0.905	1800	-0.905	200	-0.905
162	-0.912	90	-0.854	250	-0.816	1810	-0.922	250	-0.918
163	-0.92	95	-0.809	300	-0.728	1820	-0.94	300	-0.927
164	-0.927	100	-0.769	350	-0.64	1830	-0.958	350	-0.934
165	-0.935	105	-0.732	400	-0.552	1840	-0.976	400	-0.937
166	-0.942					1850	-0.995
Sensitivity = [(-0.912 + 0.897)/2]/ [0.905/161] = -1.33		Sensitivity = [(-0.854 + 0.962)/10]/ [0.905/85] = 1.01		Sensitivity = [(-0.816 + 0.993)/100]/ [0.905/200] = 0.39		Sensitivity = [(-0.922 + 0.887)/20]/ [0.905/1800] = -3.48		Sensitivity = [(-0.918 + 0.889)/100]/ [0.905/200] = -0.064
Range = 0.075		Range = 0.45		Range = 0.71		Range = 0.18		Range = 0.12

v_cloud	v_lurch	Q	v_lurch	m_frag1	v_lurch	m_frags23	v_lurch	Q_frag1	v_lurch
0	-
50	-10.231	12	-0.883	0.022	-0.893	0.005	-0.908
100	-4.319	13	-0.887	0.023	-0.895	0.006	-0.907	20	-0.938
150	-2.575	14	-0.891	0.024	-0.898	0.007	-0.907	25	-0.932
200	-1.73	15	-0.895	0.025	-0.9	0.008	-0.906	30	-0.924
250	-1.232	16	-0.9	0.026	-0.902	0.009	-0.905	35	-0.915
300	-0.905	17	-0.905	0.027	-0.905	0.01	-0.905	40	-0.905
350	-0.674	18	-0.91	0.028	-0.907	0.011	-0.904	45	-0.893
400	-0.504	19	-0.915	0.029	-0.909	0.012	-0.903	50	-0.881
450	-0.374	20	-0.921	0.03	-0.912	0.013	-0.903	55	-0.868
500	-0.272	21	-0.927	0.031	-0.914	0.014	-0.902	60	-0.854
550	-0.19	22	-0.933	0.032	-0.916	0.015	-0.901
600	-0.123
Sensitivity = [(-0.674 + 1.232)/100]/ [0.905/300] = 1.85		Sensitivity = [(-0.91 + 0.9)/2]/ [0.905/17] = -0.094		Sensitivity = [(-0.907 + 0.902)/0.002]/ [0.905/0.027] = -0.075		Sensitivity = [(-0.904 + 0.905)/0.002]/ [0.905/0.01] = 0.006		Sensitivity = [(-0.893 + 0.915)/10]/ [0.905/40] = 0.097
Range = 4.20		Range = 0.05		Range = 0.023		Range = 0.007		Range = 0.084

Q_frags23	v_lurch	f_xcl	v_lurch	f_kecl	v_lurch	f_kefrag1	v_lurch	f_kebody	v_lurch
50	-0.926			1	-2.467
55	-0.921	0.4	-0.236	1.25	-1.842
60	-0.916	0.5	-0.459	1.5	-1.426	1	-0.928	1.0	-0.905
65	-0.91	0.6	-0.682	1.75	-1.128	1.1	-0.919	1.1	-0.905
70	-0.905	0.7	-0.905	2	-0.905	1.25	-0.905	1.2	-0.905
75	-0.899	0.8	-1.127	2.25	-0.731	1.4	-0.891	1.3	-0.905
80	-0.893	0.9	-1.35	2.5	-0.592	1.5	-0.882	1.4	-0.905
85	-0.887	1.0	-1.572	2.75	-0.479
90	-0.881			3	-0.384
Sensitivity = [(-0.899 + 0.91)/10]/ [0.905/70] = 0.085		Sensitivity = [(-1.127 + 0.682)/0.2]/ [0.905/0.7] = -1.72		Sensitivity = [(-0.731 + 1.128)/0.5]/ [0.905/2] = 1.75		Sensitivity = [(-0.891 + 0.919)/0.3]/ [0.905/1.25] = 0.129		Sensitivity = [(-0.905 + 0.905)/0.2]/ [0.905/1.2] = 0.000
Range = 0.045		Range = 1.34		Range = 2.08		Range = 0.046		Range = 0.000

The sensitivities of the variables in Lurch 6 Linear are very similar to those in Lurch 5 Linear, although with some minor rearrangements attributable to equating some of the variables and solving for m_cloud instead of v_cloud. The most important variables remained those of the diffuse cloud.

Summary
Linking the speed of the cloud to the speed of the fragments allowed us to solve for the mass of the cloud, which was nearly the same as that previously assumed. The default speed of the lurch was also very similar to previous calculations. Sensitivities to the incoming bullet increased by factors of two to seven, however. As in previous cases, the greatest sensitivities and effects were to properties of the diffuse cloud such as its speed and its spatial properties.

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	v_cloud = 300	f_kefrag1 = 1.25
Q = 17°	Q_frag1 = 40°	f_kebody = 1.2
PE = 200 ft-lb	Q_frags23 = 70°	f_xcl = 0.7
		f_kecl = 2

Momentum		Energy, ft-lb
Before	After	Before	After
P_bullet = 1.24	P_bulletafter = 0.14	KE_bullet = 1164	KE_bulletafter = 14
	P_bodyafter = -1.20		KE_bodyafter = 0.3
	P_cloud = 2.06		KE_cloud = 885
	P_frag1 = 0.19		KE_frag1 = 47
	P_frags23 = 0.03		KE_frags23 = 18
			PE = 200

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	0.006	0.03	Small
Q_frags23	0.085	0.04	Small
f_kefrag1	0.129	0.05	Small
Q_frag1	0.097	0.08	Small
m_body	1.01	0.45	Small
PE	0.39	0.71	Medium
f_kecl	1.75	2.08	Large
v_cloud	1.85	4.20	Large
Negative effect on lurch (increases rearward velocity)
m_bullet	-0.13	0.01	Small
m_frag1	-0.075	0.02	Small
Q	-0.094	0.05	Small
v_bulletafter	-0.064	0.12	Small
v_bullet	-3.48	0.18	Small
f_xcl	-1.72	1.34	Large