13. Lurch 5 linear

The model and its justification
Having added 3-D motion to the large fragments in Lurch 4 Linear, we now try the same for the body. We can neglect the component of the momentum into or out of the plane of the Z-film because are dealing only with X-momentum. But we cannot neglect that component of the kinetic energy, because the equation for conservation of total energy uses total kinetic energy. As happened when adding the 3-D component of the fragments, the 3-D component of the body's KE hardly changes the answer at all. In fact, the previous -1.02 ft s^-1 for the default values of variables remains the same here. This the 3-D motion of the body is an insignificant term in the overall equation.
The equations calculate the speed of the rearward lurch of the upper body from the bullet, the body (in 3-D), the cloud (in 3-D), and the large fragments (also in 3-D).

Solving the simultaneous equations
The three-dimensionality of the body's kinetic energy is expressed by the term f_kebody, which is similar in form and function to f_kefrag. We assume a range of 1–1.4 and a starting value of 1.2.

Conservation of X-momentum

Conservation of total energy

Default solutions to simultaneous equations

Distributions of momentum and energy

Sensitivity analysis
This sensitivity analysis is practically identical to the one for Lurch 4 Linear. The new term, f_kebody, has undetectable sensitivity and range of effect.

m_bullet	v_lurch	m_body	v_lurch	m_cloud	v_lurch	PE	v_lurch	v_bullet	v_lurch
						0	-1.343
156	-1.004					50	-1.293	1750	-0.970
157	-1.007	65	-1.336			100	-1.242	1760	-0.980
158	-1.011	70	-1.240			150	-1.189	1770	-0.990
159	-1.014	75	-1.157	0.1	-0.351	200	-1.135	1780	-1.000
160	-1.017	80	-1.085	0.2	-0.729	250	-1.079	1790	-1.011
161	-1.021	85	-1.021	0.3	-1.021	300	-1.021	1800	-1.021
162	-1.024	90	-0.964	0.4	-1.267	350	-0.960	1810	-1.030
163	-1.027	95	-0.913	0.5	-1.484	400	-0.897	1820	-1.040
164	-1.030	100	-0.867	0.6	-1.681	450	-0.832	1830	-1.050
165	-1.033	105	-0.826	0.7	-1.863	500	-0.763	1840	-1.060
166	-1.036			0.8	-2.033	550	-0.690	1850	-1.070
						600	-0.613
Sensitivity = [(-1.024 + 1.017)/2]/ [1.021/161] = -0.55		Sensitivity = [(-0.964 + 1.085)/10]/ [1.021/85] = 1.01		Sensitivity = [(-1.267 + 0.729)/0.2]/ [1.021/0.3] = -0.79		Sensitivity = [(-0.960 + 1.079)/100]/ [1.021/300] = 0.35		Sensitivity = [(-1.030 + 1.011)/20]/ [1.021/1800] = -1.68
Range = 0.03		Range = 0.51		Range = 1.68		Range = 0.73		Range = 0.10

v_bulletafter	v_lurch	Q	v_lurch	Q_cl	v_lurch	m_frag1	v_lurch	m_frags23	v_lurch
		7	-1.006	20	-2.005	0.022	-1.004	0.005	-1.029
0	-0.929	8	-1.009	30	-1.876	0.023	-1.007	0.006	-1.027
50	-0.955	9	-1.011	40	-1.705	0.024	-1.011	0.007	-1.026
100	-0.979	10	-1.014	50	-1.499	0.025	-1.014	0.008	-1.024
150	-1.001	11	-1.017	60	-1.267	0.026	-1.017	0.009	-1.022
200	-1.021	12	-1.021	70	-1.021	0.027	-1.021	0.010	-1.021
250	-1.038	13	-1.024	80	-0.769	0.028	-1.024	0.011	-1.019
300	-1.053	14	-1.028	90	-0.522	0.029	-1.027	0.012	-1.017
350	-1.066	15	-1.033	100	-0.289	0.030	-1.030	0.013	-1.015
400	-1.077	16	-1.037	110	-0.078	0.031	-1.033	0.015	-1.013
		17	-1.042	120	+0.105	0.032	-1.037	0.015	-1.012
Sensitivity = [(-1.038 + 1.001)/100]/ [1.021/200] = -0.07		Sensitivity = [(-1.028 + 1.014)/4]/ [1.021/12] = -0.04		Sensitivity = [(-0.769 + 1.267)/20]/ [1.021/70] = 1.71		Sensitivity = [-1.027 + 1.014)/0.004]/ [1.021/0.027] = -0.09		Sensitivity = [(-1.017 + 1.024)/0.004]/ [1.021/0.010] = 0.02
Range = 0.15		Range = 0.04		Range = 2.11		Range = 0.03		Range = 0.02

v_frag1	v_lurch	v_frags23	v_lurch	Q_frag1	v_lurch	Q_frags23	v_lurch
250	-1.011	250	-1.043
300	-1.020	300	-1.041	20	-1.076	50	-1.056
350	-1.025	350	-1.038	25	-1.065	55	-1.048
400	-1.027	400	-1.033	30	-1.052	60	-1.039
450	-1.025	450	-1.027	35	-1.037	65	-1.030
500	-1.021	500	-1.021	40	-1.021	70	-1.021
550	-1.012	550	-1.012	45	-1.002	75	-1.011
600	-0.999	600	-1.003	50	-0.981	80	-1.001
650	-0.982	650	-0.992	55	-0.959	85	-0.990
700	-0.961	700	-0.980	60	-0.936	90	-0.980
700	-0.934	750	-0.967
Sensitivity = [(-1.012 + 1.025)/100]/ [1.021/500] = 0.06		Sensitivity = [(-1.012 + 1.027)/100]/ [1.021/500] = 0.07		Sensitivity = [(-1.002 + 1.037)/10]/ [1.021/40] = 0.14		Sensitivity = [(-1.011 + 1.030)/10]/ [1.021/70] = 0.13
Range = 0.08		Range = 0.08		Range = 0.14		Range = 0.08

f_kefrag1	v_lurch	f_kefrags23	v_lurch	f_kebody	v_lurch
1.00	-1.051	1.00	-1.032
1.05	-1.045	1.05	-1.030	1.00	-1.021
1.10	-1.039	1.10	-1.027	1.05	-1.021
1.15	-1.033	1.15	-1.025	1.10	-1.021
1.20	-1.027	1.20	-1.023	1.15	-1.021
1.25	-1.021	1.25	-1.021	1.20	-1.021
1.30	-1.014	1.30	-1.018	1.25	-1.021
1.35	-1.008	1.35	-1.016	1.30	-1.020
1.40	-1.002	1.40	-1.014	1.35	-1.020
1.45	-0.995	1.45	-1.011	1.40	-1.020
1.50	-0.989	1.50	-1.009
Sensitivity = [(-1.014 + 1.027)/0.10]/ [1.021/1.25] = 0.16		Sensitivity = [(-1.018 + 1.023)/0.10]/ [1.021/1.25] = 0.06		Sensitivity = [(-1.020 + 1.021)/0.4]/ [1.021/1.20] = 0.00
Range = 0.06		Range = 0.02		Range = 0.00

These sensitivities and effects are virtually the same as for Lurch 4 Linear. The new term, f_kebody, has undetectable sensitivity and range of effect, i.e., counts for nothing. This is a consequence of the fact that the body gets virtually none of the final kinetic energy.

Summary
Adding three-dimensionality to the body's KE does not change the solution perceptibly, presumably because the body gets almost none of the KE after the explosion (<0.1%). This is confirmed by the undetectable sensitivity and range of effect for f_kebody. The two variables with the largest effects remain Q_cl (2.11 ft s^-1) and m_cloud (1.68 ft s^-1). Only two variables have medium effects, PE (0.73 ft s^-1) and m_body (0.51 ft s^-1). Thus the cloud remains all-important to the solution.

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°	v_frag1 = 500 ft s^-1
Q = 12°	Q_frags23 = 70°	v_frags23 = 500 ft s^-1
PE = 300 ft-lb	Q_cl = 70°	f_kefrag1 = 1.25
f_kebody = 1.2	f_xcl = 0.59 (calc.)	f_kefrags23 = 1.25

Momentum		Energy, ft-lb
Before	After	Before	After
P_bullet = 1.26	P_bulletafter = 0.14	KE_bullet = 1164	KE_bulletafter = 14
	P_bodyafter = -1.35		KE_bodyafter = 0.4
	P_cloud = 2.10		KE_cloud = 669
	P_frag1 = 0.32		KE_frag1 = 132
	P_frags23 = 0.05		KE_frags23 = 49
			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
m_frags23	0.02	0.02	Small
f_kefrags23	0.06	0.02	Small
f_kefrag1	0.16	0.06	Small
v_frag1	0.06	0.08	Small
v_frags23	0.07	0.08	Small
Q_frags23	0.13	0.08	Small
Q_frag1	0.14	0.14	Small
m_body	1.01	0.51	Medium
PE	0.35	0.73	Medium
Q_cl	1.71	2.11	Large
Negative effect on lurch (increases rearward velocity)
m_bullet	-0.55	0.03	Small
m_frag1	-0.09	0.03	Small
Q	-0.04	0.04	Small
v_bullet	-1.68	0.10	Small
v_bulletafter	-0.07	0.15	Small
m_cloud	-0.79	1.68	Large