17. Lurch 2 angular

The model and its justification
Lurch 2 Angular is the rotational analog of Lurch 2 Linear. It calculates the speed of the rearward lurch from the bullet, the body, the diffuse cloud of fragments, and the large fragments. Its result for the lurch, -2.94 ft s^-1, is within 1% of Lurch 1 Angular (-2.92 ft s^-1) and 35% greater than for Lurch 2 Linear (-2.18 ft s^-1).

Solving the simultaneous equations
As with Lurch 2 Linear, the large fragments are represented by six terms, two for masses, two for exit speeds, and two for exit angles.

Conservation of X-momentum

Conservation of total energy

Default values of variables

Solutions to simultaneous equations

Distributions of momentum and energy

Angular momentum		Energy, ft-lb
Before	After	Before	After
Ω_bullet = 3.80	Ω_bulletafter = 0.43	KE_bullet = 1164	KE_bulletafter = 14
	Ω_bodyafter = -8.64		KE_bodyafter = 4
	Ω_cloud = 10.88		KE_cloud = 701
	Ω_frag1 = 0.97		KE_frag1 = 105
	Ω_frags23 = 0.16		KE_frags23 = 39
			PE = 300

The angular momentum and energy are distributed similarly to the linear momentum and energy. The momentum is partitioned roughly equally between the forward movement of the cloud and the rearward movement of the body. By contrast, nearly all the kinetic energy goes to the cloud, except for the 300 units consumed in the potential energy of breaking the skull and scalp.

Sensitivity analysis
The sensitivity analysis for Lurch 2 Angular is very similar to that for Lurch 2 Linear. Only three terms matter much, the mass of the body, the mass of the cloud, and the potential energy. The first and last of these are negatively related to the strength of the lurch, whereas the second is positively related. The latter (the mass of the cloud), is also the strongest.

m_bullet	v_lurch	m_body	v_lurch	m_cloud	v_lurch	PE	v_lurch	v_bullet	v_lurch
						0	-3.660
156	-2.883					50	-3.549	1750	-2.807
157	-2.896	65	-3.849			100	-3.435	1760	-2.835
158	-2.908	70	-3.574			150	-3.317	1770	-2.862
159	-2.919	75	-3.335	0.1	-1.379	100	-3.196	1780	-2.889
160	-2.931	80	-3.127	0.2	-2.264	250	-3.072	1790	-2.916
161	-2.943	85	-2.943	0.3	-2.943	300	-2.943	1800	-2.943
162	-2.955	90	-2.779	0.4	-3.515	350	-2.809	1810	-2.970
163	-2.966	95	-2.633	0.5	-4.018	400	-2.671	1820	-2.996
164	-2.978	100	-2.501	0.6	-4.472	450	-2.526	1830	-3.023
165	-2.989	105	-2.382	0.7	-4.890	500	-2.375	1840	-3.049
166	-3.000			0.8	-5.278	550	-2.216	1850	-3.075
						600	-2.048
Sensitivity = [(-2.955 + 2.931)/2]/ [2.943/161] = -0.66		Sensitivity = [(-2.779 + 3.127)/10]/ [2.943/85] = 1.00		Sensitivity = [(-3.515 + 2.264)/0.2]/ [2.943/0.3] = -0.64		Sensitivity = [(-2.809 + 3.072)/100]/ [2.943/300] = 0.27		Sensitivity = [(-2.970 + 2.916)/20]/ [2.943/1800] = -1.74
Range = 0.12		Range = 1.47		Range = 3.90		Range = 1.61		Range = 0.27

v_bulletafter	v_lurch	Q	v_lurch	m_frag1	v_lurch	m_frags23	v_lurch	v_frag1	v_lurch
		7	-2.924	0.022	-2.933	0.005	-2.967	250	-2.981
0	-2.835	8	-2.927	0.023	-2.935	0.006	-2.962	300	-2.985
50	-2.869	9	-2.930	0.024	-2.937	0.007	-2.957	350	-2.983
100	-2.898	10	-2.934	0.025	-2.939	0.008	-2.952	400	-2.976
150	-2.923	11	-2.938	0.026	-2.941	0.009	-2.948	450	-2.962
200	-2.943	12	-2.943	0.027	-2.943	0.010	-2.943	500	-2.943
250	-2.958	13	-2.948	0.028	-2.945	0.011	-2.938	550	-2.917
300	-2.968	14	-2.953	0.029	-2.947	0.012	-2.933	600	-2.885
350	-2.974	15	-2.959	0.030	-2.949	0.013	-2.928	650	-2.845
400	-2.974	16	-2.965	0.031	-2.950	0.014	-2.924	700	-2.798
		17	-2.972	0.032	-2.952	0.015	-2.919	750	-2.743
Sensitivity = [(-2.958 + 2.923)/100]/ [2.943/200] = -0.02		Sensitivity = [(-2.953 + 2.934)/4]/ [2.943/12] = -0.02		Sensitivity = [(-2.945 + 2.941)/0.002]/ [2.943/0.027] = -0.02		Sensitivity = [(-2.938 + 2.948)/0.002]/ [2.943/0.010] = 0.02		Sensitivity = [(-2.917 + 2.962)/100]/ [2.943/500] = 0.08
Range = 0.14		Range = 0.05		Range = 0.02		Range = 0.05		Range = 0.24

v_frags23	v_lurch	Q_frag1	v_lurch	Q_frags23	v_lurch	f_I	v_lurch
250	-2.992					1.06	-3.081
300	-2.986	20	-3.017	50	-2.991	1.07	-3.053
350	-2.979	25	-3.003	55	-2.980	1.08	-3.024
400	-2.969	30	-2.986	60	-2.968	1.09	-2.997
450	-2.957	35	-2.966	65	-2.956	1.10	-2.970
500	-2.943	40	-2.943	70	-2.943	1.11	-2.943
550	-2.927	45	-2.918	75	-2.930	1.12	-2.917
600	-2.908	50	-2.890	80	-2.916	1.13	-2.891
650	-2.888	55	-2.861	85	-2.903	1.14	-2.866
700	-2.865	60	-2.829	90	-2.889	1.15	-2.841
750	-2.840					1.16	-2.817
Sensitivity = [(-2.927 + 2.957)/100]/ [2.943/500] = 0.05		Sensitivity = [(-2.918 + 2.966)/10]/ [2.943/40] = 0.06		Sensitivity = [(-2.930 + 2.956)/10]/ [2.943/70] = 0.06		Sensitivity = [(-2.917 + 2.970)/0.02]/ [2.943/1.11] = 1.00
Range = 0.15		Range = 0.19		Range = 0.10		Range = 0.26

Adding the large fragments and their six variables does not affect the sensitivity analysis materially, for all six terms have small sensitivities (<0.1) and small effects (<0.25 ft s^-1). All the other terms remain as before, with the body, the cloud, and the potential energy still being the only large terms.

Summary
Adding the large fragments hardly changes the lurch or any of the basic sensitivities. The six new variables all have small sensitivities and effects.

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	f_I = 1.11

Variable	Sensitivity of v_lurch	*Range of v_lurch, ft s^-1*	Magnitude
Positive effect on lurch (reduces rearward velocity)
m_frags23	0.02	0.05	Small
v_frags23	0.05	0.15	Small
Q_frags23	0.06	0.10	Small
Q_frag1	0.06	0.19	Small
v_frag1	0.08	0.24	Small
f_I	1.00	0.26	Small
m_body	1.00	1.47	Large
PE	0.27	1.61	Large
Negative effect on lurch (increases rearward velocity)
m_frag1	-0.02	0.02	Small
Q	-0.02	0.05	Small
m_bullet	-0.66	0.12	Small
v_bulletafter	-0.02	0.14	Small
v_bullet	-1.74	0.27	Small
m_cloud	-0.64	3.90	Large