17--L2A

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, -3.29 ft s^-1, is 10% less than from Lurch 1 Angular (-3.53 ft s^-1) and 50% greater than for Lurch 2 Linear (-2.44 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 = 3.71	Ω_bulletafter = 0.43	KE_bullet = 1164	KE_bulletafter = 14
	Ω_bodyafter = -9.67		KE_bodyafter = 5
	Ω_cloud = 12.27		KE_cloud = 893
	Ω_frag1 = 0.58		KE_frag1 = 38
	Ω_frags23 = 0.10		KE_frags23 = 14
			PE = 200

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 200 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
156	-3.24							1750	-3.18
157	-3.25	65	-4.31			0	-3.73	1760	-3.20
158	-3.26	70	-4.00			50	-3.63	1770	-3.22
159	-3.27	75	-3.73	0.1	-1.13	100	-3.52	1780	-3.25
160	-3.28	80	-3.50	0.2	-2.53	150	-3.41	1790	-3.27
161	-3.29	85	-3.29	0.3	-3.29	200	-3.29	1800	-3.29
162	-3.30	90	-3.11	0.4	-3.94	250	-3.18	1810	-3.32
163	-3.31	95	-2.95	0.5	-4.51	300	-3.05	1820	-3.34
164	-3.32	100	-2.80	0.6	-5.02	350	-2.93	1830	-3.36
165	-3.33	105	-2.67	0.7	-5.49	400	-2.80	1840	-3.38
166	-3.34			0.8	-5.93			1850	-3.41
Sensitivity = [(-3.30 + 3.28)/2]/ [3.29/161] = -0.49		Sensitivity = [(-3.11 + 3.50)/10]/ [3.29/85] = 1.01		Sensitivity = [(-3.94 + 2.53)/0.2]/ [3.29/0.3] = -0.64		Sensitivity = [(-3.18 + 3.41)/100]/ [3.29/200] = 0.14		Sensitivity = [(-3.34 + 3.25)/40]/ [3.29/1800] = -1.23
Range = 0.10		Range = 1.64		Range = 4.80		Range = 0.93		Range = 0.23

v_bulletafter	v_lurch	Q	v_lurch	m_frag1	v_lurch	m_frags23	v_lurch	v_frag1	v_lurch
		12	-3.26	0.022	-3.27	0.005	-3.29
0	-3.18	13	-3.27	0.023	-3.28	0.006	-3.29	100	-3.24
50	-3.22	14	-3.28	0.024	-3.28	0.007	-3.29	150	-3.26
100	-3.24	15	-3.28	0.025	-3.29	0.008	-3.29	200	-3.28
150	-3.27	16	-3.29	0.026	-3.29	0.009	-3.29	250	-3.29
200	-3.29	17	-3.29	0.027	-3.29	0.010	-3.29	300	-3.29
250	-3.31	18	-3.30	0.028	-3.30	0.011	-3.29	350	-3.29
300	-3.32	19	-3.31	0.029	-3.30	0.012	-3.29	400	-3.29
350	-3.33	20	-3.32	0.030	-3.30	0.013	-3.29	450	-3.28
400	-3.34	21	-3.32	0.031	-3.31	0.014	-3.29	500	-3.26
		22	-3.33	0.032	-3.31	0.015	-3.29
Sensitivity = [(-3.31 + 3.27)/100]/ [3.29/200] = -0.024		Sensitivity = [(-3.32 + 3.28)/6]/ [3.29/17] = -0.034		Sensitivity = [(-3.31 + 3.27)/0.01]/ [3.29/0.027] = -0.033		Sensitivity = [(-3.29 + 3.29)/0.01]/ [3.29/0.01] = 0.00		Sensitivity = [(-3.26 + 3.24)/400]/ [3.29/300] = -0.0046
Range = 0.16		Range = 0.07		Range = 0.04		Range = 0.00		Range = 0.02

v_frags23	v_lurch	Q_frag1	v_lurch	Q_frags23	v_lurch

100	-3.30	20	-3.34	50	-3.32
150	-3.30	25	-3.33	55	-3.32
200	-3.30	30	-3.32	60	-3.31
250	-3.30	35	-3.31	65	-3.30
300	-3.29	40	-3.29	70	-3.29
350	-3.29	45	-3.28	75	-3.28
400	-3.28	50	-3.26	80	-3.28
450	-3.27	55	-3.24	85	-3.27
500	-3.26	60	-3.22	90	-3.26
Sensitivity = [(-3.28 + 3.30)/200]/ [3.29/300] = 0.0091		Sensitivity = [(-3.26 + 3.32)/20]/ [3.29/40] = 0.036		Sensitivity = [(-3.26 + 3.32)/40]/ [3.29/70] = 0.032
Sensitivity = [(-3.28 + 3.30)/200]/ [3.29/300] = 0.0091		Sensitivity = [(-3.26 + 3.32)/20]/ [3.29/40] = 0.036		Sensitivity = [(-3.26 + 3.32)/40]/ [3.29/70] = 0.032
Range = 0.04		Range = 0.12		Range = 0.06

Adding the large fragments and their four associated variables does not affect the sensitivity analysis much, for their six terms all have small sensitivities and small effects. The body, the cloud, and the potential energy are the things that really matter.

Summary
Adding the large fragments decreases the intensity of the lurch by 10%, but hardly changes any of the basic sensitivities.

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 = 300 ft s-1
Q = 17°	Q_frags23 = 70°	v_frags23 = 300 ft s-1
PE = 200 ft-lb

Variable	Sensitivity of v_lurch	*Range of v_lurch, ft s^-1*	Magnitude
Positive effect on lurch (reduces rearward velocity)
v_frags23	0.091	0.04	Small
Q_frags23	0.032	0.06	Small
Q_frag1	0.036	0.12	Small
PE	0.14	0.93	Medium
m_body	1.01	1.64	Medium
Negative effect on lurch (increases rearward velocity)
m_frags23	-0.000	0.00	Small
v_frag1	-0.0046	0.02	Small
m_frag1	-0.033	0.04	Small
Q	-0.034	0.07	Small
m_bullet	-0.49	0.10	Small
v_bulletafter	-0.024	0.16	Small
v_bullet	-1.23	0.23	Small
m_cloud	-0.64	4.80	Large