25a--Mhead = 5

    We begin with potential energy because it offers the greatest constraints on the solution. The PE may absolutely not be zero or negative. It may also not exceed some high value, which here is taken to be 600 ft-lb, double the conservative (high) estimate of 300 ft-lb from Dr. Steve Cogswell of AFIP.
    The most important feature of the result for PE and m_head = 5 lb is that the light head requires a very high PE in order to balance the equation, so high as to exceed the threshold of 600 ft-lb in nearly all of the cases (78 of 81, or 96%). The disallowed values are shown in boldface. The three allowed cases, all borderline 510–570 ft-lb), fall in the upper right. In essence, a head weighing 5 lb is too light to allow any real solutions.
    The pattern of disallowed values from the PE form a sort of mask that can be applied to the tables for the other variables. This is done below.

R_bullet, in ↓ R_head, in →	*3.50*	*3.75*	*4.00*	*4.25*	*4.50*	*4.75*	*5.00*	*5.25*	*5.50*
*4.75*	689	683	675	664	651	632	606	568	507
*5.00*	693	686	681	673	663	650	632	608	572
*5.25*	696	691	686	680	672	662	649	632	609
*5.50*	698	694	690	685	679	671	661	649	632
*5.75*	700	697	693	689	684	677	670	660	648
*6.00*	701	699	696	692	686	682	676	669	659
*6.25*	703	700	698	695	691	686	681	675	668
*6.50*	704	702	699	697	693	690	685	680	674
*6.75*	705	703	701	698	696	692	689	684	679

The velocity of the fragments is considered next because it has the potential to provide additional constraints on the other variables—values of v_frags that are <0 or >1100 ft s^-1 are disallowed. In this case, however, all values fell within the acceptable limits, so that the fragments did not provide additional constraints. The masking shown below comes from the PE. The allowed values for v_frags ranged from 460 to 550 ft s^-1.

R_bullet, in ↓ R_head, in →	*3.50*	*3.75*	*4.00*	*4.25*	*4.50*	*4.75*	*5.00*	*5.25*	*5.50*
*4.75*	231	250	271	296	326	363	410	471	552
*5.00*	220	236	254	274	299	328	363	407	464
*5.25*	211	224	240	257	277	301	329	363	405
*5.50*	203	215	228	244	261	281	304	331	363
*5.75*	197	207	219	232	247	264	283	306	332
*6.00*	191	201	211	223	236	250	267	286	308
*6.25*	186	195	204	215	226	239	254	270	288
*6.50*	182	190	198	208	218	230	242	257	273
*6.75*	178	185	193	202	211	221	233	245	259

The third constraining variable is the time span of the lurch, t_lurch. By this we mean the beginning part of the lurch, while the body is being accelerated backward by the force of the particles of cloud being accelerated forward. The key to this constraint is that the initial lurch is limited to the time that frame 313 is open, 1/40 s, because the lurch began after 313 opened and ended before it closed. In other words, t_lurch < 25 ms. In actuality, it is probably considerably less than this, but the exact amount less is difficult or impossible to determine accurately. In general, t_lurch constrains the other variables less than PE and v_frags do, but there are circumstances where it is the only constraint.
The constraints provided by the lurch are shown in the table below. As with the other constraints, they form a diagonal pattern, with the disallowed values in the lower left. Even though about half the values are disallowed, that is considerably less stringent than for the PE shown above (>95% disallowed).

R_bullet, in ↓ R_head, in →	*3.50*	*3.75*	*4.00*	*4.25*	*4.50*	*4.75*	*5.00*	*5.25*	*5.50*
*4.75*	25.9	24.0	27.2	20.3	18.4	16.5	14.6	12.8	10.9
*5.00*	27.2	25.5	23.7	21.9	20.1	18.3	16.5	14.7	12.9
*5.25*	28.4	26.7	25.0	23.3	21.6	19.9	18.2	16.5	14.8
*5.50*	29.5	27.9	26.3	24.6	23.0	21.4	19.8	18.1	16.5
*5.75*	30.5	29.0	27.4	25.8	24.3	22.7	21.2	19.6	18.1
*6.00*	31.4	29.9	28.4	26.9	25.5	24.0	22.5	21.0	19.5
*6.25*	32.2	30.8	29.4	28.0	26.5	25.1	23.7	22.2	20.8
*6.50*	33.0	31.6	30.3	28.9	27.5	26.1	24.8	23.4	22.0
*6.75*	33.7	32.4	31.1	29.8	28.4	27.1	25.8	24.5	23.1

The high PE keeps the velocity of the lurch within narrow limits, 0.7–1.0 ft s^-1 in particular. Most of these are disallowed. The only ones that remain, three values in the upper right, range from -0.9 to -1.0 ft s^-1. These values are close to the observed -0.8 ft s^-1.

R_bullet, in ↓ R_head, in →	*3.50*	*3.75*	*4.00*	*4.25*	*4.50*	*4.75*	*5.00*	*5.25*	*5.50*
*4.75*	-0.73	-0.75	-0.76	-0.78	-0.81	-0.83	-0.87	-0.92	-0.98
*5.00*	-0.72	-0.74	-0.75	-0.76	-0.78	-0.81	-0.83	-0.87	-0.91
*5.25*	-0.72	-0.73	-0.74	-0.75	-0.77	-0.79	-0.81	-0.83	-0.86
*5.50*	-0.71	-0.72	-0.73	-0.74	-0.75	-0.77	-0.79	-0.81	-0.83
*5.75*	-0.70	-0.71	-0.72	-0.73	-0.74	-0.76	-0.77	-0.79	-0.81
*6.00*	-0.70	-0.71	-0.72	-0.72	-0.74	-0.75	-0.76	-0.77	-0.79
*6.25*	-0.70	-0.70	-0.71	-0.72	-0.73	-0.74	-0.75	-0.76	-0.78
*6.50*	-0.60	-0.70	-0.71	-0.71	-0.72	-0.73	-0.74	-0.75	-0.76
*6.75*	-0.69	-0.70	-0.70	-0.71	-0.72	-0.72	-0.73	-0.74	-0.75

A fringe benefit of the constraining process with R_head and R_bullet is that it can offer constraints on the snap as well, because R_head and R_bullet control the speed of the snap. The results for m_head = 5 lb show that the snap is constrained to speeds between 4.4 and 4.7 ft s^-1. This range is consistent with the observed v_snap > 3.3 ft s^-1.

R_bullet, in ↓ R_head, in →	*3.50*	*3.75*	*4.00*	*4.25*	*4.50*	*4.75*	*5.00*	*5.25*	*5.50*
*4.75*	6.96	6.50	6.09	5.73	5.41	5.13	4.87	4.64	4.43
*5.00*	7.33	6.84	6.41	6.03	5.70	5.40	5.13	4.88	4.66
*5.25*	7.69	7.18	6.73	6.33	5.98	5.67	5.38	5.13	4.90
*5.50*	8.06	7.52	7.05	6.64	6.27	5.94	5.64	5.37	5.13
*5.75*	8.42	7.86	7.37	6.94	6.55	6.21	5.90	5.62	5.36
*6.00*	8.79	8.20	7.69	7.24	6.84	6.48	6.15	5.86	5.59
*6.25*	9.16	8.55	8.01	7.54	7.12	6.75	6.41	6.10	5.83
*6.50*	9.52	8.89	8.33	7.84	7.41	7.02	6.67	6.35	6.06
*6.75*	9.89	9.23	8.65	8.14	7.69	7.29	6.92	6.59	6.29