25. Preconstraining

   These four variables in this group are related to each other, to the snap and lurch, but not directly to the group of variables most important to the lurch (Θ_cl, m_cloud, v_cloud, and PE). That allows us to consider the two groups separately. Because the first group relates to the snap, it may be thought of as preceding the lurch. We therefore use it to set the stage for the next group.
    We do this by determining which combinations of m_head and d_snap produce allowable values of t_lurch and v_frags. (Recall that the allowable values for t_lurch are 0–25 ms and for d_snap are >2.2 in.) Scenario SL6L was used for these calculations in order to fix the PE and to avoid dealing with R_bullet and R_head, whose values are not well constrained. SL6A gave very similar results for default values of R_bullet and R_head, and virtually the same constraints overall.
    The results, as shown in the table below, are interesting and clear. Allowed values are shown in bold green, disallowed values in red. If any of d_snap, t_lurch, or v_frags is disallowed, the entire combination is disallowed. The overall status of the combination is shown in the last column.
    The biggest feature of the table is that the allowed combinations of m_head and d_snap cluster near the middle. Except for the borderline combination of m_head = 4 and d_snap = 3.0, which we will not consider further, they comprise d_snap = 2.4–2.8 for m_head = 5–7 lb, plus d_snap = 3.0 for m_head = 6 and 7 lb. There is also a single acceptable value of d_snap = 2.4 for m_head = 8. These combinations mean in essence that m_head is limited to 5–7 lb, which can greatly simplify the future calculations.
    All things considered, it is remarkable that physics can eliminate heads of 4, 8, 9, and 10 lb, but these results have been confirmed by other calculations, some of which are shown below. Based on these results, we should probably shift the default m_head from 7 to 6 lb.

m_head, lb	d_snap, in	t_lurch, ms	v_frags, ft s^-1	Allowed?
4	2.2	31.8	188	No
	2.4	30.0	200	No
	2.6	28.1	213	No
	2.8	26.3	228	No
	3.0	24.4	246	Yes
5	2.2	26.7	224	No
	2.4	24.4	246	Yes
	2.6	22.1	272	Yes
	2.8	19.8	304	Yes
	3.0	17.4	344	Yes
6	2.2	21.6	277	No
	2.4	18.8	319	Yes
	2.6	16.0	374	Yes
	2.8	13.3	452	Yes
	3.0	10.5	573	Yes
7	2.2	16.5	363	No
	2.4	13.3	452	Yes
	2.6	10.0	599	Yes
	2.8	6.8	887	Yes
	3.0	3.5	1708	No
8	2.2	11.4	526	No
	2.4	7.7	780	Yes
	2.6	4.0	1508	No
	2.8	0.3	22,760	No
	3.0	-3.5	-1739	No
9	2.2	6.3	952	No
	2.4	2.1	2829	No
	2.6	-2.1	-2915	No
	2.8	-6.2	-962	No
	3.0	-10.4	-576	No
10	2.2	1.2	5033	No
	2.4	-3.5	-1739	No
	2.6	-8.1	-741	No
	2.8	-12.7	-471	No
	3.0	-17.3	-345	No