29d. Summary of the constraints

29d. Summary of the constraints

The three tables summarizing the constraints for angles of 30°, 70°, and 105° are shown below. They collectively offer these important conclusions:

There are no constraints on m_cloud. Allowable solutions for each angle are found for the full range of 0.1–1.0 lb used.
There are moderate constraints for v_cloud, namely that its values for the allowable masses of head (5–7 lb, as seen under "Prescreening") do not fall below 200 ft s^-1.
All lurches for the acceptable heads and angles of 30° and 70° are negative. Only at the extremely wide 105° do some lurches become positive.

For Θ_cl = 30°

M_head, lb	m_cloud, lb	v_cloud, ft s^-1	v_lurch, ft s^-1	PE, ft-lb
5	0.1–1.0	200–600	-0.8 to -6.4	110–600
6	0.1–1.0	200–600	-0.8 to -5.7	110–580
7	0.1–1.0	200–600	-0.8 to -5.8	100–570
8	0.1–1.0	100–600	-0.2 to -4.7	100–600
9	—	—	—	—
10	—	—	—	—

For Θ_cl = 70°

M_head, lb	m_cloud, lb	v_cloud, ft s^-1	v_lurch, ft s^-1	PE, ft-lb
5	0.1–1.0	200–600	-0.2 to -3.8	110–540
6	0.1–1.0	200–600	-0.2 to -3.4	110–590
7	0.1–1.0	200–600	-0.2 to -3.5	110–580
8	0.1–1.0	100–600	+0.1 to -2.9	110–590
9	—	—	—	—
10	—	—	—	—

For Θ_cl = 105°

M_head, lb	m_cloud, lb	v_cloud, ft s^-1	v_lurch, ft s^-1	PE, ft-lb
5	0.1–1.0	200–600	+0.4 to -1.3	110–540
6	0.1–1.0	200–600	+0.4 to -1.1	130–590
7	0.1–1.0	200–600	+0.3 to -1.2	120–580
8	0.1–1.0	100–600	+0.3 to -1.1	100–590
9	—	—	—	—
10	—	—	—	—

The pseudo-inverse relation of m_cloud and v_cloud
The previous three subsections have revealed a pseudo-inverse relation between m_cloud and v_cloud. Here we explore this relation directly to see how close it comes to being inverse.
The easiest way to show the relation is in a simple table of m_cloud and v_cloud. We do this below for Θ_cl = 70°, PE = 300 ft-lb, d_snap = 2.4 in, and m_head = 5 and 6 lb, as calculated with SL6A. The results show that for both heads, increasing m_cloud by 10x decreases v_cloud by 3x. This makes their product increase by 3x. Thus v_cloud changes by one-third as much as m_cloud (and in the opposite direction).

M_head, lb	m_cloud, lb	v_cloud, ft s^-1	**m_cl x v_cl**	M_head, lb	m_cloud, lb	v_cloud, ft s^-1	**m_cl x v_cl**
5	0.1	713.2	71.3	6	0.1	689.8	69.0
	0.2	504.2	100.8		0.2	487.6	97.5
	0.3	411.5	123.4		0.3	398.0	119.4
	0.4	356.2	142.5		0.4	344.5	137.8
	0.5	318.5	159.2		0.5	308.0	154.0
	0.6	290.6	174.4		0.6	281.0	168.6
	0.7	268.9	188.2		0.7	260.0	182.0
	0.8	251.4	201.1		0.8	243.1	194.5
	0.9	236.9	213.2		0.9	229.1	206.2
	1.0	224.6	224.6		1.0	217.2	217.2

Ahead to Grand Summary of Constraints
Back to Constraints with Θ_cl = 105°
Back to Intro to Constraints
Back to Physics of the Head Shot