27. Constraints fr theta-cl and mcloud

The previous section discussed the constraints on Θ_cl and PE. This section begins the discussion of the other two important variables to the lurch, m_cloud and v_cloud. It will turn out that no significant constraints are available for either, but also that none are needed because their product, the momentum of the cloud, is constrained.
We begin by fixing PE (via SL6A) and examining how v_cloud varies in response to m_cloud and Θ_cl . Only two cases are shown below, for m_head = 6 and 7 lb (each with d_snap = 2.2 in and PE = 300 ft-lb), because they are enough to show the general result.

For m_head = 6 lb
The results for v_lurch are shown here for general interest only. They range from -6.2 ft s^-1 for heaviest and narrowest cloud to 0.9 ft s^-1 for widest cloud (spherical). The effects of m_cloud are much larger at small angles (range of 5 ft s^-1) than at large angles (range of 0.01–0.05 ft s^-1).

m_cl, lb ↓ Θ_cl →	~0°	15°	30°	45°	60°	75°	90°	105°	120°	135°	150°	165°	180°
0.1	-1.34	-1.30	-1.15	-0.92	-0.64	-0.32	-0.01	0.27	0.51	0.70	0.82	0.88	0.90
0.2	-2.27	-2.20	-2.00	-1.67	-1.27	-0.83	-0.39	0.02	0.36	0.61	0.78	0.87	0.90
0.3	-2.98	-2.90	-2.64	-2.25	-1.76	-1.22	-0.68	-0.18	0.23	0.55	0.76	0.87	0.90
0.4	-3.58	-3.48	-3.19	-2.74	-2.17	-1.55	-0.92	-0.35	0.13	0.49	0.74	0.86	0.90
0.5	-4.11	-4.00	-3.67	-3.17	-2.54	-1.84	-1.14	-0.50	0.04	0.45	0.72	0.86	0.90
0.6	-4.58	-4.46	-4.11	-3.56	-2.87	-2.10	-1.34	-0.63	-0.04	0.40	0.70	0.86	0.90
0.7	-5.02	-4.89	-4.51	-3.92	-3.17	-2.35	-1.52	-0.76	-0.12	0.36	0.68	0.86	0.90
0.8	-5.42	-5.28	-4.88	-4.25	-3.45	-2.57	-1.69	-0.87	-0.19	0.33	0.67	0.85	0.90
0.9	-5.80	-5.66	-5.23	-4.56	-3.72	-2.78	-1.84	-0.98	-0.26	0.29	0.66	0.85	0.90
1.0	-6.16	-6.01	-5.56	-4.86	-3.97	-2.98	-2.00	-1.08	-0.32	0.26	0.64	0.85	0.91

The results for v_cloud are of direct interest here. They are nearly constant for a given m_cloud. That means that for one PE and one m_cloud, there is only one v_cloud. This independence from Θ_cl is consistent with the previous section.

m_cl, lb ↓ Θ_cl →	~0°	15°	30°	45°	60°	75°	90°	105°	120°	135°	150°	165°	180°
0.1	704	704	704	704	704	704	705	704	704	704	704	704	704
0.2	497	497	497	498	498	498	498	498	498	498	498	498	498
0.3	405	405	406	406	406	407	407	407	407	407	407	407	407
0.4	351	351	351	351	352	352	352	352	352	352	352	352	352
0.5	313	313	313	314	314	315	315	315	315	315	315	315	315
0.6	285	285	286	286	287	287	287	288	288	288	288	288	288
0.7	264	264	264	265	265	266	266	266	266	266	266	266	266
0.8	246	246	247	247	248	248	249	249	249	249	249	249	249
0.9	232	232	232	233	234	234	235	235	235	235	235	235	235
1.0	220	220	220	221	221	222	222	223	223	223	223	223	223

For m_head = 7 lb
Here are the corresponding results for a 7-lb head. They are nearly the same as for the 6-lb head, that is, that one PE and one m_cloud give only one v_cloud.

m_cl, lb ↓ Θ_cl →	~0°	15°	30°	45°	60°	75°	90°	105°	120°	135°	150°	165°	180°
0.1	-1.34	-1.29	-1.15	-0.93	-0.66	-0.36	-0.07	0.20	0.43	0.60	0.71	0.77	0.79
0.2	-2.22	-2.15	-1.95	-1.65	-1.27	-0.85	-0.43	-0.04	0.28	0.52	0.68	0.77	0.79
0.3	-2.89	-2.81	02.57	-2.20	-1.73	-1.22	-0.70	-0.23	0.16	0.46	0.66	0.76	0.79
0.4	-3.46	-3.36	-3.09	-2.66	-2.12	-1.53	-0.94	-0.39	0.06	0.41	0.64	0.76	0.79
0.5	-3.96	-3.85	-3.54	-3.07	-2.47	-1.81	-1.14	-0.53	-0.02	0.36	0.62	0.76	0.80
0.6	-4.41	-4.29	-3.96	-3.44	-2.78	-2.06	-1.33	-0.66	-0.10	0.32	0.60	0.75	0.80
0.7	-4.82	-4.70	-4.34	-3.77	-3.07	-2.28	-1.50	-0.78	-0.17	0.29	0.59	0.75	0.80
0.8	-5.20	-5.07	-4.69	-4.09	-3.34	-2.50	-1.66	-0.89	-0.24	0.25	0.58	0.75	0.80
0.9	-5.56	-5.42	-5.02	-4.38	-3.59	-2.70	-1.81	-0.99	-0.30	0.22	0.56	0.75	0.80
1.0	-5.90	-5.76	-5.33	-4.67	-3.82	-2.89	-1.95	-1.09	-0.36	0.19	0.55	0.74	0.80

m_cl, lb ↓ Θ_cl →	~0°	15°	30°	45°	60°	75°	90°	105°	120°	135°	150°	165°	180°
0.1	668	668	668	668	668	668	668	668	668	668	668	668	668
0.2	472	472	472	472	472	473	473	473	473	473	473	473	473
0.3	385	385	385	385	385	386	386	386	386	386	386	386	386
0.4	333	333	333	333	334	334	334	334	334	334	334	334	334
0.5	297	297	297	298	298	299	299	299	299	299	299	299	299
0.6	271	271	271	272	272	272	273	273	273	273	273	273	273
0.7	250	250	251	251	252	252	252	253	253	253	253	253	253
0.8	234	234	234	235	235	236	236	236	236	236	236	236	236
0.9	220	220	220	221	222	222	223	223	223	223	223	223	223
1.0	208	208	209	209	210	211	211	211	211	211	211	211	211

Summary
These results can be summarized very simply: They show that for a fixed PE, a fixed mass of cloud (m_cloud) gives a fixed velocity of cloud (v_cloud), independent of the width of the cloud (Θ_cl).