28. Constraints on m_cloud and v_cloud from Θ_cl and PE

    We now examine the same four important variables in a different way, by fixing m_cloud and calculating v_cloud for various combinations of Θ_cl and PE. We also show the results for v_lurch for general interest, even though they do not contribute directly to the logic of this section. The results show that fixed values of m_cloud and PE produce fixed values of v_cloud, again nearly independent of Θ_cl . The tables are shown for m_head = 7, 5, 8, and 6 lb, default values for the other variables, and calculations with SL6A.

For m_head = 7 lb

v_bodyafter, ft s^-1

    The values for v_lurch here display several features of note. The most obvious is the progression of values from negative on the left (small angles) to positive on the right (large angles). Coupled closely to this are rapid changes at the left side that gradually give way to smaller changes nearer the right. Coupled to that are the greater changes with low PE than with high (nearly 4.5 vs. 2.8 ft s^-1, respectively). Perhaps the most important feature, however, is that more than 60% of the values are negative. The positive values are restricted to half-angles of 120° and greater, which are unrealistically broad. Every value except one for angles ≤105° is negative. This means that for all practical purposes (at least for m_head = 7 lb) the lurch is always rearward.
    Another interesting feature is the effect of potential energy at various angles. For small angles, PE can change v_lurch by as much as 1.5 ft s^-1, with greater PEs decreasing the intensity of the lurch. At Θ_cl = 90°, the effect drops to 0.6 ft s^-1. As the angles approach 180°, the effect shrinks further, until at 180° it disappears altogether. The explanation for this trend is straightforward: The PE represents energy of deformation that takes away from the energy available to explode the head and create a cloud and a lurch. The more energy it takes to break the head, the less is available for the explosion. For narrow clouds, most of the momentum is in the X-direction, so any changes in its intensity will require similar changes in the recoil (the lurch). In wider clouds, however, more of the momentum is perpendicular to the X-axis, and so changes in it affect the X-component less. The ultimate in this effect is for a spherical cloud (Θ_cl = 180°), where there is as much momentum backward as forward. Changing this momentum will therefore not affect the lurch one way or the other.
    Although all the PEs here are possible in theory, the values of 0, 50,and 100 ft-lb should be excluded in practice, and so are indicated in red. Values outside the reasonable range of 30°–90° for Θ_cl are also shown in red. The allowed velocities for the lurch range from -0.3 to -2.9 ft s^-1, which nicely bracket the observed -0.8 ft s^-1. They are shown in bold green.

v_cloud, ft s^-1

    As discussed above, widening the cloud while holding PE and other variables constant requires v_cloud to increase slightly. This effect is shown in the table above. This table offers the additional information that the effect is greater for small PE than for large PE, presumably because large PEs extract  so much energy from the system that the effect becomes undetectable. (The effect is 1–2 ft s^-1 for low PE and 0–1 ft s^-1 for high PE.) The reasonable ranges of PE and Θ_cl from the table above produce a range of 290–420 ft s^-1 for v_cloud, or roughly 300–400 ft s^-1.

For m_head = 5 lb
    In order to see how representative these effects for m_head = 7 lb are, we examine them for other values of m_head. We begin with the lightest feasible head, 5 lb.

v_bodyafter, ft s^-1

    The values for v_lurch here display the same basic features as with m_head = 7lb: negative values on the left and positive values on the right, rapid changes on the left giving way to smaller changes on the right, greater changes with low PE than with high, and the same 60% of the values being negative. Masking the same ranges of values as before leads to the same conclusion that for all practical purposes the lurch is always rearward (-0.3 to -3.0 ft s^-1, virtually the same range as before).

v_cloud, ft s^-1

    As above, v_cloud increases only slightly to the right. It ranges from 330 to 490 ft s^-1 overall, but only 330 to 450 ft s^-1 within the allowed zone. This range is similar to the previous 290–420 ft s^-1, for m_head = 7 lb, but 40 to 70 ft s^-1 lower.

For m_head = 8 lb
    We continue by examining the patterns for m_head > 7 lb. Since the solutions with m_head = 9 and 10 lb are all disallowed because of negative or too-high v_frags, we are left with only m_head = 8 lb.

v_bodyafter, ft s^-1

    These values for v_lurch display the same basic features as before, and with the same 60% of the values being negative. Masking the same ranges of values as before leads to the same conclusion that for all practical purposes the lurch is always rearward (0.0 to -2.4 ft s^-1, slightly higher values than the ranges with lighter heads).

v_cloud, ft s^-1

    As before, v_cloud increases only slightly to the right. It ranges from 140 to 380 ft s^-1 overall, but only 140 to 340 ft s^-1 within the allowed zone. This range is broader and lower than the previous ranges.

For m_head = 6 lb
    We conclude this series of patterns with the ones for m_head = 6 lb. The results are very similar to those with m_head = 5 and 7 lb.

v_bodyafter, ft s^-1

    Applying the same masking as before yields a range of -0.3 to -3.0 ft s^-1, for the lurch, virtually the same as for m_head = 5 and 7 lb..

v_cloud, ft s^-1

    The pattern for v_cloud is also very similar to the previous ones. It ranges from 320 to 880 ft s^-1 overall, and 320 to 440 ft s^-1 within the allowed zone.

Summary of constraints
    Three major results can be seen in the summary table below. The first is that there are no allowable solutions for m_head = 9 and 10 lb. This is a major constraint, which together with the mass of the brain restricts m_head to 5–8 lb. The second is that virtually all the values for v_lurch were negative, meaning a rearward lurch. None of the cases considered here gave a positive lurch, and only one gave zero. The third major result is the similarity between lurches for m_head = 5, 6, and 7 lb. They are virtually indistinguishable. This seems to mean that 5, 6, and 7 lb are the most probable values for m_head.

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