28. The critical role of Θ_cl, the half-angle of the cloud

   Section 23 summarized the major variables and noted that they had to do with the head, the cloud, and the time intervals. This section examines in more detail the important role of the cloud, and in particular the critical role of its half-angle, Θ_cl, in determining whether the lurch is forward or to the rear. It will turn out that one need only have a cloud with a modest net forward component in order to require that the lurch be rearward. Furthermore, once a forward cloud is present, other constraints come into play to fix the limits of other important variables. There seems no way to avoid this important conclusion.
    We begin by seeing what happens to the lurch when the individual variables are removed from the solutions. (This is another way to evaluate their importance.) The tables below shows the results with Lurch 8 Angular ( default conditions), which was used because it allows us to separate v_cloud from v_frags and control the values of m_cloud. The results are extraordinarily clear—deleting the cloud (by deleting any of its variables) reverses the lurch, from -0.82 ft s-1 to +0.74 ft s-1, but deleting any of the other variables increases or decreases the speed while keeping it rearward (speeds of 0.44–1.71 ft s-1). This means that the cloud alone is responsible for the rearward lurch.

    But these figures are for standard conditions, which include Θ_cl = 45°. What happens at other half-angles? The answer is that any half-angle less than 90° gives a rearward lurch, any half-angle greater than that gives a forward lurch, and angles near 90° give small or nonexistent lurches. A quick survey, for standard conditions, shows this effect clearly (table below).

    We need to examine this effect in more detail, however, because it is so important to the basic question of the lurch. We do this by including four other critical variables: the mass of the cloud, the mass of the lurch, the mass of the head, and the potential energy (energy of deformation). Specifically, we present tables of vbodyafter and PE as functions of m_cloud and Θ_cl , and we do this for m_head = 4–9 lb. These detailed results show that the pattern in the table above holds for the entire range of conditions considered, i.e., that a forward-moving cloud inevitably gives a rearward lurch.

For m_head = 4 lb
    We begin with the results for a mass of 4 lb, which is at the very lower limit of consideration, and almost certainly too small. Constraints on PE are used to establish acceptable values for the lurch. The allowed values for both variables are shown in boldface.

PE, ft-lb

The excluded values of PE fall into two categories: >600 and <100 ft-lb. The allowed values form a symmetrical arch centered on Θ_cl = 90° (meaning that the conical cloud has become hemispheric). Allowed PE ranged from 112 to 570 ft-lb, allowed m_cloud from 0.3 to 1.0 lb.

v_bodyafter, ft s^-1

    The peak of the arch of allowed values for the lurch corresponds to the transition point for its velocity, i.e., where it changes from negative values at narrower angles to positive values at broader angles. At first glance one might think this to be logical, for that is where negative velocities of cloud particles begin to cancel forward velocities. But the actual situation is more complicated than that, because masses of cloud are involved as well. It may just be coincidence that the transition point occurs at 90°. Allowed values of the lurch ranged from -5.5 ft s^-1 to +1.0 ft s^-1.
    It is interesting to note that the allowed positive velocities have a much smaller range for a given angle than the allowed negative velocities do. The explanation is straightforward, though. The range in velocity represents the effect of varying mass. For narrow angles, more mass will give a greater forward impulse. But as the angles approach 180° the pattern approaches a sphere, and as much mass is going forward as backward. That effectively removes the mass from the equations.

For m_head = 5 lb
    The mass of 5 lb is about the smallest reasonable value. As before, the allowed values for PE and lurch are shown in boldface.

PE, ft-lb

The excluded values of PE fall into two categories: >600 and <100 ft-lb. The allowed values form a symmetrical arch centered on Θ_cl = 90° (meaning that the conical cloud has become hemispheric). As before, the allowed vales of m_cloud ranged from 0.3 to 1.0 lb. Allowed PE ranged from 95 to 591 ft-lb, very similar to before.

v_bodyafter, ft s^-1

    The peak of the arch of allowed values for the lurch again corresponds to the transition point for its velocity. Allowed values of the lurch ranged from -5.5 ft s^-1 to +0.9 ft s^-1, also very similar to the case for m_head = 4 lb.

For m_head = 6 lb
    The mass of 6 lb is very near the default (optimum) value of 7 lb. As before, the allowed values for PE and lurch are shown in boldface.

PE, ft-lb

As before, the arch of allowed values centered on Θ_cl = 90°, and the allowed vales of m_cloud ranged from 0.3 to 1.0 lb. Allowed PE ranged from 102 to 593 ft-lb, very similar to before.

v_bodyafter, ft s^-1

    The peak of the arch of allowed values for the lurch again corresponds to the transition point for its velocity. Allowed values of the lurch ranged from -5.6 ft s^-1 to +0.9 ft s^-1, also very similar to the case for m_head = 5 lb.

For m_head = 7 lb
    The mass of 7 lb is the default (optimum) value of 7 lb. As before, the allowed values for PE and lurch are shown in boldface.

PE, ft-lb

As with all the cases before, the arch of allowed values center on Θ_cl = 90°. This time, however, the allowed vales of m_cloud range from 0.2 to 1.0 lb. The allowed PE ranges from 90 to 600 ft-lb, very similar to before.

v_bodyafter, ft s^-1

    The peak of the arch of allowed values for the lurch again corresponds to the transition point for its velocity. Allowed values of the lurch ranged from -5.6 ft s^-1 to +0.9 ft s^-1, also very similar to the case for m_head = 5 lb.

Logic

1. Several of the 30 variables are much more important than the others.

2. The most important variable among them is the forward-moving cloud. It is the only thing that keeps the lurch negative.

3. Without a cloud, all the lurches are positive.

4. The solutions are also few and tenuous. They seem to lie at a sharp transition between one kind of exclusion and another. They are occasional points, not broad zones.

5. With the cloud, all the lurches are negative.

6. Furthermore, once the cloud is admitted, the most important variables are constrained to well-defined zones of "good" solutions, very different from the no-cloud case.

7. Thus once the existence of a forward-moving cloud is admitted, a rearward lurch very similar to the observed one is required.

8. To cement the argument, a frontal shot cannot explain even the first part of the lurch.