27. Constraints from d_snap vs. d_frag

(N.B. These values were the first try at constraints. There were calculated with L7AC, not L8AC. That is why there is no PE here. Need to think about how there results relate to those obtained with L8AC and PE.)

Calculations for m_head = 7 lb
    Here is the first set of three tables, constructed with default values of all variables. That makes m_head = 7 lb. We first consider the tables individually, then show how the results for the mass and velocity of the cloud constrain the answers for the lurch.
    The first table (below) is for v_lurch. The values in the upper left are the most negative (meaning the strongest rearward lurch), and reach 7.66 ft s^-1. The size of the lurch decreases to the lower right, then passes through zero into positive values (a forward lurch), then passes through a zone of negative values, and finally on the far right returns to positive values.

v_lurch, ft s^-1

    The second table (below) is for m_cloud. It has highest values in its upper left (exceeding 4 lb). The values decrease systematically toward its lower right, and reach a broad triangular zone of -0.01 to -0.02 lb. These negative masses are of course impossible.

m_cloud, lb

    The third table (below) is for v_frag = v_cloud. It has smallest values in its upper left (as low as 75 ft s^-1), highest values toward its lower right (up to 8900 ft s^-1), but then a column of negative values on the far right.

v_frag, ft s^-1 = v_cloud

    External constraints on v_lurch
    The most important external constraints on the lurch are provided by the mass of the cloud, m_cloud, which may not be negative or greater than 1 lb (the mass of the right hemisphere that was found to be blown out). Here is the table of masses with those constraints shown in boldface. 

m_cloud, lb

    Most of the forbidden solutions lie at the lower right of the table, and represent negative masses of the cloud. Adjacent to this zone lies a diagonal zone of very low masses, zero or tiny positive values, which probably also should be excluded. If we arbitrarily remove solutions with m_cloud <0.1 lb, which is a mild constraint because 1 lb was blown out, we get the table below, with the new deletions shown in boldface italic. This produces a considerably narrower range of acceptable solutions that forms a rough diagonal from lower left to upper right.

m_cloud, lb

    The next external constraint comes from the velocity of the fragments and the cloud, here set equal to each other. The fundamental constraint here is that neither velocity can be negative, because both cloud and large fragments moved forward. A second level of constraint is the sound barrier—it is highly unlikely that the fragments from the explosion could be given enough energy to exceed the speed of sound, given the extreme turbulence induced at that speed. That allows us to exclude speeds greater than 1100 ft s^-1. The table with these two constraints (below) shows that they eliminate many solutions in the lower right, as did the negative and small masses of cloud. This is equivalent to saying that the high and negative velocities of the cloud correspond to zero or negative masses.

v_frag, ft s^-1 = v_cloud

    Also note that the external constraints work to eliminate most of the positive solutions, or make nearly all the acceptable answers rearward lurches. In other words, forward or zero lurches are all but forbidden by external constraints from the mass and velocity of the cloud. If this result holds up, it will be very powerful, for it says that rearward lurches are the rule, not the exception.

    Internal constraints on v_lurch
    I have mixed feelings about using internal constraints on the lurch, for they refer to what ought to be rather than to what can possibly be, as the external constraints do. But if we accept the fact that the computed rearward lurch may not exceed 1 ft s^-1 because the observed one did not, we get the table below.

v_lurch, ft s^-1

    There is another internal constraint on the lurch, however, namely that it was to the rear. That allows us to remove from consideration all positive values. Those deleted cases are shown in boldface italic in the table below. The two sets of internal constraints give a diagonal pattern of acceptable solutions that resembles the one from the masses but is displaced downward from it.

v_lurch, ft s^-1

    Combined external constraints on v_lurch
    Now we combine the constraints and see what solutions remain. First we combine the external constraints from m_cloud and v_cloud, and darken the corresponding cells for v_lurch. That gives:

v_lurch, ft s^-1

These constraints limit the values of v_lurch to -0.4 to -2.7 ft s^-1 (i.e., all negative). This is equivalent to saying that the external constraints imposed by the properties of the cloud remove forward lurches from consideration and leave moderate to strong rearward lurches as the only allowed solutions. Again, this is an extremely important result, which essentially says that as long as there is a cloud of diffuse fragments that moves forward, the head and body will mechanically recoil rearward with a speed similar to the one that was observed.

    Combined external and internal constraints on v_lurch
    Last, we combine the internal constraints for the lurch with the external constraints in order to get the full picture of allowed vs. forbidden solutions. The full list of constraints is:

• The speed of the lurch, v_lurch, may not be positive or more negative than -1 ft s^-1 (internal).
• The mass of the cloud, m_cloud, may not exceed 1 lb or be less than 0.1 lb (external).
• The speed of the fragments and the cloud, v_frags and v_cloud, may not be negative or exceed the speed of sound, 1100 ft s^-1 (external).

Since the internal constraints allow the same general diagonal shape of values but positioned lower on the table, the result is a narrower band of acceptable solutions:

v_lurch, ft s^-1

The remaining values of v_lurch are -0.4 to -1.0 ft s^-1, which make a narrow diagonal band from lower left to upper right. The diagonality means that d_snap and d_frag have similar-sized effects, as we saw above. Since their effects are both positive (increases in the variable increase the forward components of the lurch, or decrease its rearward speed), the effective slope of their relation must be negative, as shown here. In other words, the observed lurch of 0.4–1.0 ft s^-1 in the rearward direction represents a delicate balance between d_snap and d_frag. As one increases, the other must decrease to maintain the balance, and vice versa.
    The main feature of the answer, however, is that the acceptable solutions fall in the narrow band between 0.4 and 1.0 ft s^-1 in the rearward direction. The external constraints cut off the low end of the lurch (speeds less than 0.4 ft s^-1); the internal constraints cut off the high end (speeds greater than 1 ft s^-1). The combination leaves only the narrow band between 0.4 and 1 ft s^-1 in the rearward direction. If you are uncomfortable with internal constraints, you can say that the external constraints limit the lurch to values of 0.4 to 2.7 ft s^-1, all in the rearward direction, and that the observed lurch is in the lower end of that range.
    The table for m_cloud with the same cells boldfaced is:

m_cloud, lb

The range of acceptable masses for the cloud is 0.1–0.2 lb.

    The table for v_frags = v_cloud with the same cells boldfaced is:

v_frag, ft s^-1 = v_cloud

The range of acceptable speeds of the fragments is 350–500 ft s^-1.

Calculations for m_head = 6 lb
    Here is the first set of three tables, constructed with default values of all variables except m_head, which was set to 6 lb. The boldface cells at the upper left of the table for v_lurch (below) represent the cases eliminated because their values exceed the observed speed of <1.0 ft s^-1 rearward (the internal constraint). The cells at the lower right are the ones with positive velocities, which of course were also not observed for the lurch.

v_lurch, ft s^-1

    The external constraints of masses in > 1 lb or < 0.1 lb are shown in the table below. They form a pattern similar to the case with m_head = 7 lb.

m_cloud, lb

    The external constraints from high or negative velocities of the fragments and cloud are minimal in this case, and confined to a small area in the lower right of the table below. Their pattern is nearly the same as that for negative masses of cloud shown above.

v_frag, ft s^-1 = v_cloud

   Shown below are the three tables from above, but with cells for solutions made impossible by any of the above constraints indicated in boldface.

v_lurch, ft s^-1

The range of acceptable values of v_lurch is now narrowed to 0.4–1.0 ft s^-1, just the same as in the case above, with m_head = 7 lb.
    The table for m_cloud with the same cells boldfaced is:

m_cloud, lb

This makes the range of acceptable values for m_cloud to be 0.1–0.2 lb, again the same as in the previous case.
    The table for v_frags = v_cloud with the same cells boldfaced is:

v_frag, ft s^-1 = v_cloud

The range of acceptable values for v_frags = v_cloud becomes 300–450 ft s^-1, the same as in the previous case.

Calculations for m_head = 8 lb
    Here is the basic set of three tables, constructed with default values of all variables except m_head, which was set to 8 lb. All tables are constructed as before. The table for v_lurch and the internal constraints is:

v_lurch, ft s^-1

It leaves a narrow diagonal zone of acceptable values that fall between 0.1 and 1.0 ft s^-1 rearward. The two isolated values near the bottom center will be eliminated by the external constraints.
    The external constraints of masses in excess of 1 lb or below 0.1 lb are shown in the table below. They form a pattern similar to the cases with m_head = 6 and 7 lb, except for a few acceptable cases in the upper right.

m_cloud, lb

    The external constraints from high or negative velocities of the fragments and cloud are substantial in this case (below), and occupy nearly the entire right half of the table.

v_frag, ft s^-1 = v_cloud

    Shown below are the three tables from above, but with cells for solutions made impossible by any of the above constraints indicated in boldface. Note how the increased mass of the head has "squeezed" the zone of acceptable solutions to a very small one.

v_lurch, ft s^-1

The range of acceptable values of v_lurch is restricted to 0.5–1.0 ft s^-1, similar to the ranges of the two cases above.
    The table for m_cloud with the same cells boldfaced is:

m_cloud, lb

This makes the range of acceptable values for m_cloud 0.1–0.2 lb, again the same as in the previous cases.
    The table for v_frags = v_cloud with the same cells boldfaced is:

v_frag, ft s^-1 = v_cloud

The range of acceptable values for v_frags = v_cloud is 300–450 ft s^-1, very similar to the previous cases.

Conclusions
    The above analysis leads to a series of interesting and important conclusions.

• The range of allowed values for v_lurch, m_cloud, and v_frags = v_cloud is severely limited by internal and external factors that include forbidden values of the other variables.
• For example, the default values of all variables except d_frags and d_snap yield a wide range of answers, but they are limited by v_lurch <-1, m_cloud <0, v_frags <0, and m_cloud <0.1 lb.
• Eliminating these cases narrows the range of combinations of d_snap and d_frags, as well as the range of values for v_lurch, m_cloud, and v_frags.
• Changing the values of m_head shifts the range of answers for the three variables, but not the ranges of acceptable answers, which are still controlled by the "veto power" of each of v_lurch, m_cloud, and v_frags over the others.
• Summary table for Lurch 7 Angular conical:

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