27. Constraints from R_head vs. R_bullet

For m_head = 5 lb
    

PE, ft-lb

    The potential energy is constrained to not exceed 600 ft-lb. The is the strongest constraint in the calculations for m_head = 5 lb. Given the best estimates of 200–300 ft-lb for PE, this constraint is mild. The disallowed values are shown in boldface. Only 10 values are <600, and they are in the upper right.

v_bodyafter, ft s^-1

    The ranges of R_bullet and R_head keep the velocity of the lurch within narrow limits, 0.6–0.9 ft s^-1 in particular. The largest variations are in the upper right, where the PE falls below 600 ft-lb. The values disallowed by the PE are shown in boldface. They represent the only constraints on the lurch, and allow velocities of -0.7 to -0.9 ft s^-1.

v_frags, ft s^-1

    The velocity of the fragments also stays within reasonably narrow limits, 180–610 ft s^-1. None approach the disallowed zones of <0 or >1100 ft s^-1. As with the lurch, their only constraint comes from the potential energy. The disallowed values are shown in boldface. The allowed values range from 380 to 610 ft s^-1.

v_snap, ft s^-1

    The high PE creates high values of v_snap. The values disallowed by PE > 600 are shown in boldface. The allowed values are all 4.3–5.0 ft s^-1. 

Summary of allowed values

For m_head = 6 lb
    

PE, ft-lb

    This set of potential energies has two major constraints, values >600 ft-lb and values <0. In addition, one value of 11 ft-lb was deleted because it is far too small to be real. The disallowed values are shown in boldface. The allowed values form a zone that moves from the upper left to the lower right in the table. Their values are 320–590 ft-lb.

v_bodyafter, ft s^-1

    Increasing the mass of the head to 6 lb appears to greatly expand the range of velocities of the lurch from various combinations of R_bullet and R_head. Whereas with the head weighing 5 lb, the range was limited to 0.6–0.9 ft s^-1, it now reaches 4 ft s^-1 to the rear. All values above 1 ft s-1 are disallowed by PEs near or below zero, however, leaving an allowed range of 0.7–1.0 ft s-1 to the rear, which is essentially the same as the previous one.

v_frags, ft s^-1

    The velocity of the fragments also expands its range with a 6-lb head, reaching maxima well past the speed of sound in the upper right of the table. There three values are disallowed on those grounds alone. Most of the disallowed values come from high and low PE, however. The disallowed values in boldface represent both these effects. The final range of allowed values is 370–730 ft s^-1, which is similar to the previous 380–610 ft s^-1.

v_snap, ft s^-1

    The overall range for v_snap remains abut the same, 3.6–8.0 ft s^-1. The values disallowed by PE > 600 are shown in boldface. The allowed values are 4.2–5.1 ft s^-1, very similar to the previous 4.3–5.0 ft s^-1. 

Summary of allowed values

For m_head = 7 lb
    

PE, ft-lb

    The potential energies with m_head = 7 lb are still lower than before, and again form disallowed high values at the lower left and disallowed low values at the upper right. The allowed values range between 180 and 600 ft-lb.

v_bodyafter, ft s^-1

    The velocities of the lurch now display a more complex pattern, with values abut -0.6 to -0.7 ft s^-1 at the lower left, more-negative values toward the upper right, a line of indeterminable values beyond that, and a small zone of positive values in the far upper right. The values disallowed by the PE are shown in boldface. They leave an allowed range of -0.7 to -1.1 ft s-1, which is very similar to previous ranges.

v_frags, ft s^-1

    The velocity of the fragments also shows a complex pattern that has the earmarks of instabilities. It increases from the lower left to the upper right. About two-thirds of the way up, it passes through the speed of sound, then increases to about 6500 ft s^-1, flips back to large negative values, and continues to increase but remain negative. The negative and supersonic values are disallowed of their own accord, but also by negative PE. Since the negation by PE covers all that by PE and more, its disallowed values are shown here. The allowed values are 370–850 ft s^-1, very similar to previous ranges.

v_snap, ft s^-1

    The overall range for v_snap decreases slightly, from 3.6–8.0 ft s^-1.to the current 3.1–6.9. The values disallowed by high and low PE are shown in boldface. The allowed values are 4.1–5.1 ft s^-1, practically the same as the previous 4.2–5.1 ft s^-1. 

Summary of allowed values

For m_head = 8 lb
    

PE, ft-lb

    The potential energies with m_head = 8 lb are lower and more irregular than before. They begin in the lower left with values around 600 ft-lb, and decrease gradually toward the upper right. Just past the middle, they plummet to extreme negative values, and then recover to nearly 600 in the upper right corner. The allowed values are 150–600 ft-lb. The disallowed values are shown in boldface.

v_bodyafter, ft s^-1

    The velocities of the lurch are also more complex, and generally follow the PE. They have values about -0.6 to -0.7 ft s^-1 at the lower left, increasingly negative values toward the upper right, an abrupt switch to positive values, and a return to negative values at the far upper right. The values disallowed by the PE leave a small allowed zone with values near zero in the upper right. They are disallowed by negative velocities for v_frags, as shown in the table below. The values shown in boldface represent both sets of disallowed values. They leave an allowed range of -0.7 to -1.1 ft s-1, the same as in the previous case.

v_frags, ft s^-1

    The velocity of the fragments also shows a pattern similar to those of the lurch and the potential energy. Its values start low at the lower left, increase rapidly into supersonic values just past the midpoint toward the upper right, then plummet to high negative values, and finally increase to less-negative values at the upper left. The values at the lower left are disallowed by the PE. The upper-right half of the values are disallowed by their own negative and supersonic values and to a certain extent by the PE. Both sets of disallowed values are shown in boldface. The allowed values are 370–870 ft s^-1, very similar to previous ranges.

v_snap, ft s^-1

    The overall range for v_snap again decreases slightly, from 3.1–6.9 ft s^-1 to the current 2.7–6.0. The values disallowed by PE and v_frags are shown in boldface. The allowed values are 4.0–5.0 ft s^-1, practically the same as the previous 4.1–5.1 ft s^-1. 

Summary of allowed values

For m_head = 9 lb
    

PE, ft-lb

    The potential energies with m_head = 9 lb are lower and more complex than before. There are three values of disallowed values, two of values >600 ft-lb in the lower left and upper right corners, and a zone of negative values that stretches from the upper left to the lower right. The allowed values are 140–590 ft-lb (200–590 if negative v_frags are considered). The disallowed values are shown in boldface. (Apply constraints from v_frags to PE?)

v_bodyafter, ft s^-1

    The velocities of the lurch are follow a striped pattern similar to that of the PE. They have values about -0.7 to -0.8 ft s^-1 at the lower left, much deeper minima switching abruptly to high positive values across the middle, and increasing values toward the upper right. The values disallowed by the PE leave a small allowed zone with values near zero in the upper right. They are disallowed by a large zone of negative velocities for v_frags, as shown in the table below. The values shown in boldface represent both sets of disallowed values. They leave a small zone of allowed values in the lower left, with a range of -0.7 to -1.1 ft s^-1, the same as in the previous case.

v_frags, ft s^-1

    The velocity of the fragments also shows a pattern like the previous ones, with moderate positive values at the lower left, much higher values in the middle flipping rapidly to very low negative values, and less-negative values toward the upper right. Two of the values at the lower left are disallowed by the PE, as are the high positive and negative values in the center. The supersonic values and the negative values in the upper-right half self-disallowed. Both sets of disallowed values are shown in boldface. The allowed values are 380–830 ft s^-1, similar to previous ranges.

v_snap, ft s^-1

    The overall range for v_snap again decreases slightly, from 2.7–6.0 ft s^-1 to the current 2.4–5.4. The values disallowed by PE and v_frags are shown in boldface. The allowed values are 4.1–5.0 ft s^-1, practically the same as the previous 4.0–5.0 ft s^-1.

Summary of allowed values

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