27. Constraints from Θcl vs. mcl

 

 

For mhead = 5 lb
    We begin with the results for a mass of 5 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

mcloud, lb ↓ Θcl, degrees →  ~0 15 30 45 60 75 90 105 120 135 150 165 180
0.1 965 965 965 965 965 965 965 965 965 965 965 965 965
0.2 824 824 824 824 825 825 825 825 824 824 824 824 824
0.3 682 682 682 683 683 684 684 684 684 684 684 684 683
0.4 539 539 539 541 542 543 543 543 543 543 543 543 543
0.5 394 395 396 398 400 401 402 403 403 403 402 402 402
0.6 249 249 251 254 257 259 261 262 262 262 262 262 262
0.7 102 103 106 110 114 117 120 121 121 121 121 121 121
0.8 -46 -44 -41 -35 -30 -25 -22 -20 -19 -19 -19 -20 -20
0.9 -194 -193 -188 -181 -174 -168 -164 -161 -160 -160 -160 -160 -160
1.0 -344 -342 -336 -328 -319 -311 -305 -302 -301 -300 -301 -301 -301

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

vbodyafter, ft s-1

mcloud, lb ↓ Θcl, degrees →  ~0 15 30 45 60 75 90 105 120 135 150 165 180
0.1 0.32 0.34 0.38 0.44 0.52 0.61 0.70 0.78 0.85 0.90 0.93 0.95 0.96
0.2 -0.32 -0.29 -0.20 -0.08 0.09 0.26 0.44 0.60 0.74 0.84 0.91 0.95 0.96
0.3 -0.96 -0.91 -0.79 -0.59 -0.35 -0.08 0.18 0.43 0.63 0.79 0.89 0.94 0.96
0.4 -1.60 -1.54 -1.37 -1.11 -0.79 -0.43 -0.08 0.25 0.52 0.73 0.87 0.94 0.96
0.5 -2.24 -2.17 -1.96 -1.63 -1.23 -0.78 -0.34 0.07 0.42 0.67 0.84 0.94 0.96
0.6 -2.88 -2.79 -2.54 -2.15 -1.67 -1.13 -0.60 -0.10 0.31 0.62 0.82 0.93 0.96
0.7 -3.53 -3.42 -3.13 -2.68 -2.11 -1.48 -0.86 -0.28 0.20 0.56 0.80 0.93 0.96
0.8 -4.17 -4.06 -3.72 -3.20 -2.55 -1.83 -1.12 -0.46 0.09 0.50 0.78 0.92 0.96
0.9 -4.82 -4.69 -4.31 -3.72 -2.99 -2.18 -1.38 -0.64 -0.02 0.44 0.76 0.92 0.97
1.0 -5.47 -5.32 -4.90 -4.25 -3.44 -2.54 -1.64 -0.82 -0.13 0.39 0.73 0.92 0.97

    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.