28. Constraints on mcloud and vcloud from Θcl and PE

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

For mhead = 7 lb

vbodyafter, ft s-1

PE, ft-lb Θcl   ~0 15 30 45 60 75 90 105 120 135 150 165 180
0 -3.61 -3.51 -3.22 -2.78 -2.22 -1.61 -1.00 -0.43 0.04 0.40 0.63 0.76 0.79
50 -3.50 -3.40 -3.12 -2.69 -2.15 -1.55 -0.95 -0.40 0.06 0.40 0.64 0.76 0.79
100 -3.38 -3.29 -3.02 -2.60 -2.07 -1.49 -0.90 -0.37 0.08 0.42 0.64 0.76 0.79
150 -3.27 -3.18 -2.91 -2.50 -1.99 -1.42 -0.86 -0.34 0.10 0.43 0.64 0.76 0.79
200 -3.14 -3.06 -2.80 -2.40 -1.91 -1.36 -0.81 -0.30 0.12 0.44 0.65 0.76 0.79
250 -3.02 -2.93 -2.69 -2.30 -1.82 -1.29 -0.76 -0.27 0.14 0.45 0.65 0.76 0.79
300 -2.89 -2.81 -2.57 -2.20 -1.73 -1.22 -0.70 -0.23 0.16 0.46 0.66 0.76 0.79
350 -2.76 -2.68 -2.45 -2.09 -1.64 -1.14 -0.65 -0.20 0.18 0.47 0.66 0.76 0.79
400 -2.62 -2.54 -2.32 -1.98 -1.54 -1.07 -0.59 -0.16 0.21 0.48 0.67 0.77 0.79
450 -2.47 -2.40 -2.19 -1.86 -1.44 -0.99 -0.53 -0.12 0.23 0.50 0.67 0.77 0.79
500 -2.32 -2.25 -2.05 -1.73 -1.34 -0.91 -0.47 -0.07 0.26 0.51 0.68 0.77 0.79
550 -2.16 -2.09 -1.90 -1.60 -1.23 -0.82 -0.41 -0.03 0.29 0.53 0.68 0.77 0.79
600 -1.99 -1.93 -1.75 -1.47 -1.11 -0.73 -0.34 0.02 0.32 0.54 0.69 0.77 0.79

    The values for vlurch 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 mhead = 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 vlurch by as much as 1.5 ft s-1, with greater PEs decreasing the intensity of the lurch. At Θcl
= 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 (Θcl = 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 3090 for Θcl 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.

vcloud, ft s-1

PE, ft-lb Θcl   ~0 15 30 45 60 75 90 105 120 135 150 165 180
0 460 460 460 460 461 461 461 461 461 461 461 461 461
50 448 448 448 449 449 449 450 450 450 450 450 450 450
100 436 436 436 437 437 437 438 438 438 438 438 438 438
150 424 424 424 424 425 425 425 425 425 425 425 425 425
200 411 411 411 412 412 412 413 413 413 413 413 413 413
250 398 398 398 399 399 399 399 399 399 399 399 399 399
300 385 385 385 385 385 386 386 386 386 386 386 386 386
350 371 371 371 371 371 372 372 372 372 372 372 372 372
400 356 356 356 357 357 357 357 357 357 357 357 357 357
450 341 341 341 341 342 342 342 342 342 342 342 342 342
500 325 325 325 325 326 326 326 326 326 326 326 326 326
550 308 308 309 309 309 309 309 309 309 309 309 309 309
600 291 291 291 291 291 291 291 291 291 291 291 291 291

    As discussed above, widening the cloud while holding PE and other variables constant requires vcloud 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 12 ft s-1 for low PE and 01 ft s-1 for high PE.) The reasonable ranges of PE and Θcl from the table above produce a range of 290420 ft s-1 for vcloud, or roughly 300400 ft s-1.

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

vbodyafter, ft s-1

PE, ft- lb Θcl   ~0 15 30 45 60 75 90 105 120 135 150 165 180
0 -3.68 -3.58 -3.27 -2.81 -2.22 -1.58 -0.93 -0.34 0.16 0.54 0.79 0.92 0.96
50 -3.57 -3.47 -3.18 -2.72 -2.15 -1.52 -0.89 -0.31 0.18 0.55 0.79 0.92 0.96
100 -3.46 -3.37 -3.08 -2.63 -2.08 -1.46 -0.84 -0.28 0.20 0.56 0.79 0.92 0.96
150 -3.35 -3.26 -2.98 -2.54 -2.00 -1.40 -0.80 -0.25 0.22 0.57 0.80 0.92 0.96
200 -3.24 -3.15 -2.87 -2.45 -1.92 -1.34 -0.75 -0.21 0.24 0.58 0.80 0.92 0.96
250 -3.12 -3.03 -2.77 -2.36 -1.84 -1.27 -0.70 -0.18 0.26 0.59 0.81 0.92 0.96
300 -3.00 -2.91 -2.66 -2.26 -1.76 -1.20 -0.65 -0.15 0.28 0.60 0.81 0.92 0.96
350 -2.88 -2.79 -2.54 -2.16 -1.67 -1.14 -0.60 -0.11 0.30 0.61 0.82 0.92 0.96
400 -2.75 -2.67 -2.42 -2.05 -1.58 -1.07 -0.55 -0.08 0.32 0.62 0.82 0.93 0.96
450 -2.62 -2.54 -2.30 -1.94 -1.49 -0.99 -0.50 -0.04 0.34 0.63 0.82 0.93 0.96
500 -2.48 -2.40 -2.18 -1.83 -1.40 -0.92 -0.44 -0.00 0.37 0.65 0.83 0.93 0.96
550 -2.33 -2.26 -2.04 -1.71 -1.30 -0.84 -0.38 0.04 0.39 0.66 0.84 0.93 0.96
600 -2.18 -2.11 -1.91 -1.59 -1.19 -0.76 -0.32 0.08 0.42 0.67 0.84 0.93 0.96

    The values for vlurch here display the same basic features as with mhead = 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).

vcloud, ft s-1

PE, ft- lb Θcl   ~0 15 30 45 60 75 90 105 120 135 150 165 180
0 484 484 484 485 485 485 486 486 486 486 486 486 486
50 473 473 473 474 474 474 475 475 475 475 475 475 474
100 462 462 462 462 463 463 463 463 463 463 463 463 463
150 450 450 450 451 451 451 451 452 452 452 451 451 451
200 438 438 438 439 439 439 440 440 440 440 440 439 439
250 426 426 426 426 427 427 427 427 427 427 427 427 427
300 413 413 413 414 414 414 414 415 415 415 415 415 414
350 400 400 401 401 401 401 402 402 402 402 401 401 401
400 387 387 387 387 388 388 388 388 388 388 388 388 388
450 373 373 373 373 374 374 374 374 374 374 374 374 374
500 358 359 359 359 359 359 359 360 360 359 359 359 359
550 343 343 344 344 344 344 344 344 344 344 344 344 344
600 328 328 328 328 328 328 328 329 328 328 328 328 328

    As above, vcloud 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 290420 ft s-1, for mhead = 7 lb, but 40 to 70 ft s-1 lower.

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

vbodyafter, ft s-1

PE, ft- lb Θcl   ~0 15 30 45 60 75 90 105 120 135 150 165 180
0 -3.12 -3.04 -2.80 -2.44 -1.97 -1.46 -0.96 -0.49 -0.10 -.20 0.40 0.50 0.53
50 -2.99 -2.91 -2.68 -2.32 -1.88 -1.39 -0.90 -0.45 -0.07 0.21 0.40 0.50 0.53
100 -2.85 -2.77 -2.55 -2.21 -1.78 -1.31 -0.84 -0.41 -0.05 0.22 0.41 0.50 0.53
150 -2.70 -2.63 -2.42 -2.09 -1.68 -1.23 -0.78 -0.37 -0.02 0.24 0.41 0.50 0.53
200 -2.55 -2.48 -2.28 -1.97 -1.58 -1.15 -0.72 -0.33 0.00 0.25 0.42 0.50 0.53
250 -2.38 -2.32 -2.13 -1.84 -1.47 -1.06 -0.66 -0.28 0.03 0.27 0.42 0.51 0.53
300 -2.21 -2.15 -1.97 -1.70 -1.35 -0.97 -0.58 -0.24 0.06 0.28 0.43 0.51 0.53
350 -2.03 -1.97 -1.81 -1.55 -1.22 -0.87 -0.51 -0.18 0.09 0.30 0.44 0.51 0.53
400 -1.83 -1.78 -1.63 -1.39 -1.09 -0.76 -0.43 -0.13 0.12 0.32 0.44 0.51 0.53
450 -1.62 -1.57 -1.43 -1.21 -0.94 -0.64 -0.34 -0.07 0.16 0.34 0.45 0.51 0.53
500 -1.38 -1.34 -1.21 -1.02 -0.78 -0.51 -0.25 -0.00 0.20 0.36 0.46 0.52 0.53
550 -1.10 -1.07 -0.96 -0.80 -0.59 -0.36 -0.13 0.08 0.25 0.38 0.47 0.52 0.53
600 -0.77 -0.74 -0.66 -0.53 -0.36 -0.18 0.00 0.17 0.31 0.41 0.48 0.52 0.53

    These values for vlurch 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).

vcloud, ft s-1

PE, ft- lb Θcl   ~0 15 30 45 60 75 90 105 120 135 150 165 180
0 381 381 382 382 383 383 383 383 383 383 383 383 383
50 367 367 368 368 368 368 369 369 369 369 369 369 369
100 353 353 353 353 353 354 354 354 354 354 354 354 354
150 337 337 337 338 338 338 338 339 339 339 339 339 339
200 321 321 322 322 322 322 322 322 322 322 322 322 322
250 304 304 305 305 305 305 305 305 305 305 305 305 305
300 286 286 287 287 287 287 287 287 287 287 287 287 287
350 267 267 268 268 268 268 268 268 268 268 268 268 268
400 247 247 247 247 247 247 248 248 248 248 248 248 248
450 224 224 224 225 225 225 225 225 225 225 225 225 225
500 199 199 199 200 200 200 200 200 200 200 200 200 200
550 171 171 171 171 171 171 171 171 171 171 171 171 171
600 136 136 136 136 136 136 137 136 136 136 136 136 136

    As before, vcloud 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 mhead = 6 lb
    We conclude this series of patterns with the ones for mhead = 6 lb. The results are very similar to those with mhead = 5 and 7 lb.

vbodyafter, ft s-1

PE, ft- lb Θcl   ~0 15 30 45 60 75 90 105 120 135 150 165 180
0 -3.67 -3.57 -3.27 -2.81 -2.23 -1.60 -0.96 -0.38 0.12 0.49 0.73 0.86 0.90
50 -3.56 -3.46 -3.17 -2.72 -2.16 -1.54 -0.92 -0.35 0.13 0.50 0.74 0.86 0.90
100 -3.45 -3.36 -3.07 -2.64 -2.08 -1.48 -0.87 -0.32 0.15 0.50 0.74 0.86 0.90
150 -3.34 -3.25 -2.97 -2.54 -2.01 -1.42 -0.82 -0.28 0.17 0.52 0.74 0.86 0.90
200 -3.22 -3.13 -2.86 -2.45 -1.93 -1.35 -0.78 -0.25 0.19 0.53 0.75 0.86 0.90
250 -3.11 -3.02 -2.76 -2.35 -1.85 -1.29 -0.73 -0.22 0.21 0.54 0.75 0.87 0.90
300 -2.98 -2.90 -2.64 -2.25 -1.76 -1.22 -0.68 -0.18 0.23 0.55 0.76 0.87 0.90
350 -2.86 -2.77 -2.53 -2.15 -1.68 -1.15 -0.63 -0.15 0.26 0.56 0.76 0.87 0.90
400 -2.72 -2.64 -2.41 -2.04 -1.58 -1.08 -0.57 -0.11 0.28 0.57 0.77 0.87 0.90
450 -2.59 -2.51 -2.28 -1.93 -1.49 -1.00 -0.52 -0.07 0.30 0.58 0.77 0.87 0.90
500 -2.45 -2.37 -2.15 -1.82 -1.39 -0.93 -0.46 -0.03 0.32 0.60 0.78 0.87 0.90
550 -2.30 -2.23 -2.02 -1.70 -1.29 -0.85 -0.40 0.01 0.35 0.61 0.78 0.87 0.90
600 -2.14 -2.07 -1.88 -1.57 -1.18 -0.76 -0.34 0.05 0.38 0.62 0.79 0.87 0.90

    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 mhead = 5 and 7 lb..

vcloud, ft s-1

PE, ft- lb Θcl   ~0 15 30 45 60 75 90 105 120 135 150 165 180
0 477 477 478 478 478 479 479 479 479 479 479 479 479
50 466 466 466 467 467 467 468 468 468 468 468 468 468
100 455 455 455 455 456 456 456 456 456 456 456 456 456
150 443 443 443 443 444 444 444 444 444 444 444 444 444
200 431 431 431 431 432 432 432 432 432 432 432 432 432
250 418 418 419 419 419 419 420 420 420 420 420 420 420
300 405 405 406 406 406 407 407 407 407 407 407 407 407
350 392 392 392 393 393 393 393 393 393 393 393 393 393
400 378 378 379 379 379 379 380 380 380 380 380 380 379
450 364 364 364 365 365 365 365 365 365 365 365 365 365
500 349 349 350 350 350 350 350 350 350 350 350 350 350
550 334 334 334 334 335 335 335 335 335 335 335 335 335
600 318 318 318 318 318 318 318 319 318 318 318 318 318

    The pattern for vcloud 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 mhead = 9 and 10 lb. This is a major constraint, which together with the mass of the brain restricts mhead to 58 lb. The second is that virtually all the values for vlurch 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 mhead = 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 mhead.

Mhead, lb vlurch, ft s-1 vcloud, ft s-1 PE, ft-lb Θcl
5 -0.3 to -3.0 330450 100600 3090
6 -0.3 to -3.0 320440 100600 3090
7 -0.3 to -2.9 290420 100600 3090
8 0.0 to -2.4 140340 100600 3090
9
10

Ahead to Constraints on mcloud and vcloud from mcloud vs. vcloud
Back to Constraints on mcloud and vcloud from Θcl and mcloud

Back to Intro to Constraints

Back to Physics of the Head Shot