14. Snap/Lurch 6 Linearadd four time intervals

The model and its justification
Adding the four time intervals to the model represents a significant complication to it. It represents the fact that we know only what happened from one frame to the next, not anything within a frame. But the details within a frame can strongly change the answer, and we must take account of their effect on the answer and on its uncertainties.
This procedure calculates the speed of the rearward lurch of JFK's upper body from the bullet, the body, the cloud, and the large fragments, and incorporates 3-D motions of the cloud, the large fragments, and the body: Before doing that, however, it interposes three steps having to do with the times. (1) It first calculates vsnap and uses it to calculate tsnap. (2) Then it calculates ttransit and combines it with tsnap with tdelay to calculate tlurch. (3) Then it uses tlurch and dfrags to calculate vfrags. At this point it returns to the former procedure and (4) solves for vbodyafter and vcloud. To simplify matters a bit, it sets vfrags23 = vfrag1 and fkefrags23 = fkefrag1.
Including these time intervals leaves the default solution about the same: -1.04 and 401 ft s-1 for lurch and cloud, respectively, versus -1.02 and 378 ft s-1 in Lurch 5 Linear. The sensitivities and effects change, however, presumably to better reflect the true situation.

(1) Equations for tsnap  (2) Equations for tlurch  (3) Equations for vfrags  (4) Simultaneous equations for vbodyafter and vcloud

##### fkefrags23 = fkefrag1

##### Conservation of X-momentum ##### Conservation of total energy Default values of variables

 mbullet = 161 gr vbullet = 1800 ft s-1 mfrag1 = 0.027 gr mbody = 85 lb vbulletafter = 200 ft s-1 mfrags23 = 0.01 gr mcloud = 0.3 lb Qfrag1 = 40° fkefrag1 = 1.25 mhead = 7 lb Qfrags23 = 70° fkefrags23 = fkefrag1 Q = 12° Qcl = 70° fkebody = 1.2 PE = 300 ft-lb tdelay = 0.002 s fxcl = 0.59 (calc.) dtransit = 4 in dsnap = 2.2 in dfrag1 = 6 ft

### Solutions to simultaneous equations

vbodyafter = -1.04 ft s-1  vcloud = 401 ft s-1
vfrag1
= vfrags23 = 363 ft s-1
vsnap
= 5.1 ft s-1

ttransit
= 0.33 ms
tsnap
= 35.8 ms
tlurch
= 16.5 ms

## Distributions of momentum and energy

 Momentum Energy Before After Before After Pbullet = 1.26 Pbulletafter = 0.14 KEbullet = 1164 KEbulletafter = 14 Pbodyafter = -1.38 KEbodyafter = 0.4 Pcloud = 2.22 KEcloud = 754 Pfrag1 = 0.24 KEfrag1 = 70 Pfrags23 = 0.04 KEfrags23 = 26 PE = 300

Sensitivity analysis

Sensitivity analysis, Snap/Lurch 6 Linear
(Standard conditions in boldface)

 mbullet vlurch mbody vlurch mcloud vlurch mhead vlurch vbullet vlurch 4.0 -1.017 156 -1.025 4.5 -1.021 1750 -0.993 157 -1.029 65 -1.362 5.0 -1.026 1760 -1.003 158 -1.032 70 -1.265 5.5 -1.031 1770 -1.012 159 -1.035 75 -1.180 0.1 -0.329 6.0 -1.035 1780 -1.022 160 -1.038 80 -1.106 0.2 -0.731 6.5 -1.039 1790 -1.031 161 -1.041 85 -1.041 0.3 -1.041 7.0 -1.041 1800 -1.041 162 -1.043 90 -0.983 0.4 -1.302 7.5 -1.035 1810 -1.050 163 -1.046 95 -0.931 0.5 -1.533 8.0 -1.012 1820 -1.059 164 -1.048 100 -0.884 0.6 -1.742 8.5 -0.931 1830 -1.068 165 -1.051 105 -0.842 0.7 -1.935 9.0 -0.548 1840 -1.076 166 -1.053 0.8 -2.115 9.5 — 1850 -1.085 10.0 — Sensitivity =  [(-1.043 + 1.038)/2]/ [1.041/161] = -0.39 Sensitivity =  [(-0.983 + 1.106)/10]/ [1.041/85] = 1.00 Sensitivity =  [(-1.302 + 0.731)/0.2]/ [1.041/0.3] = -0.82 Sensitivity =  [(-1.012 + 1.035)/2]/ [1.041/7.0] = 0.08 Sensitivity =  [(-1.050 + 1.031)/20]/ [1.041/1800] = -1.64 Range = 0.03 Range = 0.52 Range = 1.79 Range = 0.47 Range = 0.09

 vbulletafter vlurch PE vlurch Q vlurch Qcl vlurch mfrag1 vlurch 0 -1.347 50 -1.299 7 -1.027 20 -2.086 0.022 -1.022 0 -0.944 100 -1.250 8 -1.029 30 -1.949 0.023 -1.026 50 -0.972 150 -1.200 9 -1.031 40 -1.767 0.024 -1.029 100 -0.997 200 -1.149 10 -1.034 50 -1.549 0.025 -1.033 150 -1.020 250 -1.095 11 -1.037 60 -1.303 0.026 -1.037 200 -1.041 300 -1.041 12 -1.041 70 -1.041 0.027 -1.041 250 -1.057 350 -0.984 13 -1.044 80 -0.773 0.028 -1.044 300 -1.070 400 -0.925 14 -1.048 90 -0.511 0.029 -1.048 350 -1.074 450 -0.864 15 -1.052 100 -0.264 0.030 -1.052 400 -1.066 500 -0.801 16 -1.057 110 -0.040 0.031 -1.055 550 -0.734 17 -1.061 120 +0.155 0.032 -1.059 600 -0.664 Sensitivity =  [(-1.057 + 1.020)/100]/ [1.041/200] = -0.07 Sensitivity =  [(-0.984 + 1.095)/100]/ [1.041/300] = 0.32 Sensitivity =  [(-1.048 + 1.034)/4]/ [1.041/12] = -0.04 Sensitivity =  [(-0.773 + 1.303)/20]/ [1.041/70] = 1.78 Sensitivity =  [-1.048 + 1.033)/0.004]/ [1.041/0.027] = -0.10 Range = 0.12 Range = 0.68 Range = 0.03 Range = 2.24 Range = 0.04

 mfrags23 vlurch Qfrag1 vlurch Qfrags23 vlurch fkefrags vlurch fkebody vlurch 0.005 -1.040 1.00 -1.062 0.006 -1.040 20 -1.081 50 -1.066 1.05 -1.058 1.00 -1.041 0.007 -1.040 25 -1.073 55 -1.060 1.10 -1.053 1.05 -1.041 0.008 -1.040 30 -1.064 60 -1.054 1.15 -1.049 1.10 -1.041 0.009 -1.041 35 -1.053 65 -1.048 1.20 -1.045 1.15 -1.041 0.010 -1.041 40 -1.041 70 -1.041 1.25 -1.041 1.20 -1.041 0.011 -1.041 45 -1.027 75 -1.033 1.30 -1.036 1.25 -1.041 0.012 -1.041 50 -1.012 80 -1.026 1.35 -1.032 1.30 -1.041 0.013 -1.041 55 -0.996 85 -1.019 1.40 -1.028 1.35 -1.041 0.015 -1.041 60 -0.979 90 -1.011 1.45 -1.024 1.40 -1.041 0.015 -1.041 1.50 -1.019 Sensitivity =  [(-1.041 + 1.040)/0.010]/ [1.041/0.010] = -0.00 Sensitivity =  [(-1.027 + 1.053)/10]/ [1.041/40] = 0.10 Sensitivity =  [(-1.033 + 1.048)/10]/ [1.041/70] = 0.10 Sensitivity =  [(-1.036 + 1.045)/0.10]/ [1.041/1.25] = 0.11 Sensitivity =  [(-1.041 + 1.041)/0.40]/ [1.041/1.20] = 0.00 Range = 0.00 Range = 0.10 Range = 0.06 Range = 0.04 Range = 0.00

 dtransit vlurch dsnap vlurch dfrag1 vlurch tdelay vlurch 0 -1.041 2 -0.995 1 -1.041 1.6 -1.027 3 -1.015 2 -1.041 1.8 -1.033 4 -1.030 0.000 -1.040 3 -1.041 2.0 -1.038 5 -1.038 0.001 -1.041 4 -1.041 2.2 -1.041 6 -1.041 0.002 -1.041 5 -1.041 2.4 -1.032 7 -1.036 0.003 -1.040 6 -1.041 2.6 -0.981 8 -1.025 0.004 -1.037 7 -1.040 2.8 -0.684 9 -1.005 0.005 -1.033 8 -1.040 10 -0.977 0.006 -1.026 0.007 -1.014 Sensitivity =  [(-1.040 + 1.041)/8]/ [1.041/4] = 0.00 Sensitivity =  [(-1.032 + 1.038)/0.4]/ [1.041/2.2] = 0.03 Sensitivity =  [(-1.025 + 1.030)/4]/ [1.041/6] = 0.01 Sensitivity =  [(-1.037 + 1.040)/0.004]/ [1.041/0.002] = 0.00 Range = 0.00 Range = 0.34 Range = 0.02 Range = 0.03

Ordered summary of sensitivities

 Variable Sensitivity of vlurch Range of vlurch, ft s-1 Magnitude Positive effect on lurch (reduces rearward velocity) fkebody 0.00 0.00 Small dtransit 0.00 0.00 Small dfrag1 0.01 0.02 Small tdelay 0.00 0.03 Small fkefrag1 0.11 0.04 Small Qfrags23 0.10 0.06 Small Qfrag1 0.10 0.10 Small dsnap 0.03 0.34 Small mhead 0.08 0.47 Medium mbody 1.00 0.52 Medium PE 0.32 0.68 Medium Qcl 1.78 2.24 Large Negative effect on lurch (increases rearward velocity) mfrags23 -0.00 0.00 Small mbullet -0.39 0.03 Small Q -0.04 0.03 Small mfrag1 -0.10 0.04 Small vbullet -1.64 0.09 Small vbulletafter -0.07 0.12 Small mcloud -0.82 1.79 Large

This ordered summary of sensitivities is strikingly similar to the previous one, given the differences in calculation introduced here. All the previous variables maintained very similar sensitivities and ranges, with Qcl and mcloud continuing to dominate. The five new variables related to the timing and the snap all had positive sensitivities, that is, they worked to increase the velocity of the lurch (to reduce its rearward velocity). In spite of this, the lurch hardly changed, and if anything, increased in rearward speed (from -1.02 ft s-1 to -1.04 ft s-1). This tiny effect is probably because the sensitivities and effects of the new variables were so smallthree were negligible (0.000.03 ft s-1), and two were small to medium (0.34 and 0.47 ft s-1). Thus, adding the four time intervals did not change the basic result.

Summary
The basic solution remained virtually unchanged when the four time intervals were added: The lurch of -1.02 ft s-1 in Lurch 5 Linear increased (in magnitude) only to -1.04 ft s-1 in Snap/Lurch 6 Linear. This important result was probably a consequence of the small sensitivities and effects of the five new variables.