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.

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