12. Lurch 4 Linearadd 3-D motion of the large fragments

The model and its justification
    Like Lurch 3 Linear, Lurch 4 Linear calculates the speed of the rearward lurch of the upper body from the bullet, the body, the cloud, the large fragments, and three-dimensional motion of the cloud. To these it adds three-dimensional motion of the large fragments. This addition hardly changes the answer (from -0.88 ft s-1 to -0.87 ft s-1 for default values of the variables). This shows that the 3-D motion of the large fragments is not an important consideration.

Solving the simultaneous equations
    As before, the equations for conservation of X-momentum and total energy are solved simultaneously. To the version in Lurch 3 are added two terms that express the three-dimensionality of the motion of the two large fragmentsfkefrag1 and fkefrags23. These terms are roughly analogous to the earlier fkecl, and range upward from 1 because the the kinetic energy of a fragment will always be greater than the value calculated from the X- and Z-components of its velocity that are derived from the Zapruder film. To this must be added a component perpendicular to the film, which we can call the Y-component. A fkefrag of 2 would mean that the fragment has twice the kinetic energy of that calculated from its X- and Z-velocities.
    There is no reliable way to tell how much more kinetic energy the two large fragments had than is shown in the film's XZ plane, because we can't determine their velocities perpendicular to the film. Thus we can only estimate values for fkefrag1 and fkefrags23. To be conservative, we use default values of 1.25 for both and ranges of 1 to 1.5 It turns out not to matter much, for the speed of the lurch is very insensitive to both terms.

Conservation of X-momentum

 

Conservation of total energy

Preset 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  vfrag1 = 500 ft s-1
Q = 12  Qfrags23 = 70  vfrags23 = 500 ft s-1
PE = 300 ft-lb Qcl = 70  fkefrag1 = 1.25
  fxcl = 0.59 (calc.) fkefrags23 = 1.25

Default solutions to the simultaneous equations

vbodyafter = -1.02 ft s-1  vcloud = 378 ft s-1

Distributions of momentum and energy

Momentum Energy, ft-lb
Before After Before After
Pbullet = 1.26 Pbulletafter = 0.14 KEbullet = 1164  KEbulletafter = 14
  Pbodyafter = -1.35   KEbodyafter = 0.3
  Pcloud = 2.10   KEcloud = 669
  Pfrag1 = 0.32   KEfrag1 = 132
  Pfrags23 = 0.05    KEfrags23 = 49
      PE = 300

Sensitivity analysis
    This sensitivity analysis is practically identical to the one for Lurch 3 Linear. The two new terms, fkefrag1 and fkefrags23, have minimal sensitivities and minimal effects on the lurch. Thus they are unimportant variables.

Sensitivity analysis, Lurch 4 linear
(Standard conditions in boldface)

mbullet

vlurch

mbody

vlurch

mcloud

vlurch

PE

vlurch

vbullet

vlurch

 

 

 

 

 

 

0

-1.343

 

 

156

-1.004

 

 

 

 

50

-1.293

1750

-0.970

157

-1.007

65

-1.336

 

 

100

-1.242

1760

-0.980

158

-1.011

70

-1.240

 

 

150

-1.189

1770

-0.990

159

-1.014

75

-1.157

0.1

-0.351

200

-1.135

1780

-1.000

160

-1.017

80

-1.085

0.2

-0.729

250

-1.079

1790

-1.011

161

-1.021

85

-1.021

0.3

-1.021

300

-1.021

1800

-1.021

162

-1.024

90

-0.964

0.4

-1.267

350

-0.960

1810

-1.031

163

-1.027

9

-0.913

0.5

-1.484

400

-0.898

1820

-1.040

164

-1.030

100

-0.867

0.6

-1.682

450

-0.832

1830

-1.050

165

-1.033

105

-0.826

0.7

-1.863

500

-0.763

1840

-1.060

166

-1.036

 

 

0.8

-2.033

550

-0.690

1850

-1.070

 

 

 

 

 

 

600

-0.613

 

 

Sensitivity = 
[(-1.027 + 1.014)/4]/
[1.021/161] = -0.51

Sensitivity = 
[(-0.964 + 1.085)/10]/
[1.021/85] = 1.01

Sensitivity = 
[(-1.267 + 0.729)/0.2]/
[1.021/0.3] = -0.79

Sensitivity = 
[(-0.960 + 1.079)/100]/
[1.021/300] = 0.35

Sensitivity = 
[(-1.031 + 1.011)/20]/
[1.021/1800] = -1.76

Range = 0.03

Range = 0.51

Range = 1.68

Range = 0.73

Range = 0.10

vbulletafter

vlurch

Q

vlurch

Qcl

vlurch

mfrag1

vlurch

mfrags23

vlurch

 

 

7

-1.007

20

-2.005

0.022

-1.004

0.005

-1.029

0

-0.929

8

-1.009

30

-1.877

0.023

-1.007

0.006

-1.027

50

-0.955

9

-1.011

40

-1.705

0.024

-1.011

0.007

-1.026

100

-0.979

10

-1.014

50

-1.499

0.025

-1.014

0.008

-1.024

150

-1.001

11

-1.017

60

-1.268

0.026

-1.017

0.009

-1.022

200

-1.021

12

-1.021

70

-1.021

0.027

-1.021

0.010

-1.021

250

-1.038

13

-1.024

80

-0.769

0.028

-1.024

0.011

-1.019

300

-1.053

14

-1.028

90

-0.522

0.029

-1.027

0.012

-1.017

350

-1.067

15

-1.033

100

-0.289

0.030

-1.030

0.013

-1.015

400

-1.077

16

-1.037

110

-0.078

0.031

-1.034

0.015

-1.014

 

 

17

-1.042

120

+0.015

0.032

-1.037

0.015

-1.012

Sensitivity = 
[(-1.038 + 1.001)/100]/
[1.021/200] = -0.07

Sensitivity = 
[(-1.028 + 1.014)/4]/
[1.021/12] = -0.04

Sensitivity = 
[(-0.769 + 1.268)/20]/
[1.021/70] = 1.71

Sensitivity = 
[-1.024 + 1.017)/0.002]/
[1.021/0.027] = -0.09

Sensitivity = 
[(-1.017 + 1.024)/0.004]/
[1.021/0.010] = 0.02

Range = 0.15

Range = 0.04

Range = 2.11

Range = 0.03

Range = 0.02

vfrag1

vlurch

vfrags23

vlurch

Qfrag1

vlurch

Qfrags23

vlurch

250

-1.011

250

-1.043

 

 

 

 

300

-1.020

300

-1.041

20

-1.076

50

-1.056

350

-1.025

350

-1.038

25

-1.065

55

-1.048

400

-1.027

400

-1.033

30

-1.052

60

-1.039

450

-1.026

450

-1.028

35

-1.038

65

-1.030

500

-1.021

500

-1.021

40

-1.021

70

-1.021

550

-1.012

550

-1.012

45

-1.002

75

-1.011

600

-0.999

600

-1.003

50

-0.981

80

-1.001

650

-0.982

650

-0.992

55

-0.959

85

-0.991

700

-0.961

700

-0.980

60

-0.936

90

-0.980

700

-0.934

750

-0.967

 

 

 

 

Sensitivity = 
[(-1.012 + 1.026)/100]/
[1.021/500] = 0.07

Sensitivity = 
[(-1.012 + 1.028)/100]/
[1.021/500] = 0.08

Sensitivity = 
[(-1.002 + 1.038)/10]/
[1.021/40] = 0.14

Sensitivity = 
[(-1.011 + 1.030)/10]/
[1.021/70] = 0.13

Range = 0.08

Range = 0.08

Range = 0.14

Range = 0.08

fkefrag1

vlurch

fkefrags23

vlurch

1.00

-1.051

1.00

-1.032

1.05

-1.045

1.05

-1.030

1.10

-1.039

1.10

-1.028

1.15

-1.033

1.15

-1.025

1.20

-1.027

1.20

-1.023

1.25

-1.021

1.25

-1.021

1.30

-1.014

1.30

-1.018

1.35

-1.008

1.35

-1.016

1.40

-1.002

1.40

-1.014

1.45

-0.995

1.45

-1.011

1.50

-0.989

1.50

-1.009

Sensitivity = 
[(-1.014 + 1.027)/0.10]/
[1.021/1.25] = 0.16

Sensitivity = 
[(-1.018 + 1.023)/0.1]/
[1.021/1.25] = 0.06

Range = 0.06

Range = 0.02

Ordered summary of sensitivities

Variable Sensitivity of vlurch Range of vlurch, ft s-1 Magnitude
Positive effect on lurch (reduces rearward velocity)
mfrags23 0.02 0.02 Small
fkefrags23 0.06 0.02 Small
fkefrag1 0.16 0.06 Small
vfrag1 0.07 0.08 Small
vfrags23 0.08 0.08 Small
Qfrags23 0.13 0.08 Small
Qfrag1  0.14 0.14 Small
mbody 1.01 0.51 Medium
PE 0.35 0.73 Medium
Qcl 1.71 2.11 Large
Negative effect on lurch (increases rearward velocity)
mbullet -0.51 0.03 Small
mfrag1 -0.09 0.03 Small
Q  -0.04 0.04 Small
vbullet -1.76 0.10 Small
vbulletafter -0.07 0.15 Small
mcloud -0.79 1.68 Large

    This list of sensitivities is virtually the same as for Lurch 3 Linear. The two additional variables from the three-dimensionality of the fragments hardly changed the solutio, and had two of the three smallest effects of any of the ten intensifying variables.

Summary
    Adding three-dimensional effects for the two large fragments hardly changed anything about the solution, presumably because the fragments were minor parts of the solution to begin with. The dominant contributors to the solution remained the mass and half-angle of the cloud.

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