13. Lurch 5 Linearadd 3-D motion of body

The model and its justification
    Having added 3-D motion to the large fragments in Lurch 4 Linear, we now try the same for the body. We can neglect the component of the momentum into or out of the plane of the Z-film because are dealing only with X-momentum. But we cannot neglect that component of the kinetic energy, because the equation for conservation of total energy uses total kinetic energy. As happened when adding the 3-D component of the fragments, the 3-D component of the body's KE hardly changes the answer at all. In fact, the previous -1.02 ft s-1 for the default values of variables remains the same here. This the 3-D motion of the body is an insignificant term in the overall equation.
    The equations calculate the speed of the rearward lurch of the upper body from the bullet, the body (in 3-D), the cloud (in 3-D), and the large fragments (also in 3-D).

Solving the simultaneous equations
    The three-dimensionality of the body's kinetic energy is expressed by the term fkebody, which is similar in form and function to fkefrag. We assume a range of 11.4 and a starting value of 1.2.

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  vfrag1 = 500 ft s-1
Q = 12   Qfrags23 = 70  vfrags23 = 500 ft s-1
PE = 300 ft-lb Qcl = 70  fkefrag1 = 1.25
fkebody = 1.2 fxcl = 0.59 (calc.) fkefrags23 = 1.25

Default solutions to 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 = 0.4
  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 4 Linear. The new term, fkebody, has undetectable sensitivity and range of effect.

Sensitivity analysis, Lurch 5 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

-1.030

163

-1.027

95

-0.913

0.5

-1.484

400

-0.897

1820

-1.040

164

-1.030

100

-0.867

0.6

-1.681

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 = 
[(-1.024 + 1.017)/2]/
[1.021/161] = -0.55

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 = 
[(-1.030 + 1.011)/20]/
[1.021/1800] = -1.68

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

-1.006

20

-2.005

0.022

-1.004

0.005

-1.029

0

-0.929

8

-1.009

30

-1.876

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

-1.267

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

-1.066

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

-1.033

0.015

-1.013

 

 

17

-1.042

120

+0.105

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 + 1.267)/20]/
[1.021/70] = 1.71

Sensitivity = 
[-1.027 + 1.014)/0.004]/
[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

-1.025

450

-1.027

35

-1.037

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

-0.990

700

-0.961

700

-0.980

60

-0.936

90

-0.980

700

-0.934

750

-0.967

 

 

 

 

Sensitivity = 
[(-1.012 + 1.025)/100]/
[1.021/500] = 0.06

Sensitivity = 
[(-1.012 + 1.027)/100]/
[1.021/500] = 0.07

Sensitivity = 
[(-1.002 + 1.037)/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

fkebody

vlurch

1.00

-1.051

1.00

-1.032

 

 

1.05

-1.045

1.05

-1.030

1.00

-1.021

1.10

-1.039

1.10

-1.027

1.05

-1.021

1.15

-1.033

1.15

-1.025

1.10

-1.021

1.20

-1.027

1.20

-1.023

1.15

-1.021

1.25

-1.021

1.25

-1.021

1.20

-1.021

1.30

-1.014

1.30

-1.018

1.25

-1.021

1.35

-1.008

1.35

-1.016

1.30

-1.020

1.40

-1.002

1.40

-1.014

1.35

-1.020

1.45

-0.995

1.45

-1.011

1.40

-1.020

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)/0.10]/
[1.021/1.25] = 0.06

Sensitivity = 
[(-1.020 + 1.021)/0.4]/
[1.021/1.20] = 0.00

Range = 0.06

Range = 0.02

Range = 0.00

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
mfrags23 0.02 0.02 Small
fkefrags23 0.06 0.02 Small
fkefrag1 0.16 0.06 Small
vfrag1 0.06 0.08 Small
vfrags23 0.07 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 -0.55 0.03 Small
mfrag1 -0.09 0.03 Small
Q  -0.04 0.04 Small
vbullet -1.68 0.10 Small
vbulletafter -0.07 0.15 Small
mcloud -0.79 1.68 Large

    These sensitivities and effects are virtually the same as for Lurch 4 Linear. The new term, fkebody, has undetectable sensitivity and range of effect, i.e., counts for nothing. This is a consequence of the fact that the body gets virtually none of the final kinetic energy.

Summary
    Adding three-dimensionality to the body's KE does not change the solution perceptibly, presumably because the body gets almost none of the KE after the explosion (<0.1%). This is confirmed by the undetectable sensitivity and range of effect for fkebody. The two variables with the largest effects remain Qcl (2.11 ft s-1) and mcloud (1.68 ft s-1). Only two variables have medium effects, PE (0.73 ft s-1) and mbody (0.51 ft s-1). Thus the cloud remains all-important to the solution.

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