Axiomatic development of classical mechanics...2015/08/25 · Lecture 1 Tue. 8.25.2015 Axiomatic development of classical mechanics (Ch. 1 and Ch. 2 of Unit 1) Geometry of momentum

Lecture 1 Tue. 8.25.2015

Axiomatic development of classical mechanics(Ch. 1 and Ch. 2 of Unit 1)

Geometry of momentum conservation axiomTotally Inelastic “ka-runch”collisions*Perfectly Elastic “ka-bong” and Center Of Momentum (COM) symmetry*Comments on idealization in classical models

Geometry of Galilean translation symmetry45° shift in (V1,V2)-spaceTime reversal symmetry...of COM collisions

Algebra,Geometry, and Physics of momentum conservation axiom Vector algebra of collisionsMatrix or tensor algebra of collisions Deriving Energy Conservation Theorem

Numerical details of collision tensor algebra

*Download Superball Collision Simulator http://www.uark.edu/ua/modphys/testing/markup/BounceItWeb.html

*Car Collision simulator http://www.uark.edu/ua/modphys/testing/markup/CMMotionWeb.html

1Wednesday, August 26, 2015

http://www.uark.edu/rso/modphys/animations/BounceItWeb.html


http://www.uark.edu/ua/modphys/testing/markup/CMMotionWeb.html




Ka-bong!Ka-bong! Ka-runch!Ka-runch!

Totallyinelastic

case

Perfectlyelasticcase

??????????

1060

????????

1060

0-0.2-0.4-0.6-0.8-1-6 sec.-12 sec.

-24 sec.

-36 sec.

-48 sec.

1 mile

1minute(60sec.) (b) Collision!

After collision...what velocities?Before collision.....A problem in ssppaaccee--ttiimmee : (6600mph Cell-faxing 4ton SUV rear-ends 1100mph 1ton VW)

6600mph1100mph








Totallyinelasticcase


??????????

1060

????????

1060

0-0.2-0.4-0.6-0.8-1-6 sec.-12 sec.

-24 sec.

-36 sec.

-48 sec.

1 mile



6600mph

Conventional solution:Get out formulas:ΣmV(before)=ΣmV(before) [momentum conservation]ΣmV2(before)=ΣmV2(before)[energy conservation]etc.

Conventional solution: Look up the usualmomentum and energy formulas/axioms:ΣimVi(initial) = ΣimVi(final)ΣimV2i(initial) = ΣimV2i(final)and solve...










??????????

1060

????????

1060

0-0.2-0.4-0.6-0.8-1-6 sec.-12 sec.

-24 sec.

-36 sec.

-48 sec.

1 mile



6600mph


...But an UNconventional way is quicker and slicker.......... (Just have to draw 2 lines!)











??????????

1060

????????

1060

0-0.2-0.4-0.6-0.8-1-6 sec.-12 sec.

-24 sec.

-36 sec.

-48 sec.

1 mile



50mph20 40 60 90

20

40

60

80

50mph

100mph

100mph

VVVW

0

INITIAL

10 30 70 80

90

10

30

70

I

VVSUV

6600mph1100mph

VVeelloocciittyy-vveelloocciittyy PPlloott

6600mph

1100mph




*Car Collision simulator http://www.uark.edu/ua/modphys/testing/markup/CMMotionWeb.html5Wednesday, August 26, 2015








??????????

1060

????????

1060

0-0.2-0.4-0.6-0.8-1-6 sec.-12 sec.

-24 sec.

-36 sec.

-48 sec.

1 mile



50mph20 40 60 90

20

40

60

80

50mph

100mph

100mph

VVVW

0

INITIAL

10 30 70 80

90

10

30

70

I

VVSUV

6600mph1100mph


6600mph

1100mph

Conventional solution:Get out formulas:ΣmV(before)=ΣmV(before) [momentum conservation]ΣmV2(before)=ΣmV2(before)[energy conservation]etc.MSUVVVSUV+MVWVVVW =constant is AAxxiioomm ##11


Conventional solution: Look up the usualmomentum and energy formulas/axioms:ΣimVi(initial) = ΣimVi(final)ΣimV2i(initial) = ΣimV2i(final)









??????????

1060

????????

1060

0-0.2-0.4-0.6-0.8-1-6 sec.-12 sec.

-24 sec.

-36 sec.

-48 sec.

1 mile



50mph20 40 60 90

20

40

60

80

50mph

100mph

100mph

VVVW

0

INITIAL

10 30 70 80

90

10

30

70

I

VVSUV

4

-1

slope =-4

6600mph1100mph


6600mph

1100mph


50mph20 40 60 90

20

40

60

80

50mph

100mph

100mph

VVVW

0

INITIAL

10 30 70 80

90

10

30

70

I

VVSUV6600mph

1100mph

= ΔVSUV

ΔVVWMSUV-mVW

4-1 =

MSUVVVSUV+MVWVVVW =constant is AAxxiioomm ##11











Totally

inelastic

case

Perfectly

elastic

case

??????????

1060

????????

1060

0-0.2-0.4-0.6-0.8-1

-6 sec.

-12 sec.

-24 sec.

-36 sec.

-48 sec.

1 mile


After collision...what velocities?Before collision.....

A problem in ssppaaccee--ttiimmee : (6600mph Cell-faxing 4ton SUV rear-ends 1100mph 1ton VW)

50mph20 40 60 90

20

40

60

80

50mph

100mph

100mph

VVVW

0

INITIAL

10 30 70 80

90

10

30

70

I

VVSUV

“Ka-Runch!”

V VW

=V SUV

line

FINAL

Ka-runch!

4

-1

slope =-4

6600mph1100mph


6600mph

1100mph

Conventional solution:

Get out formulas:

ΣmV(before)=ΣmV(before) [momentum conservation]ΣmV2(before)=ΣmV2(before)[energy conservation]etc.

50mph20 40 60 90

20

40

60

80

50mph

100mph

100mph

VVVW

0

INITIAL

10 30 70 80

90

10

30

70

F

45°

45°

I

VVSUV

6600mph

1100mph

= ΔVSUV

ΔVVW

MSUV

-mVW

4

-1 =

MSUVVVSUV

+MVWVVVW

=constant is AAxxiioomm ##11

BINGO!slope=+1


Totally Inelastic (Ka − Runch!)






Totally

inelastic

case

Perfectly

elastic

case

??????????

1060

????????

1060

0-0.2-0.4-0.6-0.8-1

-6 sec.

-12 sec.

-24 sec.

-36 sec.

-48 sec.

1 mile




50mph20 40 60 90

20

40

60

80

50mph

100mph

100mph

VVVW

0

INITIAL

10 30 70 80

90

10

30

70

I

VVSUV

“Ka-Runch!”

V VW

=V SUV

line

FINAL

Ka-runch!

4

-1

slope =-4

6600mph1100mph


6600mph

1100mph


Get out formulas:


50mph20 40 60 90

20

40

60

80

50mph

100mph

100mph

VVVW

0

INITIAL

10 30 70 80

90

10

30

70

F

45°

45°

I

VVSUV

6600mph

1100mph

= ΔVSUV

ΔVVW

MSUV

-mVW

4

-1 =

MSUVVVSUV

+MVWVVVW


BINGO!slope=+1


Totally Inelastic (Ka − Runch!)




Totally

inelastic

case

Perfectly

elastic

case

??????????

1060

????????

1060

0-0.2-0.4-0.6-0.8-1

-6 sec.

-12 sec.

-24 sec.

-36 sec.

-48 sec.

1 mile




50mph20 40 60 90

20

40

60

80

50mph

100mph

100mph

VVVW

0

INITIAL

10 30 70 80

90

10

30

70

I

VVSUV

“Ka-Runch!”

V VW

=V SUV

line

FINAL

Ka-runch!

4

-1

slope =-4

6600mph1100mph


6600mph

1100mph


Get out formulas:


50mph20 40 60 90

20

40

60

80

50mph

100mph

100mph

VVVW

0

INITIAL

10 30 70 80

90

10

30

70

F

45°

45°

I

VVSUV

6600mph

1100mph

= ΔVSUV

ΔVVW

MSUV

-mVW

4

-1 =

FINAL

Ka-bong!

MSUVVVSUV

+MVWVVVW


BINGO!

DOUBLE

BINGO!

“!hnuR-aK”

...But an UNconventional way is quicker and slicker.......... (Just have to draw 2 lines!) ... (and a circle...)

Totally Elastic (Ka − Bong!)




Totally

inelastic

case

Perfectly

elastic

case

??????????

1060

????????

1060

0-0.2-0.4-0.6-0.8-1

-6 sec.

-12 sec.

-24 sec.

-36 sec.

-48 sec.

1 mile




50mph20 40 60 90

20

40

60

80

50mph

100mph

100mph

VVVW

0

INITIAL

10 30 70 80

90

10

30

70

I

VVSUV

“Ka-Runch!”

V VW

=V SUV

line

FINAL

Ka-runch!

4

-1

slope =-4

6600mph1100mph


6600mph

1100mph


Get out formulas:


50mph20 40 60 90

20

40

60

80

50mph

100mph

100mph

VVVW

0

INITIAL

10 30 70 80

90

10

30

70

F

45°

45°

I

VVSUV

6600mph

1100mph

= ΔVSUV

ΔVVW

MSUV

-mVW

4

-1 =

FINAL

Ka-bong!

MSUVVVSUV

+MVWVVVW


BINGO!

DOUBLE

BINGO!

“!hnuR-aK”

Superball Collision Simulator


*Download Superball Collision Simulator http://www.uark.edu/ua/modphys/testing/markup/BounceItWeb.html13Wednesday, August 26, 2015




Totally

inelastic

case

Perfectly

elastic

case

??????????

1060

????????

1060

0-0.2-0.4-0.6-0.8-1

-6 sec.

-12 sec.

-24 sec.

-36 sec.

-48 sec.

1 mile




50mph20 40 60 90

20

40

60

80

50mph

100mph

100mph

VVVW

0

INITIAL

10 30 70 80

90

10

30

70

I

VVSUV

“Ka-Runch!”

V VW

=V SUV

line

FINAL

Ka-runch!

4

-1

slope =-4

6600mph1100mph


6600mph

1100mph


Get out formulas:


50mph20 40 60 90

20

40

60

80

50mph

100mph

100mph

VVVW

0

INITIAL

10 30 70 80

90

10

30

70

F

45°

45°

I

VVSUV

6600mph

1100mph

= ΔVSUV

ΔVVW

MSUV

-mVW

4

-1 =

FINAL

Ka-bong!

MSUVVVSUV

+MVWVVVW


BINGO!

DOUBLE

BINGO!

“!hnuR-aK”

( )VSUVIN=60mphVVWIN =10mph

50mph20 40 60 90

20

40

60

80

50mph

100mph

100mph

VVW

VSUV

( )VSUVFIN=50mphVVWFIN =50mph

VVW=VSUVline

0

INITIAL

10 30 70 80

90

10

30

70

F FINAL

45°45° “K

a-Runch!”

I

“Ka-Runch!”(Extreme inelastic collision)4

-1

= ΔVSUV

ΔVVWMSUV-mVW

4-1 =slope

Fig. 2.1in Unit 1





Totally

inelastic

case

Perfectly

elastic

case

??????????

1060

????????

1060

0-0.2-0.4-0.6-0.8-1

-6 sec.

-12 sec.

-24 sec.

-36 sec.

-48 sec.

1 mile




50mph20 40 60 90

20

40

60

80

50mph

100mph

100mph

VVVW

0

INITIAL

10 30 70 80

90

10

30

70

I

VVSUV

“Ka-Runch!”

V VW

=V SUV

line

FINAL

Ka-runch!

4

-1

slope =-4

6600mph1100mph


6600mph

1100mph


Get out formulas:


50mph20 40 60 90

20

40

60

80

50mph

100mph

100mph

VVVW

0

INITIAL

10 30 70 80

90

10

30

70

F

45°

45°

I

VVSUV

6600mph

1100mph

= ΔVSUV

ΔVVW

MSUV

-mVW

4

-1 =

FINAL

Ka-bong!

MSUVVVSUV

+MVWVVVW


BINGO!

DOUBLE

BINGO!

“!hnuR-aK”

( )VSUVIN=60mphVVWIN =10mph

50mph20 40 60 90

20

40

60

80

50mph

100mph

100mph

VVW

VSUV

( )VSUVFIN=50mphVVWFIN =50mph

VVW=VSUVline

0

INITIAL

10 30 70 80

90

10

30

70

F FINAL

45°45° “K

a-Runch!”

I

“Ka-Runch!”(Extreme inelastic collision)4

-1

= ΔVSUV

ΔVVWMSUV-mVW

4-1 =slope

Fig. 2.1in Unit 1

Fig. 2.2in Unit 1

Notice “Ka-Bong”Figure 2.2 scaling(ft./min. is more realistic)

Conventional solution: momentum and energy formulas:ΣimVi(initial) = ΣimVi(final)ΣimV2i(initial) = ΣimV2i(final) ( )V

SUV

IN=60ft per min

VVW

IN =10ft per min

50ft per min20 40 60 90

20

40

60

80

50ft per min

100ft per min

100ft per min

VVW

VSUV

( )VSUV

COM=50ft per min

VVW

COM =50ft per minVVW=V

SUV

line

0

“Ka-Bong!” (Ideal elastic collision)

INITIAL

10 30 70 80

90

10

30

70

45°

45°

“Ka-Runch!”

( )VSUV

FIN=40ft per min

VVW

FIN =90ft per min

F FINAL

Center

of

Momentum

“!hnuR-aK”

I




The SUV and VW Idealized thought experiments

Idealization 1. Ignore background.(No rolling friction, air resistance, etc.)

Idealization 2.Make each 1-dimensional.(Cars “constrained” to ride on frictionless rail)

System now hasjust two “dimensions”or “degrees-of-freedom”

Landscape 1.1 Idealized model for collision model and thought experiments


3 translationdimensions


6 translationaldegrees of freedomfor SUV and VW.

Translation (Each body has 3 translational degrees of freedom) (Intoduced in Units 1 and 2)

3 rotationaldimensions

yaw-pitch-rollEuler angles



6 rotationaldegrees of freedomfor SUV and VW.

Rotation (Each body has 3 rotationaldegrees of freedom) (Intoduced in Units 3 and 7)

SUV and VW system involves12 rigid-body degrees of freedom

Summary of Classical Mechanical Degrees of Freedom

Vibration (Each body has many vibrational degrees of freedom) (Intoduced in Units 3-8)

An N-atom molecule has3N-6 vibrational degrees of freedom































Landscape 1.2 Some idealized classical model degrees of freedom

Generalized Curvilinear Coordinates (GCC)introduced in Unit 1 Chapters 10 -12




A problem in ssppaaccee--ttiimmee : (6600mph Cell-faxing 4ton SUV rear-ends 1100mph 1ton VW)Geometry of Galilean translation (A ssyymmmmeettrryy ttrraannssffoorrmmaattiioonn)

5020 40 60 90

10010 30 70 80

FEarth

10

20

30

40

50

60

70

80

90

100

-100

-90

-80

-70

-60

-50

-40

-30

-20

-10-100-90-80-70-60

-50-40-30-20-10

FCOM

ICOM

IEarth

VSUV

VVW

subtracting

(50,50)

(a) Galileo transforms to COM frame

Fig. 2.5ain Unit 1

If you increase your velocity by 50 mph,...

...the rest of the world appears to be 50 mph sslloowweerr

(In some direction x,y, or z…)(In that direction…)



5020 40 60 90

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FEarth

10

20

30

40

50

60

70

80

90

100

-100

-90

-80

-70

-60

-50

-40

-30

-20

-10-100-90-80-70-60

-50-40-30-20-10

FCOM

ICOM

IEarth

VSUV

VVW

subtracting

(50,50)




5020 40 60 90

10010 30 70 80

FEarth

10

20

30

40

50

60

70

80

90

100

-100

-90

-80

-70

-60

-50

-40

-30

-20

-10-100-90-80-70-60

-50-40-30-20-10

FVW(I)

FSUV(F)

ISUV(F)

FCOM

ICOM

FSUV(I)

ISUV(I)

FVW(F)

IVW(F)

F

I

IEarth

IVW(I)

VVW

VSUV

(b) ... and to five or six other reference frames

Fig. 2.5ain Unit 1

Fig. 2.5bin Unit 1

(In some direction x,y, or z…)(In that direction…)





5020 40 60 90

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FEarth

10

20

30

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50

60

70

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100

-100

-90

-80

-70

-60

-50

-40

-30

-20

-10-100-90-80-70-60

-50-40-30-20-10

FCOM

ICOM

IEarth

VSUV

VVW

subtracting

(50,50)


Fig. 2.5ain Unit 1



5020 40 60 90

10010 30 70 80

FEarth

10

20

30

40

50

60

70

80

90

100

-100

-90

-80

-70

-60

-50

-40

-30

-20

-10-100-90-80-70-60

-50-40-30-20-10

FVW(I)

FSUV(F)

ISUV(F)

FCOM

ICOM

FSUV(I)

ISUV(I)

FVW(F)

IVW(F)

F

I

IEarth

IVW(I)

VVW

VSUV


5020 40 60 90

10010 30 70 80

FEarth

10

20

30

40

50

60

70

80

90

100

-100

-90

-80

-70

-60

-50

-40

-30

-20

-10-100-90-80-70-60

-50-40-30-20-10

FVW(I)

ISUV(F)

FCOM

ICOM

FSUV(I)

ISUV(I)

FVW(F)

IVW(F)

F

I

IEarth

IVW(I)

VVW

VSUVFig. 2.5b

in Unit 1

TTiimmee--rreevveerrssaall ((FF--II))ssyymmmmeettrryy ppaaiirrss((FFoouurr eexxaammpplleess))

FSUV(F)

Time-reversal means flip t to -t…(Run a movie backwards)

Final F and Initial Itrade places…



5020 40 60 90

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FEarth

10

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30

40

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60

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-100

-90

-80

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-60

-50

-40

-30

-20

-10-100-90-80-70-60

-50-40-30-20-10

FCOM

ICOM

IEarth

VSUV

VVW

subtracting

(50,50)


Fig. 2.5ain Unit 1



5020 40 60 90

10010 30 70 80

FEarth

10

20

30

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60

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-40

-30

-20

-10-100-90-80-70-60

-50-40-30-20-10

FVW(I)

FSUV(F)

ISUV(F)

FCOM

ICOM

FSUV(I)

ISUV(I)

FVW(F)

IVW(F)

F

I

IEarth

IVW(I)

VVW

VSUV


5020 40 60 90

10010 30 70 80

FEarth

10

20

30

40

50

60

70

80

90

100

-100

-90

-80

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-60

-50

-40

-30

-20

-10-100-90-80-70-60

-50-40-30-20-10

FVW(I)

ISUV(F)

FCOM

ICOM

FSUV(I)

ISUV(I)

FVW(F)

IVW(F)

F

I

IEarth

IVW(I)

VVW

VSUVFig. 2.5b

in Unit 1

TTiimmee--rreevveerrssaall ((FF--II))ssyymmmmeettrryy ppaaiirrss((FFoouurr eexxaammpplleess))

FSUV(F)


That means you flip Velocity V to -V…(Everything goes backwards)

Final F and Initial Itrade places…





5020 40 60 90

10010 30 70 80

FEarth

10

20

30

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60

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-100

-90

-80

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-60

-50

-40

-30

-20

-10-100-90-80-70-60

-50-40-30-20-10

FCOM

ICOM

IEarth

VSUV

VVW

subtracting

(50,50)


Fig. 2.5ain Unit 1



5020 40 60 90

10010 30 70 80

FEarth

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-50-40-30-20-10

FVW(I)

FSUV(F)

ISUV(F)

FCOM

ICOM

FSUV(I)

ISUV(I)

FVW(F)

IVW(F)

F

I

IEarth

IVW(I)

VVW

VSUV


Fig. 2.5bin Unit 1

5020 40 60 90

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FEarth

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20

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-10-100-90-80-70-60

-50-40-30-20-10

FVW(I)

FSUV(F)

ISUV(F)

FCOM

ICOM

FSUV(I)

ISUV(I)

FVW(F)

IVW(F)

F

I

IEarth

IVW(I)

VVW

VSUV

TTHHEECCOOMM TTiimmee--rreevveerrssaallssyymmmmeettrryy ppaaiirr((JJuusstt 11 ccaassee))


That means you flip Velocity V to -V…(Everything goes backwards)

There is just one velocity framein which the time-reversed collisionlooks just like the original collision

That is the Center-of-Momentum

(COM)-frame


Algebra,Geometry, and Physics of momentum conservation axiom Vector algebra of collisionsMatrix or tensor algebra of collisions Deriving Energy Conservation Theorem Energy Ellipse geometry


10 20 30 40 50 60 70 80 90 100

10

20

30405060

708090

100

V IN

V FIN

VSUV

VVW

(MSUV+M

VW)V COM=M

SUVVSUV

IN+MVWVVW

IN=MSUVVSUV

FIN+MVWVVW

FIN

Conservation of momentum line:

V COM=(V IN+V FIN)/2

Algebra, Geometry, and Physics of Momentum Conservation Axiom

MSUV=4

mVW=1

MSUV

-mVW

slope =

Developing Conservation-of-Momentum The key axiom of mechanics

Wow! This is constant! (Says the axiom)


10 20 30 40 50 60 70 80 90 100

10

20

30405060

708090

100

V IN

V FIN

VSUV

VVW

(MSUV+M

VW)V COM=M

SUVVSUV

IN+MVWVVW

IN=MSUVVSUV

FIN+MVWVVW

FIN




MSUV=4

mVW=1

MSUV

-mVW

slope =

Define velocity vector points

VFIN=VSUV

FIN

VVWFIN

⎛

⎝⎜⎜

⎞

⎠⎟⎟

, VCOM =VSUV

COM

VVWCOM

⎛

⎝⎜⎜

⎞

⎠⎟⎟

, V IN =VSUV

IN

VVWIN

⎛

⎝⎜⎜

⎞

⎠⎟⎟




10 20 30 40 50 60 70 80 90 100

10

20

30405060

708090

100

V IN

V FIN

VSUV

VVW

(MSUV+M

VW)V COM=M

SUVVSUV

IN+MVWVVW

IN=MSUVVSUV

FIN+MVWVVW

FIN




MSUV=4

mVW=1

MSUV

-mVW

slope =

Points of vectorsVFIN and VCOM and V IN all lie on momentum - conservation line


VFIN=VSUV

FIN

VVWFIN

⎛

⎝⎜⎜

⎞

⎠⎟⎟

, VCOM =VSUV

COM

VVWCOM

⎛

⎝⎜⎜

⎞

⎠⎟⎟

, V IN =VSUV

IN

VVWIN

⎛

⎝⎜⎜

⎞

⎠⎟⎟




10 20 30 40 50 60 70 80 90 100

10

20

30405060

708090

100

V IN

V FIN

VSUV

VVW

(MSUV+M

VW)V COM=M

SUVVSUV

IN+MVWVVW

IN=MSUVVSUV

FIN+MVWVVW

FIN


const.=V COM=MSUVVSUV

IN+MVWVVW

IN=MSUVVSUV

FIN+MVWVVW

FIN

(MSUV+M

VW) (M

SUV+M

VW)

Mass weighted average velocity at anytime is Center of Mass velocity V COM:





VFIN=VSUV

FIN

VVWFIN

⎛

⎝⎜⎜

⎞

⎠⎟⎟

, VCOM =VSUV

COM

VVWCOM

⎛

⎝⎜⎜

⎞

⎠⎟⎟

, V IN =VSUV

IN

VVWIN

⎛

⎝⎜⎜

⎞

⎠⎟⎟




10 20 30 40 50 60 70 80 90 100

10

20

30405060

708090

100

V IN

V FIN

VSUV

VVW

(MSUV+M

VW)V COM=M

SUVVSUV

IN+MVWVVW

IN=MSUVVSUV

FIN+MVWVVW

FIN



IN+MVWVVW

IN=MSUVVSUV

FIN+MVWVVW

FIN

(MSUV+M

VW) (M

SUV+M

VW)


V FIN=VSUV

FIN

VVW

FIN

V IN=VSUV

IN

VVW

IN

V COM= =V COM

=V COMu

V COM

V COM

1

1

Express this using velocity vectors:



Define funny-unit vector: u= 1

1 10 20 30 40 50 60 70 80 90 100

10

20

30405060

708090

100

V IN

V FIN

VSUV

VVW

(MSUV+M

VW)V COM=M

SUVVSUV

IN+MVWVVW

IN=MSUVVSUV

FIN+MVWVVW

FIN



IN+MVWVVW

IN=MSUVVSUV

FIN+MVWVVW

FIN

(MSUV+M

VW) (M

SUV+M

VW)


V FIN=VSUV

FIN

VVW

FIN

V IN=VSUV

IN

VVW

IN

V COM= =V COM

=V COMu

V COM

V COM

1

1




Define funny-unit vector:

u=1

1


VFIN=VSUV

FIN

VVWFIN

⎛

⎝⎜⎜

⎞

⎠⎟⎟

= 4090

⎛⎝⎜

⎞⎠⎟

V IN =VSUV

IN

VVWIN

⎛

⎝⎜⎜

⎞

⎠⎟⎟

= 6010

⎛⎝⎜

⎞⎠⎟


VFIN=VSUV

FIN

VVWFIN

⎛

⎝⎜⎜

⎞

⎠⎟⎟

, VCOM =VSUV

COM

VVWCOM

⎛

⎝⎜⎜

⎞

⎠⎟⎟

, V IN =VSUV

IN

VVWIN

⎛

⎝⎜⎜

⎞

⎠⎟⎟



10 20 30 40 50 60 70 80 90 100

10

20

30405060

708090

100

V IN

V FIN

VSUV

VVW

(MSUV+M

VW)V COM=M

SUVVSUV

IN+MVWVVW

IN=MSUVVSUV

FIN+MVWVVW

FIN



IN+MVWVVW

IN=MSUVVSUV

FIN+MVWVVW

FIN

(MSUV+M

VW) (M

SUV+M

VW)


V FIN=VSUV

FIN

VVW

FIN

V IN=VSUV

IN

VVW

IN

V COM= =V COM

=V COMu

V COM

V COM

1

1





1 10 20 30 40 50 60 70 80 90 100

10

20

30405060

708090

100

V IN

V FIN

VSUV

VVW

(MSUV+M

VW)V COM=M

SUVVSUV

IN+MVWVVW

IN=MSUVVSUV

FIN+MVWVVW

FIN



IN+MVWVVW

IN=MSUVVSUV

FIN+MVWVVW

FIN

(MSUV+M

VW) (M

SUV+M

VW)


V FIN=VSUV

FIN

VVW

FIN

V IN=VSUV

IN

VVW

IN

V COM= =V COM

=V COMu

V COM

V COM

1

1





u=1

1

Points of vectorsVFIN and VCOM and V IN all lie on momentum - conservation line Vector VCOM is along 45° line

VFIN=VSUV

FIN

VVWFIN

⎛

⎝⎜⎜

⎞

⎠⎟⎟

= 4090

⎛⎝⎜

⎞⎠⎟

V IN =VSUV

IN

VVWIN

⎛

⎝⎜⎜

⎞

⎠⎟⎟

= 6010

⎛⎝⎜

⎞⎠⎟


VFIN=VSUV

FIN

VVWFIN

⎛

⎝⎜⎜

⎞

⎠⎟⎟

, VCOM =VSUV

COM

VVWCOM

⎛

⎝⎜⎜

⎞

⎠⎟⎟

, V IN =VSUV

IN

VVWIN

⎛

⎝⎜⎜

⎞

⎠⎟⎟

Define funny-unit vector : u= 11

⎛⎝⎜

⎞⎠⎟





10 20 30 40 50 60 70 80 90 100

10

20

30405060

708090

100

V IN

V FIN

VSUV

VVW

(MSUV+M

VW)V COM=M

SUVVSUV

IN+MVWVVW

IN=MSUVVSUV

FIN+MVWVVW

FIN



IN+MVWVVW

IN=MSUVVSUV

FIN+MVWVVW

FIN

(MSUV+M

VW) (M

SUV+M

VW)


V FIN=VSUV

FIN

VVW

FIN

V IN=VSUV

IN

VVW

IN

V COM= =V COM

=V COMu

V COM

V COM

1

1





1 10 20 30 40 50 60 70 80 90 100

10

20

30405060

708090

100

V IN

V FIN

VSUV

VVW

(MSUV+M

VW)V COM=M

SUVVSUV

IN+MVWVVW

IN=MSUVVSUV

FIN+MVWVVW

FIN



IN+MVWVVW

IN=MSUVVSUV

FIN+MVWVVW

FIN

(MSUV+M

VW) (M

SUV+M

VW)


V FIN=VSUV

FIN

VVW

FIN

V IN=VSUV

IN

VVW

IN

V COM= =V COM

=V COMu

V COM

V COM

1

1





u=1

1


VFIN=VSUV

FIN

VVWFIN

⎛

⎝⎜⎜

⎞

⎠⎟⎟

= 4090

⎛⎝⎜

⎞⎠⎟

V IN =VSUV

IN

VVWIN

⎛

⎝⎜⎜

⎞

⎠⎟⎟

= 6010

⎛⎝⎜

⎞⎠⎟


VFIN=VSUV

FIN

VVWFIN

⎛

⎝⎜⎜

⎞

⎠⎟⎟

, VCOM =VSUV

COM

VVWCOM

⎛

⎝⎜⎜

⎞

⎠⎟⎟

, V IN =VSUV

IN

VVWIN

⎛

⎝⎜⎜

⎞

⎠⎟⎟


⎛⎝⎜

⎞⎠⎟



10 20 30 40 50 60 70 80 90 100

10

20

30405060

708090

100

V IN

V FIN

VSUV

VVW

(MSUV+M

VW)V COM=M

SUVVSUV

IN+MVWVVW

IN=MSUVVSUV

FIN+MVWVVW

FIN



IN+MVWVVW

IN=MSUVVSUV

FIN+MVWVVW

FIN

(MSUV+M

VW) (M

SUV+M

VW)


MSUV0

0 MVW

M=

V FIN=VSUV

FIN

VVW

FIN

V IN=VSUV

IN

VVW

IN

V COM= =V COM

=V COMu

V COM

V COM

1

1


...and matrix operators:

...that give momentum vector:

whose sum of components is constant.

MSUV0

0 MVW

P=M•V= = =VSUV

VVW

PSUV

PVW

MSUVVSUV

MVWVVW

const.=u•P=PSUV+P

VW=M

SUVVSUV+M

VWVVW=u•M•V=u•M•V IN



(by u• product)

=u•M•V FIN

TTiimmee--RReevveerrssaallSSyymmmmeettrryy AAxxiioomm

Denote Center of Momentum VCOMwith engineer’s centering symbol

10 20 30 40 50 60 70 80 90 100

10

20

30405060

708090

100

V IN

V FIN

VSUV

VVW

(MSUV+M

VW)V COM=M

SUVVSUV

IN+MVWVVW

IN=MSUVVSUV

FIN+MVWVVW

FIN



IN+MVWVVW

IN=MSUVVSUV

FIN+MVWVVW

FIN

(MSUV+M

VW) (M

SUV+M

VW)


MSUV0

0 MVW

M=

V FIN=VSUV

FIN

VVW

FIN

V IN=VSUV

IN

VVW

IN

V COM= =V COM

=V COMu

V COM

V COM

1

1





MSUV0

0 MVW

P=M•V= = =VSUV

VVW

PSUV

PVW

MSUVVSUV

MVWVVW

const.=u•P=PSUV+P

VW=M

SUVVSUV+M




(by u• product)

=u•M•V FIN




u=1

1

VFIN=VSUV

FIN

VVWFIN

⎛

⎝⎜⎜

⎞

⎠⎟⎟

= 4090

⎛⎝⎜

⎞⎠⎟

V IN =VSUV

IN

VVWIN

⎛

⎝⎜⎜

⎞

⎠⎟⎟

= 6010

⎛⎝⎜

⎞⎠⎟



VFIN=VSUV

FIN

VVWFIN

⎛

⎝⎜⎜

⎞

⎠⎟⎟

, VCOM =VSUV

COM

VVWCOM

⎛

⎝⎜⎜

⎞

⎠⎟⎟

, V IN =VSUV

IN

VVWIN

⎛

⎝⎜⎜

⎞

⎠⎟⎟


⎛⎝⎜

⎞⎠⎟



10 20 30 40 50 60 70 80 90 100

10

20

30405060

708090

100

V IN

V FIN

VSUV

VVW

(MSUV+M

VW)V COM=M

SUVVSUV

IN+MVWVVW

IN=MSUVVSUV

FIN+MVWVVW

FIN



IN+MVWVVW

IN=MSUVVSUV

FIN+MVWVVW

FIN

(MSUV+M

VW) (M

SUV+M

VW)


MSUV0

0 MVW

M=

V FIN=VSUV

FIN

VVW

FIN

V IN=VSUV

IN

VVW

IN

V COM= =V COM

=V COMu

V COM

V COM

1

1





MSUV0

0 MVW

P=M•V= = =VSUV

VVW

PSUV

PVW

MSUVVSUV

MVWVVW

const.=u•P=PSUV+P

VW=M

SUVVSUV+M




(by u• product)

=u•M•V FIN



10 20 30 40 50 60 70 80 90 100

10

20

30405060

708090

100

V IN

V FIN

VSUV

VVW

(MSUV+M

VW)V COM=M

SUVVSUV

IN+MVWVVW

IN=MSUVVSUV

FIN+MVWVVW

FIN



IN+MVWVVW

IN=MSUVVSUV

FIN+MVWVVW

FIN

(MSUV+M

VW) (M

SUV+M

VW)


MSUV0

0 MVW

M=

V FIN=VSUV

FIN

VVW

FIN

V IN=VSUV

IN

VVW

IN

V COM= =V COM

=V COMu

V COM

V COM

1

1





MSUV0

0 MVW

P=M•V= = =VSUV

VVW

PSUV

PVW

MSUVVSUV

MVWVVW

const.=u•P=PSUV+P

VW=M

SUVVSUV+M




(by u• product)

=u•M•V FIN




u=1

1

VFIN=VSUV

FIN

VVWFIN

⎛

⎝⎜⎜

⎞

⎠⎟⎟

= 4090

⎛⎝⎜

⎞⎠⎟

V IN =VSUV

IN

VVWIN

⎛

⎝⎜⎜

⎞

⎠⎟⎟

= 6010

⎛⎝⎜

⎞⎠⎟




VFIN=VSUV

FIN

VVWFIN

⎛

⎝⎜⎜

⎞

⎠⎟⎟

, VCOM =VSUV

COM

VVWCOM

⎛

⎝⎜⎜

⎞

⎠⎟⎟

, V IN =VSUV

IN

VVWIN

⎛

⎝⎜⎜

⎞

⎠⎟⎟


⎛⎝⎜

⎞⎠⎟



10 20 30 40 50 60 70 80 90 100

10

20

30405060

708090

100

V IN

V FIN

VSUV

VVW

(MSUV+M

VW)V COM=M

SUVVSUV

IN+MVWVVW

IN=MSUVVSUV

FIN+MVWVVW

FIN



IN+MVWVVW

IN=MSUVVSUV

FIN+MVWVVW

FIN

(MSUV+M

VW) (M

SUV+M

VW)


MSUV0

0 MVW

M=

V FIN=VSUV

FIN

VVW

FIN

V IN=VSUV

IN

VVW

IN

V COM= =V COM

=V COMu

V COM

V COM

1

1





MSUV0

0 MVW

P=M•V= = =VSUV

VVW

PSUV

PVW

MSUVVSUV

MVWVVW

const.=u•P=PSUV+P

VW=M

SUVVSUV+M




(by u• product)

=u•M•V FIN



10 20 30 40 50 60 70 80 90 100

10

20

30405060

708090

100

V IN

V FIN

VSUV

VVW

(MSUV+M

VW)V COM=M

SUVVSUV

IN+MVWVVW

IN=MSUVVSUV

FIN+MVWVVW

FIN



IN+MVWVVW

IN=MSUVVSUV

FIN+MVWVVW

FIN

(MSUV+M

VW) (M

SUV+M

VW)


MSUV0

0 MVW

M=

V FIN=VSUV

FIN

VVW

FIN

V IN=VSUV

IN

VVW

IN

V COM= =V COM

=V COMu

V COM

V COM

1

1





MSUV0

0 MVW

P=M•V= = =VSUV

VVW

PSUV

PVW

MSUVVSUV

MVWVVW

const.=u•P=PSUV+P

VW=M

SUVVSUV+M




(by u• product)

=u•M•V FIN




u=1

1

VFIN=VSUV

FIN

VVWFIN

⎛

⎝⎜⎜

⎞

⎠⎟⎟

= 4090

⎛⎝⎜

⎞⎠⎟

V IN =VSUV

IN

VVWIN

⎛

⎝⎜⎜

⎞

⎠⎟⎟

= 6010

⎛⎝⎜

⎞⎠⎟



VFIN=VSUV

FIN

VVWFIN

⎛

⎝⎜⎜

⎞

⎠⎟⎟

, VCOM =VSUV

COM

VVWCOM

⎛

⎝⎜⎜

⎞

⎠⎟⎟

, V IN =VSUV

IN

VVWIN

⎛

⎝⎜⎜

⎞

⎠⎟⎟



10 20 30 40 50 60 70 80 90 100

10

20

30405060

708090

100

V IN

V FIN

VSUV

VVW

(MSUV+M

VW)V COM=M

SUVVSUV

IN+MVWVVW

IN=MSUVVSUV

FIN+MVWVVW

FIN



IN+MVWVVW

IN=MSUVVSUV

FIN+MVWVVW

FIN

(MSUV+M

VW) (M

SUV+M

VW)


MSUV0

0 MVW

M=

V FIN=VSUV

FIN

VVW

FIN

V IN=VSUV

IN

VVW

IN

V COM= =V COM

=V COMu

V COM

V COM

1

1





MSUV0

0 MVW

P=M•V= = =VSUV

VVW

PSUV

PVW

MSUVVSUV

MVWVVW

const.=u•P=PSUV+P

VW=M

SUVVSUV+M




(by u• product)

=u•M•V FIN


10 20 30 40 50 60 70 80 90 100

10

20

30405060

708090

100

V IN

V FIN

VSUV

VVW

(MSUV+M

VW)V COM=M

SUVVSUV

IN+MVWVVW

IN=MSUVVSUV

FIN+MVWVVW

FIN



IN+MVWVVW

IN=MSUVVSUV

FIN+MVWVVW

FIN

(MSUV+M

VW) (M

SUV+M

VW)


MSUV0

0 MVW

M=

V FIN=VSUV

FIN

VVW

FIN

V IN=VSUV

IN

VVW

IN

V COM= =V COM

=V COMu

V COM

V COM

1

1





MSUV0

0 MVW

P=M•V= = =VSUV

VVW

PSUV

PVW

MSUVVSUV

MVWVVW

const.=u•P=PSUV+P

VW=M

SUVVSUV+M




(by u• product)

=u•M•V FIN


V COM•M•V COM=V COM•M•V IN=V COM•M•V FIN

Then: V COM=V COMu gives:

VFIN=VSUV

FIN

VVWFIN

⎛

⎝⎜⎜

⎞

⎠⎟⎟

= 4090

⎛⎝⎜

⎞⎠⎟

V IN =VSUV

IN

VVWIN

⎛

⎝⎜⎜

⎞

⎠⎟⎟

= 6010

⎛⎝⎜

⎞⎠⎟


VFIN=VSUV

FIN

VVWFIN

⎛

⎝⎜⎜

⎞

⎠⎟⎟

, VCOM =VSUV

COM

VVWCOM

⎛

⎝⎜⎜

⎞

⎠⎟⎟

, V IN =VSUV

IN

VVWIN

⎛

⎝⎜⎜

⎞

⎠⎟⎟






10 20 30 40 50 60 70 80 90 100

10

20

30405060

708090

100

V IN

V FIN

VSUV

VVW

(MSUV+M

VW)V COM=M

SUVVSUV

IN+MVWVVW

IN=MSUVVSUV

FIN+MVWVVW

FIN



IN+MVWVVW

IN=MSUVVSUV

FIN+MVWVVW

FIN

(MSUV+M

VW) (M

SUV+M

VW)


MSUV0

0 MVW

M=

V FIN=VSUV

FIN

VVW

FIN

V IN=VSUV

IN

VVW

IN

V COM= =V COM

=V COMu

V COM

V COM

1

1





MSUV0

0 MVW

P=M•V= = =VSUV

VVW

PSUV

PVW

MSUVVSUV

MVWVVW

const.=u•P=PSUV+P

VW=M

SUVVSUV+M




By TT--SSyymmmmeettrryy AAxxiioomm: V COM=1/2(V IN+V FIN). Substituting:

V COM•M•V COM=1/2(V IN+V FIN)•M•V IN=1/2(V IN+V FIN)•M•V FIN

TT--SSyymmmmeettrryy AAxxiioommV COM=(V IN+V FIN)/2


(by u• product)

=u•M•V FIN

VFIN=VSUV

FIN

VVWFIN

⎛

⎝⎜⎜

⎞

⎠⎟⎟

= 4090

⎛⎝⎜

⎞⎠⎟

V IN =VSUV

IN

VVWIN

⎛

⎝⎜⎜

⎞

⎠⎟⎟

= 6010

⎛⎝⎜

⎞⎠⎟


VFIN=VSUV

FIN

VVWFIN

⎛

⎝⎜⎜

⎞

⎠⎟⎟

, VCOM =VSUV

COM

VVWCOM

⎛

⎝⎜⎜

⎞

⎠⎟⎟

, V IN =VSUV

IN

VVWIN

⎛

⎝⎜⎜

⎞

⎠⎟⎟




10 20 30 40 50 60 70 80 90 100

10

20

30405060

708090

100

V IN

V FIN

VSUV

VVW

(MSUV+M

VW)V COM=M

SUVVSUV

IN+MVWVVW

IN=MSUVVSUV

FIN+MVWVVW

FIN



IN+MVWVVW

IN=MSUVVSUV

FIN+MVWVVW

FIN

(MSUV+M

VW) (M

SUV+M

VW)


MSUV0

0 MVW

M= =MTranspose

V FIN=VSUV

FIN

VVW

FIN

V IN=VSUV

IN

VVW

IN

V COM= =V COM

=V COMu

V COM

V COM

1

1





MSUV0

0 MVW

P=M•V= = =VSUV

VVW

PSUV

PVW

MSUVVSUV

MVWVVW

const.=u•P=PSUV+P

VW=M

SUVVSUV+M

VWVVW=u•M•V






MM--ssyymmmmeettrryyM=MTT


(by u• product)

10 20 30 40 50 60 70 80 90 100

10

20

30405060

708090

100

V IN

V FIN

VSUV

VVW

(MSUV+M

VW)V COM=M

SUVVSUV

IN+MVWVVW

IN=MSUVVSUV

FIN+MVWVVW

FIN



IN+MVWVVW

IN=MSUVVSUV

FIN+MVWVVW

FIN

(MSUV+M

VW) (M

SUV+M

VW)


MSUV0

0 MVW

M= =MTranspose

V FIN=VSUV

FIN

VVW

FIN

V IN=VSUV

IN

VVW

IN

V COM= =V COM

=V COMu

V COM

V COM

1

1





MSUV0

0 MVW

P=M•V= = =VSUV

VVW

PSUV

PVW

MSUVVSUV

MVWVVW

const.=u•P=PSUV+P

VW=M

SUVVSUV+M









(by u• product)

=u•M•V FIN

VFIN=VSUV

FIN

VVWFIN

⎛

⎝⎜⎜

⎞

⎠⎟⎟

= 4090

⎛⎝⎜

⎞⎠⎟

V IN =VSUV

IN

VVWIN

⎛

⎝⎜⎜

⎞

⎠⎟⎟

= 6010

⎛⎝⎜

⎞⎠⎟


VFIN=VSUV

FIN

VVWFIN

⎛

⎝⎜⎜

⎞

⎠⎟⎟

, VCOM =VSUV

COM

VVWCOM

⎛

⎝⎜⎜

⎞

⎠⎟⎟

, V IN =VSUV

IN

VVWIN

⎛

⎝⎜⎜

⎞

⎠⎟⎟




10 20 30 40 50 60 70 80 90 100

10

20

30405060

708090

100

V IN

V FIN

VSUV

VVW

(MSUV+M

VW)V COM=M

SUVVSUV

IN+MVWVVW

IN=MSUVVSUV

FIN+MVWVVW

FIN



IN+MVWVVW

IN=MSUVVSUV

FIN+MVWVVW

FIN

(MSUV+M

VW) (M

SUV+M

VW)


MSUV0

0 MVW

M= =MTranspose

V FIN=VSUV

FIN

VVW

FIN

V IN=VSUV

IN

VVW

IN

V COM= =V COM

=V COMu

V COM

V COM

1

1





MSUV0

0 MVW

P=M•V= = =VSUV

VVW

PSUV

PVW

MSUVVSUV

MVWVVW

const.=u•P=PSUV+P

VW=M

SUVVSUV+M

VWVVW=u•M•V








(by u• product)

10 20 30 40 50 60 70 80 90 100

10

20

30405060

708090

100

V IN

V FIN

VSUV

VVW

(MSUV+M

VW)V COM=M

SUVVSUV

IN+MVWVVW

IN=MSUVVSUV

FIN+MVWVVW

FIN



IN+MVWVVW

IN=MSUVVSUV

FIN+MVWVVW

FIN

(MSUV+M

VW) (M

SUV+M

VW)


MSUV0

0 MVW

M= =MTranspose

V FIN=VSUV

FIN

VVW

FIN

V IN=VSUV

IN

VVW

IN

V COM= =V COM

=V COMu

V COM

V COM

1

1





MSUV0

0 MVW

P=M•V= = =VSUV

VVW

PSUV

PVW

MSUVVSUV

MVWVVW

const.=u•P=PSUV+P

VW=M

SUVVSUV+M









(by u• product)

=u•M•V FIN

VFIN=VSUV

FIN

VVWFIN

⎛

⎝⎜⎜

⎞

⎠⎟⎟

= 4090

⎛⎝⎜

⎞⎠⎟

V IN =VSUV

IN

VVWIN

⎛

⎝⎜⎜

⎞

⎠⎟⎟

= 6010

⎛⎝⎜

⎞⎠⎟


VFIN=VSUV

FIN

VVWFIN

⎛

⎝⎜⎜

⎞

⎠⎟⎟

, VCOM =VSUV

COM

VVWCOM

⎛

⎝⎜⎜

⎞

⎠⎟⎟

, V IN =VSUV

IN

VVWIN

⎛

⎝⎜⎜

⎞

⎠⎟⎟




10 20 30 40 50 60 70 80 90 100

10

20

30405060

708090

100

V IN

V FIN

VSUV

VVW

(MSUV+M

VW)V COM=M

SUVVSUV

IN+MVWVVW

IN=MSUVVSUV

FIN+MVWVVW

FIN



IN+MVWVVW

IN=MSUVVSUV

FIN+MVWVVW

FIN

(MSUV+M

VW) (M

SUV+M

VW)


MSUV0

0 MVW

M= =MTranspose

V FIN=VSUV

FIN

VVW

FIN

V IN=VSUV

IN

VVW

IN

V COM= =V COM

=V COMu

V COM

V COM

1

1





MSUV0

0 MVW

P=M•V= = =VSUV

VVW

PSUV

PVW

MSUVVSUV

MVWVVW

const.=u•P=PSUV+P

VW=M

SUVVSUV+M

VWVVW=u•M•V



V COM•M•V COM - 1/2V FIN•M•V IN

=1/2V IN•M•V IN=1/2V FIN•M•V FIN

ByMM--ssyymmmmeettrryyM=MTT : V FIN•M•V IN=V IN•M•V FIN

this becomes:






(by u• product)


VFIN=VSUV

FIN

VVWFIN

⎛

⎝⎜⎜

⎞

⎠⎟⎟

, VCOM =VSUV

COM

VVWCOM

⎛

⎝⎜⎜

⎞

⎠⎟⎟

, V IN =VSUV

IN

VVWIN

⎛

⎝⎜⎜

⎞

⎠⎟⎟



Algebra,Geometry, and Physics of momentum conservation axiom Vector algebra of collisionsMatrix or tensor algebra of collisions Completing derivation of Energy Conservation Theorem Energy Ellipse geometry


10 20 30 40 50 60 70 80 90 100

10

20

30405060

708090

100

V IN

V FIN

VSUV

VVW

(MSUV+M

VW)V COM=M

SUVVSUV

IN+MVWVVW

IN=MSUVVSUV

FIN+MVWVVW

FIN



IN+MVWVVW

IN=MSUVVSUV

FIN+MVWVVW

FIN

(MSUV+M

VW) (M

SUV+M

VW)


MSUV0

0 MVW

M= =MTranspose

V FIN=VSUV

FIN

VVW

FIN

V IN=VSUV

IN

VVW

IN

V COM= =V COM

=V COMu

V COM

V COM

1

1





MSUV0

0 MVW

P=M•V= = =VSUV

VVW

PSUV

PVW

MSUVVSUV

MVWVVW

const.=u•P=PSUV+P

VW=M

SUVVSUV+M

VWVVW=u•M•V



V COM•M•V COM - 1/2V FIN•M•V IN


ByMM--ssyymmmmeettrryyM=MTT : V FIN•M•V IN=V IN•M•V FIN

this becomes:

These are equations for energy conservation ellipse:






These are equations for energy conservation ellipse:

const. = 1/2MSUVVSUV

2 + 1/2MVWVVW

2

(by u• product)

= 1/2MSUVVSUV

2 + 1/2MVWVVW

2

=Kinetic Energy=KE is now defined

and proved a constant under T-Symmetry


VFIN=VSUV

FIN

VVWFIN

⎛

⎝⎜⎜

⎞

⎠⎟⎟

, VCOM =VSUV

COM

VVWCOM

⎛

⎝⎜⎜

⎞

⎠⎟⎟

, V IN =VSUV

IN

VVWIN

⎛

⎝⎜⎜

⎞

⎠⎟⎟


leading toConservation-of-Energy Theorem




VSUV

VVW

10 20 30 40 50 60 70 80 90 100

10

20

30405060

708090

100

V IN

V FIN

Conservation of momentum line: (...one of ∞-many...)

Algebra, Geometry, and Physics of MMoommeennttuumm CCoonnsseerrvvaattiioonn AAxxiioomm

V COM•M•V COM-1/2V FIN•M•V IN


MSUV=4

mVW=1

MSUV

-mVW

slope =

All lines of slope -MSUV/mVW

...are bisected by the

(slope=1)-COM line

MMoommeennttuummCCoonnsseerrvvaattiioonnAAxxiioomm

plus

TT--SSyymmmmeettrryyAAxxiioomm(M=MTT implied)

gives

KKiinneettiicc EEnneerrggyyCCoonnsseerrvvaattiioonnTThheeoorreemm


These are equations for energy conservation ellipse:These are equations for energy conservation ellipse:

KE = 1/2MSUVVSUV

2 + 1/2MVWVVW

2

1 = VSUV

2 + VVW

2 = x2 + y2

2·KE 2·KE a2 + b2

MSUV

MVW




VSUV

VVW

10 20 30 40 50 60 70 80 90 100

10

20

30405060

708090

100

V IN

V FIN

Conservation of momentum line: (...one of ∞-many...)

Algebra, Geometry, and Physics of MMoommeennttuumm CCoonnsseerrvvaattiioonn AAxxiioomm

V COM•M•V COM-1/2V FIN•M•V IN


MSUV=4

mVW=1

MSUV

-mVW

slope =

All lines of slope -MSUV/mVW

...are bisected by the

(slope=1)-COM line

MMoommeennttuummCCoonnsseerrvvaattiioonnAAxxiioomm

plus

TT--SSyymmmmeettrryyAAxxiioomm(M=MTT implied)

gives

KKiinneettiicc EEnneerrggyyCCoonnsseerrvvaattiioonnTThheeoorreemm


These are equations for energy conservation ellipse:These are equations for energy conservation ellipse:

KE = 1/2MSUVVSUV

2 + 1/2MVWVVW

2

1 = VSUV

2 + VVW

2 = x2 + y2

2·KE 2·KE a2 + b2

MSUV

MVW



If and only if…there is T-Symmetry


VVW100110120

-120

(60,10)Initial-point

Final“Ka-Bong”-point

a=√=60.21

2·KEMSUV

b=√=120.42

2·KEmVW

ElasticKineticEnergyellipse(KE=7,250)

MomentumPTotal=250line

(50,50)

(40,90)

MSUV=4MSUV=4

mVW=1mVW=1

VSUV

Fig. 3.1 ain Unit 1

Fig. 3.1



If and only if…there is T-Symmetry


VVW

100

110

120

-120

(60,10)

Initial-point

Final“Ka-Bong”-point

a=√=60.21

2·KE

MSUV

b=√=120.42

2·KEmVW

Elastic

Kinetic

Energy

ellipse

(KE=7,250)

Momentum

PTotal=250

line

(50,50)

(40,90)

MSUV=4M

SUV=4

mVW=1m

VW=1

VSUV

Fig. 3.1 ain Unit 1

a=√=55.9

2·IE

MSUV

b=√=111.8

2·IEmVW

COM

Energy

ellipse

(ECOM=1,000)

elastic

FIN

IN

inlastic

FIN is

VCOM

point

Inelastic

Kinetic

Energy

ellipse

(IE=6,250)

Fig. 3.1 bin Unit 1

Fig. 3.1


II

Felastic

I

(a) IdealElastic(Ka-Bong!)

(b) Genericinelastic(Ka-whump!)

(c) Totallyinelastic(Ka-Runch!)

aelastic=2EM2√

belastic=2EM1

√

(1,8)

(2,2)

(3,-4)

Eelastic=35

IECOM=14

Ewhump=23.33

Fwhump(4/3,6)

FCOM

M1=6M2=1

FPass-thru

bCOM=√2·IEM1

bwhump=2EwhumpM1√

aCOM=√2·IEM2

2EwhumpM2√

√

√

√

√awhump=

As usual in physics, opposite extremes are easier to analyzethan the generic “real(er) world” in between!

Fig. 3.2in Unit 1

(This case has Bush era requisiteSUV mass of the 6 ton “Hummer”)

Next: The X-2pen-

launcher

The X-2pen-

launcher

Here T-Symmetryis best

Here T-Symmetryis less

Here T-Symmetryis least


Numerical details of collision tensor algebra


General Inertia Tensor M or inertia matrix of n2 coefficients Mjk for dimension n=2, 3,....

P1 = M11V1 + M12V2

P2 = M21V1 + M22V2

⎫⎬⎪

⎭⎪ denoted :

P =

M iV or :

P1

P2

⎛

⎝⎜⎜

⎞

⎠⎟⎟=

M11 M12

M21 M22

⎛

⎝⎜⎜

⎞

⎠⎟⎟

V1

V2

⎛

⎝⎜⎜

⎞

⎠⎟⎟

V1COM =V2

COM ≡VCOMWith 45° diagonal VCOM so: .

PTotal = M1V1IN +M2V2

IN = M1V1FIN +M2V2

FIN = M1VCOM +M2V

COM = MTotalVCOM


A product of total momentum PTotal and is expressed by tensor quadratic forms v•M•u


P1 = M11V1 + M12V2

P2 = M21V1 + M22V2

⎫⎬⎪

⎭⎪ denoted :

P =

M iV or :

P1

P2

⎛

⎝⎜⎜

⎞

⎠⎟⎟=

M11 M12

M21 M22

⎛

⎝⎜⎜

⎞

⎠⎟⎟

V1

V2

⎛

⎝⎜⎜

⎞

⎠⎟⎟

V1COM =V2



IN = M1V1FIN +M2V2

FIN = M1VCOM +M2V

COM = MTotalVCOM

VCOM

VCOMPTotal =

VCOM i

M iV IN =

VCOM i

M iVFIN =

VCOM i

M iVCOM =VCOMMTotalV

COM


Write this out with the numbers used in Fig. 3.1 where .



P1 = M11V1 + M12V2

P2 = M21V1 + M22V2

⎫⎬⎪

⎭⎪ denoted :

P =

M iV or :

P1

P2

⎛

⎝⎜⎜

⎞

⎠⎟⎟=

M11 M12

M21 M22

⎛

⎝⎜⎜

⎞

⎠⎟⎟

V1

V2

⎛

⎝⎜⎜

⎞

⎠⎟⎟

V1COM =V2



IN = M1V1FIN +M2V2

FIN = M1VCOM +M2V

COM = MTotalVCOM

VCOM

VCOMPTotal =

VCOM i

M iV IN =

VCOM i

M iVFIN =

VCOM i


COM

VCOM = 50

50PTotal = 50 50( ) i 4 00 1

⎛

⎝⎜⎞

⎠⎟i 60

10⎛

⎝⎜⎞

⎠⎟= 50 50( ) i 4 0

0 1⎛

⎝⎜⎞

⎠⎟i 40

90⎛

⎝⎜⎞

⎠⎟= 50MTotal50 = 12,500

= 100 ⋅125 = 100 ⋅125 = 50 ⋅250

(2 pages back)


PTotal =250 is the same at IN, FIN, and COM. Now use T-symmetry:



P1 = M11V1 + M12V2

P2 = M21V1 + M22V2

⎫⎬⎪

⎭⎪ denoted :

P =

M iV or :

P1

P2

⎛

⎝⎜⎜

⎞

⎠⎟⎟=

M11 M12

M21 M22

⎛

⎝⎜⎜

⎞

⎠⎟⎟

V1

V2

⎛

⎝⎜⎜

⎞

⎠⎟⎟

V1COM =V2



IN = M1V1FIN +M2V2

FIN = M1VCOM +M2V

COM = MTotalVCOM

VCOM

VCOMPTotal =

VCOM i

M iV IN =

VCOM i

M iVFIN =

VCOM i


COM

50PTotal = 50 50( ) i 4 00 1

⎛

⎝⎜⎞

⎠⎟i 60

10⎛

⎝⎜⎞

⎠⎟= 50 50( ) i 4 0

0 1⎛

⎝⎜⎞

⎠⎟i 40

90⎛

⎝⎜⎞

⎠⎟= 50MTotal50 = 12,500

= 100 ⋅125 = 100 ⋅125 = 50 ⋅250

VCOM =(

V FIN+

V IN )/2

V COM PTotal =V FIN+

V IN

2i

M iV IN =

V FIN+

V IN

2i

M iV FIN =

V FIN+

V IN

2i

M i

V FIN+

V IN

2

Write this out with the numbers used in Fig. 3.1 where . VCOM = 50 (2 pages back)





P1 = M11V1 + M12V2

P2 = M21V1 + M22V2

⎫⎬⎪

⎭⎪ denoted :

P =

M iV or :

P1

P2

⎛

⎝⎜⎜

⎞

⎠⎟⎟=

M11 M12

M21 M22

⎛

⎝⎜⎜

⎞

⎠⎟⎟

V1

V2

⎛

⎝⎜⎜

⎞

⎠⎟⎟

V1COM =V2



IN = M1V1FIN +M2V2

FIN = M1VCOM +M2V

COM = MTotalVCOM

VCOM

VCOMPTotal =

VCOM i

M iV IN =

VCOM i

M iVFIN =

VCOM i


COM

50PTotal = 50 50( ) i 4 00 1

⎛

⎝⎜⎞

⎠⎟i 60

10⎛

⎝⎜⎞

⎠⎟= 50 50( ) i 4 0

0 1⎛

⎝⎜⎞

⎠⎟i 40

90⎛

⎝⎜⎞

⎠⎟= 50MTotal50 = 12,500

= 100 ⋅125 = 100 ⋅125 = 50 ⋅250

VCOM =(

V FIN+

V IN )/2


V IN

2i

M iV IN =

V FIN+

V IN

2i

M iV FIN =

V FIN+

V IN

2i

M i

V FIN+

V IN

2

V COM PTotal −V FIN i

M iV IN

2=V IN i

M iV IN

2=V FIN i

M iV FIN

2=V IN i

M iV IN

4+V FIN i

M iV FIN

4


Subtracting:V FIN i

MiV IN

2

=V IN i

MiV FIN

2

⎛

⎝

⎜⎜⎜⎜⎜⎜⎜

⎞

⎠

⎟⎟⎟⎟⎟⎟⎟





P1 = M11V1 + M12V2

P2 = M21V1 + M22V2

⎫⎬⎪

⎭⎪ denoted :

P =

M iV or :

P1

P2

⎛

⎝⎜⎜

⎞

⎠⎟⎟=

M11 M12

M21 M22

⎛

⎝⎜⎜

⎞

⎠⎟⎟

V1

V2

⎛

⎝⎜⎜

⎞

⎠⎟⎟

V1COM =V2



IN = M1V1FIN +M2V2

FIN = M1VCOM +M2V

COM = MTotalVCOM

VCOM

VCOMPTotal =

VCOM i

M iV IN =

VCOM i

M iVFIN =

VCOM i


COM

50PTotal = 50 50( ) i 4 00 1

⎛

⎝⎜⎞

⎠⎟i 60

10⎛

⎝⎜⎞

⎠⎟= 50 50( ) i 4 0

0 1⎛

⎝⎜⎞

⎠⎟i 40

90⎛

⎝⎜⎞

⎠⎟= 50MTotal50 = 12,500

= 100 ⋅125 = 100 ⋅125 = 50 ⋅250

VCOM =(

V FIN+

V IN )/2


V IN

2i

M iV IN =

V FIN+

V IN

2i

M iV FIN =

V FIN+

V IN

2i

M i

V FIN+

V IN

2


M iV IN

2=V IN i

M iV IN

2=V FIN i

M iV FIN

2=V IN i

M iV IN

4+V FIN i

M iV FIN

4

Transpose symmetry (Mjk =Mkj) of the M-matrix implies:

V FIN i

M iV IN =

V IN i

M iV FIN

40 90( ) i 4 00 1

⎛

⎝⎜⎞

⎠⎟i 60

10⎛

⎝⎜⎞

⎠⎟= 60 10( ) i 4 0

0 1⎛

⎝⎜⎞

⎠⎟i 40

90⎛

⎝⎜⎞

⎠⎟

= 100 ⋅105 = 100 ⋅105 = 10,500


Subtracting:V FIN i

MiV IN

2

=V IN i

MiV FIN

2

⎛

⎝

⎜⎜⎜⎜⎜⎜⎜

⎞

⎠

⎟⎟⎟⎟⎟⎟⎟


With (M12 =0=M21) kinetic energy is the same at V=VIN and V=VFIN.


M iV IN

2=

V IN i

M iV IN

2=

V FIN i

M iV FIN

2= KEElastic

12,500 − 10,5002

=60 10( ) i 4 0

0 1⎛

⎝⎜⎞

⎠⎟i 60

10⎛

⎝⎜⎞

⎠⎟

2=

40 90( ) i 4 00 1

⎛

⎝⎜⎞

⎠⎟i 40

90⎛

⎝⎜⎞

⎠⎟

2= KEElastic

12,500 − 5,250 = 7,250 = 7,250


V FIN i

M iV IN =

V IN i

M iV FIN

40 90( ) i 4 00 1

⎛

⎝⎜⎞

⎠⎟i 60

10⎛

⎝⎜⎞

⎠⎟= 60 10( ) i 4 0

0 1⎛

⎝⎜⎞

⎠⎟i 40

90⎛

⎝⎜⎞

⎠⎟

= 100 ⋅105 = 100 ⋅105 = 10,500

KEElastic= 21 Vi

MiV


With (M12 =0=M21) kinetic energy is the same at V=VIN and V=VFIN.


M iV IN

2=

V IN i

M iV IN

2=

V FIN i

M iV FIN

2= KEElastic

12,500 − 10,5002

=60 10( ) i 4 0

0 1⎛

⎝⎜⎞

⎠⎟i 60

10⎛

⎝⎜⎞

⎠⎟

2=

40 90( ) i 4 00 1

⎛

⎝⎜⎞

⎠⎟i 40

90⎛

⎝⎜⎞

⎠⎟

2= KEElastic

12,500 − 5,250 = 7,250 = 7,250


V FIN i

M iV IN =

V IN i

M iV FIN

40 90( ) i 4 00 1

⎛

⎝⎜⎞

⎠⎟i 60

10⎛

⎝⎜⎞

⎠⎟= 60 10( ) i 4 0

0 1⎛

⎝⎜⎞

⎠⎟i 40

90⎛

⎝⎜⎞

⎠⎟

= 100 ⋅105 = 100 ⋅105 = 10,500

Consider kinetic energy KEinelastic=IE when V=VCOM. It is reduced by 1,000 from 7,250 to 6,250.

KEInelastic=V COM PTotal−VCOM i

MiVCOM

2= 1

2V COM PTotal=

VCOM i

MiVCOM

2=V IN i

MiV IN

4+V FIN i

MiV IN

4= 1

2KEElastic+

V FIN i

MiV IN

4

12,500 − 12,5002

= 6,250 = 6,250 = 3,625 + 2,625 = IE

KEElastic= 21 Vi

MiV


With (M12 =0=M21) kinetic energy is the same at V=VIN and V=VFIN. KEElastic= 21 Vi

MiV


M iV IN

2=

V IN i

M iV IN

2=

V FIN i

M iV FIN

2= KEElastic

12,500 − 10,5002

=60 10( ) i 4 0

0 1⎛

⎝⎜⎞

⎠⎟i 60

10⎛

⎝⎜⎞

⎠⎟

2=

40 90( ) i 4 00 1

⎛

⎝⎜⎞

⎠⎟i 40

90⎛

⎝⎜⎞

⎠⎟

2= KEElastic

12,500 − 5,250 = 7,250 = 7,250


V FIN i

M iV IN =

V IN i

M iV FIN

40 90( ) i 4 00 1

⎛

⎝⎜⎞

⎠⎟i 60

10⎛

⎝⎜⎞

⎠⎟= 60 10( ) i 4 0

0 1⎛

⎝⎜⎞

⎠⎟i 40

90⎛

⎝⎜⎞

⎠⎟

= 100 ⋅105 = 100 ⋅105 = 10,500



MiVCOM

2= 1

2V COM PTotal=

VCOM i

MiVCOM

2=V IN i

MiV IN

4+V FIN i

MiV IN

4= 1

2KEElastic+

V FIN i

MiV IN

4

12,500 − 12,5002

= 6,250 = 6,250 = 3,625 + 2,625 = IE

KEInelastic=2

1 VCOM i

MiVCOM =2

1 50 50( ) i 4 00 1

⎛

⎝⎜⎞

⎠⎟i 50

50⎛

⎝⎜⎞

⎠⎟= 6,250



MiV


M iV IN

2=

V IN i

M iV IN

2=

V FIN i

M iV FIN

2= KEElastic

12,500 − 10,5002

=60 10( ) i 4 0

0 1⎛

⎝⎜⎞

⎠⎟i 60

10⎛

⎝⎜⎞

⎠⎟

2=

40 90( ) i 4 00 1

⎛

⎝⎜⎞

⎠⎟i 40

90⎛

⎝⎜⎞

⎠⎟

2= KEElastic

12,500 − 5,250 = 7,250 = 7,250


V FIN i

M iV IN =

V IN i

M iV FIN

40 90( ) i 4 00 1

⎛

⎝⎜⎞

⎠⎟i 60

10⎛

⎝⎜⎞

⎠⎟= 60 10( ) i 4 0

0 1⎛

⎝⎜⎞

⎠⎟i 40

90⎛

⎝⎜⎞

⎠⎟

= 100 ⋅105 = 100 ⋅105 = 10,500



MiVCOM

2= 1

2V COM PTotal=

VCOM i

MiVCOM

2=V IN i

MiV IN

4+V FIN i

MiV IN

4= 1

2KEElastic+

V FIN i

MiV IN

4

12,500 − 12,5002

= 6,250 = 6,250 = 3,625 + 2,625 = IE

KEElastic − KEInelastic =V IN i

M iV IN

4−V FIN i

M iV IN

41,000 = 3,625 − 2,625 = KE − IE

The difference is inelastic “crunch” energy KE-IE or, for elastic cases, potential energy of compression. KEInelastic=2

1 VCOM i

MiVCOM =2

1 50 50( ) i 4 00 1

⎛

⎝⎜⎞

⎠⎟i 50

50⎛

⎝⎜⎞

⎠⎟= 6,250



MiV


M iV IN

2=

V IN i

M iV IN

2=

V FIN i

M iV FIN

2= KEElastic

12,500 − 10,5002

=60 10( ) i 4 0

0 1⎛

⎝⎜⎞

⎠⎟i 60

10⎛

⎝⎜⎞

⎠⎟

2=

40 90( ) i 4 00 1

⎛

⎝⎜⎞

⎠⎟i 40

90⎛

⎝⎜⎞

⎠⎟

2= KEElastic

12,500 − 5,250 = 7,250 = 7,250


V FIN i

M iV IN =

V IN i

M iV FIN

40 90( ) i 4 00 1

⎛

⎝⎜⎞

⎠⎟i 60

10⎛

⎝⎜⎞

⎠⎟= 60 10( ) i 4 0

0 1⎛

⎝⎜⎞

⎠⎟i 40

90⎛

⎝⎜⎞

⎠⎟

= 100 ⋅105 = 100 ⋅105 = 10,500



MiVCOM

2= 1

2V COM PTotal=

VCOM i

MiVCOM

2=V IN i

MiV IN

4+V FIN i

MiV IN

4= 1

2KEElastic+

V FIN i

MiV IN

4

12,500 − 12,5002

= 6,250 = 6,250 = 3,625 + 2,625 = IE

KEElastic − KEInelastic =V IN i

M iV IN

4−V FIN i

M iV IN

41,000 = 3,625 − 2,625 = KE − IE

The difference is inelastic “crunch” energy KE-IE or, for elastic cases, potential energy of compression. KEInelastic=2

1 VCOM i

MiVCOM =2

1 50 50( ) i 4 00 1

⎛

⎝⎜⎞

⎠⎟i 50

50⎛

⎝⎜⎞

⎠⎟= 6,250

This difference is the same in all reference frames including COM frame where KEinelastic=IE is zero. 66Wednesday, August 26, 2015

Fig. 3.4 Galilean Frame Views of collision like Fig. 2.5 or Fig. 3.1 with Bush (6:1) SUV. (a) Earth frame view (b) Initial VW frame (VW initially fixed) (c) COM frame view (d) Final VW frame (VW ends up fixed) Fig. 3.5 Momentum (P=const.)-lines and energy (KE=const.)-ellipses

appropriate for Fig. 3.4.

Note “crunch” energy ElasticKE-inelasticIE = 0.21 is the same in all frames including COM-frame.

COM-frame


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Axiomatic development of classical mechanics...2015/08/25 · Lecture 1 Tue. 8.25.2015 Axiomatic development of classical mechanics (Ch. 1 and Ch. 2 of Unit 1) Geometry of momentum