Analysis of 1D 2-Body Collisions · 8/27/2015  · Friday, August 28, 2015 1. V VW 100 110 120-120...

Lecture 2 Thur. 8.27.2015

Analysis of 1D 2-Body Collisions(Ch. 3 to Ch. 5 of Unit 1)

Review of elastic Kinetic Energy ellipse geometryThe X2 Superball pen launcher *

Perfectly elastic “ka-bong” velocity amplification effects (Faux-Flubber)

Geometry of X2 launcher bouncing in boxIndependent Bounce Model (IBM)Geometric optimization and range-of-motion calculation(s)Integration of (V1,V2) data to space-time plots (y1(t),t) and (y2(t),t) plotsIntegration of (V1,V2) data to space-space plots (y1, y2)

Multiple collisions calculated by matrix operator products Matrix or tensor algebra of 1-D 2-body collisions

Ellipse rescaling-geometry and reflection-symmetry analysisRescaling KE ellipse to circle

How this relates to Lagrangian, l’Etrangian, and Hamiltonian mechanics in Ch. 12

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

1Friday, August 28, 2015

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

Review of elastic Kinetic Energy ellipse geometry

2Friday, August 28, 2015

M1=70gmM1=70gm

M0=10kgM0=10kgbounceplatebounceplate

RRrrd

Superballpenetrationdepthr22Rd=

SuperballSuperball

ballpointpen

M2=10gm

ballpointpen

M2=10gm

The X-2pen-

launcher

The X-2pen-

launcher

The X-2 Pen launcher and Superball Collision Simulator*

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

3Friday, August 28, 2015

M1=70gmM1=70gm

M0=10kgM0=10kgbounceplatebounceplate

RRrrd

Superballpenetrationdepthr22Rd=

SuperballSuperball

ballpointpen

M2=10gm

ballpointpen

M2=10gm

The X-2pen-

launcher

The X-2pen-

launcherFig. 4.1 and Fig. 4.3

in Unit 1

M1

m2BANG!M1

(BiggerBANG!)

(StillBiggerBANG!)

m2

M1M1

m2

Bang1!

Bang2!M1

m2

(a) Super-elastic 2nd-body bounce (b) 2-Bang Model (c) n-BodySupernovaSuperballs

m2

-1.0

1.0

1.0

0.5

2.0

m2 Velocity axisVym2

m1 Velocity axisVym1

(0,0)

Bang-1(01)INIT point at(-1.0,-1.0)

(a)Bang-1(01)

Fig. 4.4a-bin Unit 1

Bang-1(01)FINAL point(+1.0,-1.0)

This 1st bang is a floor-bounce ofM1 off very massive plate/Earth M0

2.02.01st bang:mass (M0)vs. mass (M1)

2nd or 3rd bangs:mass (M1) vs. mass (M2)

1st bang:M1 off floor

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

4Friday, August 28, 2015

M1=70gmM1=70gm

M0=10kgM0=10kgbounceplatebounceplate

RRrrd

Superballpenetrationdepthr22Rd=

SuperballSuperball

ballpointpen

M2=10gm

ballpointpen

M2=10gm

The X-2pen-

launcher

The X-2pen-

launcherFig. 4.1 and Fig. 4.3

in Unit 1

M1

m2BANG!M1

(BiggerBANG!)

(StillBiggerBANG!)

m2

M1M1

m2

Bang1!

Bang2!M1

m2

(a) Super-elastic 2nd-body bounce (b) 2-Bang Model (c) n-BodySupernovaSuperballs

m2

-1.0

1.0

1.0

0.5

2.0

m2 Velocity axisVym2

m1 Velocity axisVym1

(0,0)

Bang-1(01)INIT point at(-1.0,-1.0)

(a)Bang-1(01)

Fig. 4.4a-bin Unit 1

Bang-1(01)FINAL point(+1.0,-1.0)

This 1st bang is a floor-bounce ofM1 off very massive plate/Earth M0

2.02.01st bang:mass (M0)vs. mass (M1)

2nd or 3rd bangs:mass (M1) vs. mass (M2)

M1 mirror reflection

thru m2 axis

1st bang:M1 off floor

5Friday, August 28, 2015

M1=70gmM1=70gm

M0=10kgM0=10kgbounceplatebounceplate

RRrrd

Superballpenetrationdepthr22Rd=

SuperballSuperball

ballpointpen

M2=10gm

ballpointpen

M2=10gm

The X-2pen-

launcher

The X-2pen-

launcherFig. 4.1 and Fig. 4.3

in Unit 1

M1

m2BANG!M1

(BiggerBANG!)

(StillBiggerBANG!)

m2

M1M1

m2

Bang1!

Bang2!M1

m2

(a) Super-elastic 2nd-body bounce (b) 2-Bang Model (c) n-BodySupernovaSuperballs

m2

-1.0

1.0

1.0

0.5

2.0

m2 Velocity axisVym2

m1 Velocity axisVym1

(0,0)

Bang-1(01)INIT point at(-1.0,-1.0)

(a)Bang-1(01)

Fig. 4.4a-bin Unit 1

Bang-1(01)FINAL point(+1.0,-1.0)

This 1st bang is a floor-bounce ofM1 off very massive plate/Earth M0

2.02.01st bang:mass (M0)vs. mass (M1)

2nd or 3rd bangs:mass (M1) vs. mass (M2)

M1 mirror reflection

thru m2 axis

1st bang:M1 off floor

2nd bang:m2 off M1

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

6Friday, August 28, 2015

M1=70gmM1=70gm

M0=10kgM0=10kgbounceplatebounceplate

RRrrd

Superballpenetrationdepthr22Rd=

SuperballSuperball

ballpointpen

M2=10gm

ballpointpen

M2=10gm

The X-2pen-

launcher

The X-2pen-

launcherFig. 4.1 and Fig. 4.3

in Unit 1

M1

m2BANG!M1

(BiggerBANG!)

(StillBiggerBANG!)

m2

M1M1

m2

Bang1!

Bang2!M1

m2

(a) Super-elastic 2nd-body bounce (b) 2-Bang Model (c) n-BodySupernovaSuperballs

m2

-1.0

1.0

1.0

0.5

2.0

m2 Velocity axisVym2

m1 Velocity axisVym1

(0,0)

Bang-1(01)INIT point at(-1.0,-1.0)

(a)Bang-1(01)

Fig. 4.4a-bin Unit 1

Bang-1(01)FINAL point(+1.0,-1.0)

This 1st bang is a floor-bounce ofM1 off very massive plate/Earth M0

2.02.01st bang:mass (M0)vs. mass (M1)

2nd or 3rd bangs:mass (M1) vs. mass (M2)

M1 mirror reflection

thru m2 axis

1st bang:M1 off floor

2nd bang:m2 off M1

3rd bang:m2 off ceiling

7Friday, August 28, 2015

M1=70gmM1=70gm

M0=10kgM0=10kg

bounce

plate

bounce

plate

RRrr

d

Superball

penetration

depth

r2

2Rd=

SuperballSuperball

ballpoint

pen

M2=10gm

ballpoint

pen

M2=10gm

The X-2pen-

launcher

The X-2pen-

launcherFig. 4.1 and Fig. 4.3

in Unit 1

M1

m2BANG!M1

(BiggerBANG!)

(StillBiggerBANG!)

m2

M1M1

m2

Bang1!

Bang2!M1

m2

(a) Super-elastic 2nd-body bounce (b) 2-Bang Model (c) n-BodySupernovaSuperballs

m2

-1.0

1.0

1.0

0.5

2.0

m2 Velocity axis

Vym2

m1 Velocity axis

Vym1

(0,0)

Bang-1(01)

INIT point at

(-1.0,-1.0)

(a)Bang-1(01)

Fig. 4.4a-bin Unit 1

INITIAL

FINAL (Elastic)

0 V1

(b)

M1<<M

0

V0

FINAL

(Totally Inelastic)COM

FI

top of a very long ellipse

Fig. 4.2bin Unit 1(slightly modified)

Bang-1(01)

FINAL point

(+1.0,-1.0)

This 1st bang is a floor-bounce of

M1off very massive plate/Earth M

0

2.02.01st bang:

mass (M0)vs. mass (M

1)

2nd or 3rd bangs:

mass (M1) vs. mass (M

2)

V1

V0Very skinny Energy ellipse for M0>>M1

M1 mirror reflection

thru m2 axis

1st bang:M1 off floor

2nd bang:m2 off M1

3rd bang:m2 off ceiling

1st bang M1 off floor “skinny-ellipse” 8Friday, August 28, 2015

M1=70gmM1=70gm

M0=10kgM0=10kg

bounce

plate

bounce

plate

RRrr

d

Superball

penetration

depth

r2

2Rd=

SuperballSuperball

ballpoint

pen

M2=10gm

ballpoint

pen

M2=10gm

The X-2pen-

launcher

The X-2pen-

launcherFig. 4.1 and Fig. 4.3

in Unit 1

M1

m2BANG!M1

(BiggerBANG!)

(StillBiggerBANG!)

m2

M1M1

m2

Bang1!

Bang2!M1

m2

(a) Super-elastic 2nd-body bounce (b) 2-Bang Model (c) n-BodySupernovaSuperballs

m2

-1.0

1.0

1.0

0.5

2.0

m2 Velocity axis

Vym2

m1 Velocity axis

Vym1

(0,0)

Bang-1(01)

INIT point at

(-1.0,-1.0)

(a)Bang-1(01)

Fig. 4.4a-bin Unit 1

INITIAL

FINAL (Elastic)

0 V1

(b)

M1<<M

0

V0

FINAL

(Totally Inelastic)COM

FI

top of a very long ellipse

INITIAL

FINAL (Elastic)

0 V1

COM

(a)

M1>>M

2

V2

FINAL

(Totally Inelastic)

F0

I0

(+) side of a

very tall

ellipse

(-) side of a

very tall

ellipse

Fig. 4.2bin Unit 1(slightly modified)

Bang-1(01)

FINAL point

(+1.0,-1.0)

This 1st bang is a floor-bounce of

M1off very massive plate/Earth M

0

Fig. 4.2ain Unit 1(slightly modified)

Later is a ceiling-bounce of

M2off ceiling/Earth M

1

2.02.01st bang:

mass (M0)vs. mass (M

1)

2nd or 3rd bangs:

mass (M1) vs. mass (M

2)

Later:

V1

V0Very skinny Energy ellipse for M0>>M1

M1 mirror reflection

thru m2 axis

1st bang:M1 off floor

2nd bang:m2 off M1

3rd bang:m2 off ceiling

1st bang M1 off floor “skinny-ellipse” 9Friday, August 28, 2015

M1=70gmM1=70gm

M0=10kgM0=10kg

bounce

plate

bounce

plate

RRrr

d

Superball

penetration

depth

r2

2Rd=

SuperballSuperball

ballpoint

pen

M2=10gm

ballpoint

pen

M2=10gm

The X-2pen-

launcher

The X-2pen-

launcherFig. 4.1 and Fig. 4.3

in Unit 1

M1

m2BANG!M1

(BiggerBANG!)

(StillBiggerBANG!)

m2

M1M1

m2

Bang1!

Bang2!M1

m2

(a) Super-elastic 2nd-body bounce (b) 2-Bang Model (c) n-BodySupernovaSuperballs

m2

-1.0

1.0

1.0

0.5

2.0

m2 Velocity axis

Vym2

m1 Velocity axis

Vym1

(0,0)

Bang-1(01)

INIT point at

(-1.0,-1.0)

(a)Bang-1(01)

Fig. 4.4a-bin Unit 1

INITIAL

FINAL (Elastic)

0 V1

(b)

M1<<M

0

V0

FINAL

(Totally Inelastic)COM

FI

top of a very long ellipse

INITIAL

FINAL (Elastic)

0 V1

COM

(a)

M1>>M

2

V2

FINAL

(Totally Inelastic)

F0

I0

(+) side of a

very tall

ellipse

(-) side of a

very tall

ellipse

Fig. 4.2bin Unit 1(slightly modified)

Bang-1(01)

FINAL point

(+1.0,-1.0)

This 1st bang is a floor-bounce of

M1off very massive plate/Earth M

0

Fig. 4.2ain Unit 1(slightly modified)

Later is a ceiling-bounce of

M2off ceiling/Earth M

1

2.02.01st bang:

mass (M0)vs. mass (M

1)

2nd or 3rd bangs:

mass (M1) vs. mass (M

2)

Later:

V1

V0Very skinny Energy ellipse for M0>>M1

V0

V2

M1 mirror reflection

thru m2 axis

1st bang:M1 off floor

2nd bang:m2 off M1

3rd bang:m2 off ceiling

1st bang M1 off floor “skinny-ellipse” 10Friday, August 28, 2015

Geometry of X2 launcher bouncing in boxIndependent Bounce Model (IBM)Geometric optimization and range-of-motion calculation(t)Integration of (V1,V2) data to space-time plots (y1(t),t) and (y2(t),t) plotsIntegration of (V1,V2) data to space-space plots (y1, y2)

11Friday, August 28, 2015

M1=70gmM1=70gm

M0=10kgM0=10kg

bounce

plate

bounce

plate

RRrr

d

Superball

penetration

depth

r2

2Rd=

SuperballSuperball

ballpoint

pen

M2=10gm

ballpoint

pen

M2=10gm

The X-2pen-

launcher

The X-2pen-

launcherFig. 4.1 and Fig. 4.3

in Unit 1

M1

m2BANG!M1

(BiggerBANG!)

(StillBiggerBANG!)

m2

M1M1

m2

Bang1!

Bang2!M1

m2

(a) Super-elastic 2nd-body bounce (b) 2-Bang Model (c) n-BodySupernovaSuperballs

m2

-1.0

1.0

1.0

0.5

2.0

m2 Velocity axis

Vym2

m1 Velocity axis

Vym1

(0,0)

Bang-1(01)

INIT point at

(-1.0,-1.0)

(a)Bang-1(01)

Fig. 4.4a-bin Unit 1

Bang-1(01)

FINAL point

(+1.0,-1.0)

This 1st bang is a floor-bounce of

M1off very massive plate/Earth M

0

2.02.01st bang:

mass (M0)vs. mass (M

1)

-7

+1

1.0

1.0

0.5

2.0

m1 Velocity axis

Vym1

(0,0)

COM-point at

(0.75,0.75))

-1.0

Bang-2(12)

INIT point at

(+1.0,-1.0)

Bang-2(12)

FINAL point

(0.5,2.5)

(b)Bang-2(12)

2nd bang:

(M1)vs.(M

2)

M1 mirror reflection

thru m2 axis

12Friday, August 28, 2015

Geometry of X2 launcher bouncing in boxIndependent Bounce Model (IBM)Geometric optimization and range-of-motion calculation(s)Integration of (V1,V2) data to space-time plots (y1(t),t) and (y2(t),t) plotsIntegration of (V1,V2) data to space-space plots (y1, y2)

13Friday, August 28, 2015

M1=70gmM1=70gm

M0=10kgM0=10kgbounceplatebounceplate

RRrrd

Superballpenetrationdepthr22Rd=

SuperballSuperball

ballpointpen

M2=10gm

ballpointpen

M2=10gm

The X-2pen-

launcher

The X-2pen-

launcherFig. 4.1 and Fig. 4.3

in Unit 1

M1

m2BANG!M1

(BiggerBANG!)

(StillBiggerBANG!)

m2

M1M1

m2

Bang1!

Bang2!M1

m2

(a) Super-elastic 2nd-body bounce (b) 2-Bang Model (c) n-BodySupernovaSuperballs

m2

Fig. 4.5ain Unit 1Start at

(1.0,-1.0)

1.0

1.0

-1.0

0.5

2.0 C

L

P

L is 15::1

P is 7::1C is 4::1

Line CPLis elastic collisionfinal pt. locus fordifferentmomentumslopesormassratiosM1::M2

M2 Velocity axisV2

M1 Velocity axis V1(0,0)

3.0

Bang-2(12)FINALpoints

(a)

14Friday, August 28, 2015

M1=70gmM1=70gm

M0=10kgM0=10kgbounceplatebounceplate

RRrrd

Superballpenetrationdepthr22Rd=

SuperballSuperball

ballpointpen

M2=10gm

ballpointpen

M2=10gm

The X-2pen-

launcher

The X-2pen-

launcherFig. 4.1 and Fig. 4.3

in Unit 1

M1

m2BANG!M1

(BiggerBANG!)

(StillBiggerBANG!)

m2

M1M1

m2

Bang1!

Bang2!M1

m2

(a) Super-elastic 2nd-body bounce (b) 2-Bang Model (c) n-BodySupernovaSuperballs

m2

Fig. 4.5a-bin Unit 1Start at

(1.0,-1.0)

1.0

1.0

-1.0

0.5

2.0 C

L

P

L is 15::1

P is 7::1C is 4::1

Line CPLis elastic collisionfinal pt. locus fordifferentmomentumslopesormassratiosM1::M2

1.0

1.0

-1.0

0.5

2.0 C

L

P

L is 15::1

P is 7::1C is 4::1

II iiss 11::::00oorr ∞∞::::11II

MM iiss 33::::11

MM

3.0M2 Velocity axisV2

M2 Velocity axisV2

M1 Velocity axis V1 M1 Velocity axis V1(0,0) (0,0)

3.0

D is 2::1

Uis 1::1

Bang-2(12)FINALpoints

(a) (b)

-1.0

U2

U1

15Friday, August 28, 2015

Geometry of X2 launcher bouncing in boxIndependent Bounce Model (IBM)Geometric optimization and range-of-motion calculation(s)Integration of (V1,V2) data to space-time plots (y1(t),t) and (y2(t),t) plotsIntegration of (V1,V2) data to space-space plots (y1, y2)

16Friday, August 28, 2015

Ceiling at y=7.1

Time

t-axis

Height

y-axisVy1

Vy2

1.0

1.0

-1.0

0.5

Velocity Vy2vs. V

y1Plot

Position y vs. Time t Plot

Vy1=+1.0

Vy2=-0.5

means M1is

somewhere

on some path of slope +1.0

slope

1/1=+11

1

slope

-0.5/1=-0.5-0.5

1means M

2is

somewhere

on some path of slope -0.5

Floor at y=0

-0.5

1

y

-0.5

+1.0

Geometric “Integration” (Converting Velocity data to Spacetime)

17Friday, August 28, 2015

Ceiling at y=7.1

Time

t-axis

Height

y-axisVy1

Vy2

1.0

1.0

-1.0

0.5

Velocity Vy2vs. V

y1Plot

Position y vs. Time t Plot

Vy1=+1.0

Vy2=-0.5

means M1is

somewhere

on some path of slope +1.0

slope

1/1=+11

1

slope

-0.5/1=-0.5-0.5

1means M

2is

somewhere

on some path of slope -0.5

Floor at y=0

-0.5

1

y

-0.5

+1.0

Geometric “Integration” (Converting Velocity data to Spacetime)

18Friday, August 28, 2015

Ceiling at y=7.1

Time

t-axis

Height

y-axisVy1

Vy2

1.0

1.0

-1.0

0.5

Velocity Vy2vs. V

y1Plot

Position y vs. Time t Plot

Vy1=+1.0

Vy2=-0.5

means M1is

somewhere

on some path of slope +1.0

slope

1/1=+11

1

slope

-0.5/1=-0.5-0.5

1means M

2is

somewhere

on some path of slope -0.5

Floor at y=0

-0.5

1

y

-0.5

+1.0

Geometric “Integration” (Converting Velocity data to Spacetime)

Until you specifyinitial conditions y0(t0)…

...you don’t know whatvy-line to use

19Friday, August 28, 2015

Ceiling at y=7.1

Time

t-axis

Height

y-axisVy1

Vy2

1.0

1.0

-1.0

0.5

Velocity Vy2vs. V

y1Plot

Position y vs. Time t Plot

Vy1=+1.0

Vy2=-0.5

means M1is

somewhere

on some path of slope +1.0

slope

1/1=+11

1

slope

-0.5/1=-0.5-0.5

1means M

2is

somewhere

on some path of slope -0.5

Floor at y=0

-0.5

1

y

-0.5

+1.0

Bang-2(12)

position

Ceiling at y=7

Time

t-axis

Height

y-axis

slope

-1

slope

+1

Bang-1(01)

position

y2(0)=3

Vy1

Vy2

1.0

-1.0

0.5

(Vy1,Vy2)=(+1.0,-1.0)

y

(a) Vy2vs. V

y1Plot of Bang-1

(01) (b) y vs. t Plot of Bang-1(01)

Bang-1(01) Bounces (-1,-1) to (+1,-1)

(Vy1,Vy2)=(-1.0,-1.0)

(y=1,t=2)(y=0,t=1)

y1(0)=1

0.5

-1.0 -0.5

-0.5

??

??Floor at y=0

Geometric “Integration” (Converting Velocity data to Spacetime)

Fig. 4.6a-bin Unit 1

Until you specifyinitial conditions y0(t0)…

...you don’t know whichvy-lines to use

initial conditions y1(0)

...and y2(0)

20Friday, August 28, 2015

Ceiling at y=7.1

Time

t-axis

Height

y-axisVy1

Vy2

1.0

1.0

-1.0

0.5

Velocity Vy2vs. V

y1Plot

Position y vs. Time t Plot

Vy1=+1.0

Vy2=-0.5

means M1is

somewhere

on some path of slope +1.0

slope

1/1=+11

1

slope

-0.5/1=-0.5-0.5

1means M

2is

somewhere

on some path of slope -0.5

Floor at y=0

-0.5

1

y

-0.5

+1.0

Bang-2(12)

position

Ceiling at y=7

Time

t-axis

Height

y-axis

slope

-1

slope

+1

Bang-1(01)

position

y2(0)=3

Vy1

Vy2

1.0

-1.0

0.5

(Vy1,Vy2)=(+1.0,-1.0)

y

(a) Vy2vs. V

y1Plot of Bang-1

(01) (b) y vs. t Plot of Bang-1(01)

Bang-1(01) Bounces (-1,-1) to (+1,-1)

(Vy1,Vy2)=(-1.0,-1.0)

(y=1,t=2)(y=0,t=1)

y1(0)=1

0.5

-1.0 -0.5

-0.5

??

??Floor at y=0

Geometric “Integration” (Converting Velocity data to Spacetime)

Fig. 4.6a-bin Unit 1

Until you specifyinitial conditions y0(t0)…

...you don’t know whichvy-lines to use

initial conditions y1(0)

...and y2(0)

21Friday, August 28, 2015

Ceiling at y=7.1

Time

t-axis

Height

y-axisVy1

Vy2

1.0

1.0

-1.0

0.5

Velocity Vy2vs. V

y1Plot

Position y vs. Time t Plot

Vy1=+1.0

Vy2=-0.5

means M1is

somewhere

on some path of slope +1.0

slope

1/1=+11

1

slope

-0.5/1=-0.5-0.5

1means M

2is

somewhere

on some path of slope -0.5

Floor at y=0

-0.5

1

y

-0.5

+1.0

Bang-2(12)

position

Ceiling at y=7

Time

t-axis

Height

y-axis

slope

-1

slope

+1

Bang-1(01)

position

y2(0)=3

Vy1

Vy2

1.0

-1.0

0.5

(Vy1,Vy2)=(+1.0,-1.0)

y

(a) Vy2vs. V

y1Plot of Bang-1

(01) (b) y vs. t Plot of Bang-1(01)

Bang-1(01) Bounces (-1,-1) to (+1,-1)

(Vy1,Vy2)=(-1.0,-1.0)

(y=1,t=2)(y=0,t=1)

y1(0)=1

0.5

-1.0 -0.5

-0.5

??

??Floor at y=0

Geometric “Integration” (Converting Velocity data to Spacetime)

Fig. 4.6a-bin Unit 1

Until you specifyinitial conditions y0(t0)…

...you don’t know whichvy-lines to use

initial conditions y1(0)

...and y2(0)

22Friday, August 28, 2015

Ceiling at y=7.1

Time

t-axis

Height

y-axisVy1

Vy2

1.0

1.0

-1.0

0.5

Velocity Vy2vs. V

y1Plot

Position y vs. Time t Plot

Vy1=+1.0

Vy2=-0.5

means M1is

somewhere

on some path of slope +1.0

slope

1/1=+11

1

slope

-0.5/1=-0.5-0.5

1means M

2is

somewhere

on some path of slope -0.5

Floor at y=0

-0.5

1

y

-0.5

+1.0

Bang-2(12)

position

Ceiling at y=7

Time

t-axis

Height

y-axis

slope

-1

slope

+1

Bang-1(01)

position

y2(0)=3

Vy1

Vy2

1.0

-1.0

0.5

(Vy1,Vy2)=(+1.0,-1.0)

y

(a) Vy2vs. V

y1Plot of Bang-1

(01) (b) y vs. t Plot of Bang-1(01)

Bang-1(01) Bounces (-1,-1) to (+1,-1)

(Vy1,Vy2)=(-1.0,-1.0)

(y=1,t=2)(y=0,t=1)

y1(0)=1

0.5

-1.0 -0.5

-0.5

??

??Floor at y=0

Geometric “Integration” (Converting Velocity data to Spacetime)

Fig. 4.6a-bin Unit 1

Until you specifyinitial conditions y0(t0)…

...you don’t know whichvy-lines to use

initial conditions y1(0)

...and y2(0)

23Friday, August 28, 2015

Ceiling at y=7.1

Time

t-axis

Height

y-axisVy1

Vy2

1.0

1.0

-1.0

0.5

Velocity Vy2vs. V

y1Plot

Position y vs. Time t Plot

Vy1=+1.0

Vy2=-0.5

means M1is

somewhere

on some path of slope +1.0

slope

1/1=+11

1

slope

-0.5/1=-0.5-0.5

1means M

2is

somewhere

on some path of slope -0.5

Floor at y=0

-0.5

1

y

-0.5

+1.0

Bang-2(12)

position

Ceiling at y=7

Time

t-axis

Height

y-axis

slope

-1

slope

+1

Bang-1(01)

position

y2(0)=3

Vy1

Vy2

1.0

-1.0

0.5

(Vy1,Vy2)=(+1.0,-1.0)

y

(a) Vy2vs. V

y1Plot of Bang-1

(01) (b) y vs. t Plot of Bang-1(01)

Bang-1(01) Bounces (-1,-1) to (+1,-1)

(Vy1,Vy2)=(-1.0,-1.0)

(y=1,t=2)(y=0,t=1)

y1(0)=1

0.5

-1.0 -0.5

-0.5

??

??Floor at y=0

Geometric “Integration” (Converting Velocity data to Spacetime)

Fig. 4.6a-bin Unit 1

Until you specifyinitial conditions y0(t0)…

...you don’t know whichvy-lines to use

initial conditions y1(0)

...and y2(0)

24Friday, August 28, 2015

Ceiling at y=7.1

Time

t-axis

Height

y-axisVy1

Vy2

1.0

1.0

-1.0

0.5

Velocity Vy2vs. V

y1Plot

Position y vs. Time t Plot

Vy1=+1.0

Vy2=-0.5

means M1is

somewhere

on some path of slope +1.0

slope

1/1=+11

1

slope

-0.5/1=-0.5-0.5

1means M

2is

somewhere

on some path of slope -0.5

Floor at y=0

-0.5

1

y

-0.5

+1.0

Bang-2(12)

position

Ceiling at y=7

Time

t-axis

Height

y-axis

slope

-1

slope

+1

Bang-1(01)

position

y2(0)=3

Vy1

Vy2

1.0

-1.0

0.5

(Vy1,Vy2)=(+1.0,-1.0)

y

(a) Vy2vs. V

y1Plot of Bang-1

(01) (b) y vs. t Plot of Bang-1(01)

Bang-1(01) Bounces (-1,-1) to (+1,-1)

(Vy1,Vy2)=(-1.0,-1.0)

(y=1,t=2)(y=0,t=1)

y1(0)=1

0.5

-1.0 -0.5

-0.5

??

??Floor at y=0

Geometric “Integration” (Converting Velocity data to Spacetime)

Fig. 4.6a-bin Unit 1

Until you specifyinitial conditions y0(t0)…

...you don’t know whichvy-lines to use

initial conditions y1(0)

...and y2(0)

25Friday, August 28, 2015

Geometric “Integration” (Converting Velocity data to Spacetime)

Start at

(-1.0,-1.0)

1.0

1.0

-1.0

0.5

2.0

Bang-2(12)

Bang-3(20)

Bang-1(01)

Floor at y=0

Bang-1(01)

Bang-2(12)

Bang-3(20)

M2

slope

+2.5

M1

slope

+0.5

Height y

y=3

y=1

Ceiling at y=7

Bang-4(12)

??

Vy1

Vy2

??

(a)

Time t

(b)

M2

slope

-2.5

26Friday, August 28, 2015

Geometric “Integration” (Converting Velocity data to Spacetime)

Start at

(-1.0,-1.0)

1.0

1.0

-1.0

0.5

2.0

Bang-2(12)

Bang-3(20)

Bang-1(01)

Floor at y=0

Bang-1(01)

Bang-2(12)

Bang-3(20)

M2

slope

+2.5

M1

slope

+0.5

Height y

y=3

y=1

Ceiling at y=7

Bang-4(12)

??

Vy1

Vy2

??

(a)

Time t Bang-1(01)

Bang-2(12)

Bang-3(20)

Bang-4(12)

Bang-5(20)

Bang-6(12)

Bang-7(01)

Bang-8(20)

Bang-9(12)

Time t

Floor at y=0

Ceiling at y=7

y

Start at

(-1.0,-1.0)

1.0

1.0

-1.0

0.5

2.0

Bang-2(12)

Bang-3(20)

Bang-4(12)

Bang-1(01)

Bang-5(20)

Bang-6(12)

Bang-7(01)

Vy1

Vy2

(b)

(c)

(d)

M2

slope

-2.5

Fig. 4.7a-din Unit 1

Kinetic Energy Ellipse

KE = 12

M1V12 + 1

2M 2V2

2 =12

+ 72= 4

1 = V12

2KE /M1

+V2

2

2KE /M 2

=x1

2

a12 +

x22

a22

27Friday, August 28, 2015

Geometric “Integration” (Converting Velocity data to Spacetime)

Start at

(-1.0,-1.0)

1.0

1.0

-1.0

0.5

2.0

Bang-2(12)

Bang-3(20)

Bang-1(01)

Floor at y=0

Bang-1(01)

Bang-2(12)

Bang-3(20)

M2

slope

+2.5

M1

slope

+0.5

Height y

y=3

y=1

Ceiling at y=7

Bang-4(12)

??

Vy1

Vy2

??

(a)

Time t Bang-1(01)

Bang-2(12)

Bang-3(20)

Bang-4(12)

Bang-5(20)

Bang-6(12)

Bang-7(01)

Bang-8(20)

Bang-9(12)

Time t

Floor at y=0

Ceiling at y=7

y

Start at

(-1.0,-1.0)

1.0

1.0

-1.0

0.5

2.0

Bang-2(12)

Bang-3(20)

Bang-4(12)

Bang-1(01)

Bang-5(20)

Bang-6(12)

Bang-7(01)

Vy1

Vy2

(b)

(c)

(d)

M2

slope

-2.5

Fig. 4.7a-din Unit 1

Kinetic Energy Ellipse

KE = 12

M1V12 + 1

2M 2V2

2 =72

+ 12= 4

1 = V12

2KE /M1

+V2

2

2KE /M 2

=x1

2

a12 +

x22

a22

Ellipse radius 1

a1 = 2KE /M1

Ellipse radius 2

a2 = 2KE /M1

28Friday, August 28, 2015

Geometric “Integration” (Converting Velocity data to Spacetime)

Start at

(-1.0,-1.0)

1.0

1.0

-1.0

0.5

2.0

Bang-2(12)

Bang-3(20)

Bang-1(01)

Floor at y=0

Bang-1(01)

Bang-2(12)

Bang-3(20)

M2

slope

+2.5

M1

slope

+0.5

Height y

y=3

y=1

Ceiling at y=7

Bang-4(12)

??

Vy1

Vy2

??

(a)

Time t Bang-1(01)

Bang-2(12)

Bang-3(20)

Bang-4(12)

Bang-5(20)

Bang-6(12)

Bang-7(01)

Bang-8(20)

Bang-9(12)

Time t

Floor at y=0

Ceiling at y=7

y

Start at

(-1.0,-1.0)

1.0

1.0

-1.0

0.5

2.0

Bang-2(12)

Bang-3(20)

Bang-4(12)

Bang-1(01)

Bang-5(20)

Bang-6(12)

Bang-7(01)

Vy1

Vy2

(b)

(c)

(d)

M2

slope

-2.5

Fig. 4.7a-din Unit 1

Kinetic Energy Ellipse

KE = 12

M1V12 + 1

2M 2V2

2 =72

+ 12= 4

1 = V12

2KE /M1

+V2

2

2KE /M 2

=x1

2

a12 +

x22

a22

Ellipse radius 1

a1 = 2KE /M1

= 2KE /7

= 8/7 = 1.07

Ellipse radius 2

a2 = 2KE /M1

= 2KE /1

= 8/1 = 2.83

29Friday, August 28, 2015

Geometric “Integration” (Converting Velocity data to Spacetime)

27825

Start Start

∞ ?

Start

4PenV2

BallV1

ErgodicFill-inat t=∞ ?

(a) (b) (c)

Fig. 4.9in Unit 1

Fig. 4.8in Unit 1

30Friday, August 28, 2015

Geometry of X2 launcher bouncing in boxIndependent Bounce Model (IBM)Geometric optimization and range-of-motion calculation(t)Integration of (V1,V2) data to space-time plots (y1(t),t) and (y2(t),t) plotsIntegration of (V1,V2) data to space-space plots (y1, y2)

31Friday, August 28, 2015

v(2)

Bang-1(01)

Start0 at

(y1(0)=1,y2(0)=3)

v(0)v(1)

Collisionliney 2=y 1

1.0

1.0

-1.0

0.5

2.0

Bang-2(12)

m2

Velocity axis

Vym2

m1

Velocity axis

Vym1

m1-Height

y1-axis

Ceilingaty1=7

Floor at y2=0.0

Flooraty1=0.0

1.0 2.0 3.0 4.0 5.0 6.0

1.0

2.0

3.0

4.0

5.0

6.0

Ceiling at y2=7

m2-Height

y2-axis

Bang-2(12)Bang-1(01)

Start0 at

(V1(0)=1, V2(0)=3)

v(1)v(0)

v(2)

Step-0: At starting position y(0)=(1,3) draw initial velocity v(0)=(-1,-1) line.Step-1: Extend v(0) line to floor point y(0)=(0,?) and draw Bang-1

(01)

velocity v(1)=(1,-1) line. (Find v(1) using V-V plot.)Step-2: Extend v(1) line to collision point y(0)=(?,?) and draw Bang-2

(12)

velocity v(2)=(0.5,2.5). (Find v(2) using V-V plot.)

Geometric “Integration” (Converting Velocity data to Space-space trajectory)

Fig. 4.11in Unit 1

32Friday, August 28, 2015

Bang-1(01)

Start0 at

(y1(0)=1,y2(0)=3)

Collisionliney 2=y 1

Bang-2(12)

Bang-3(20)

v(3)

v(4)

Bang-4(12)

v(2)

m2-Height

y2-axis

Ceiling at y2=7.1

v(5)

Bang-5(20)

Bang-6(12)

v(3)

v(4)

m1

Velocity axis

Vym1

Ceilingaty1=7.1

Floor at y2=0.0

Flooraty1=0.0

1.0 2.0 3.0 4.0 5.0 6.0

1.0

2.0

3.0

4.0

5.0

6.0

m1-Height

y1-axis

Step-2: Extend v(2) line to ceiling point y(3)=(?,7.1) and draw Bang-3(20)

velocity v(3)=(1,-1) line. (Find v(3) using V-V plot.)Step-3: Extend v(3) line to collision point y(4)=(?,?) and draw Bang-4

(12)

velocity v(4)=(0.5,2.5). (Find v(4) using V-V plot.)

Start at

(-1.0,-1.0)

1.0

1.0

-1.0

0.5

2.0

Bang-2(12)

Bang-3(20)

Bang-4(12)

m2 Velocity axis

Vym2

Step-5: Extend v(5) line to collision point y(6)=(?,?) and draw Bang-6(12)

velocity v(6)=(0.5,2.5). (Find v(6) using V-V plot.)

Step-4: Extend v(4) line to ceiling point y(4)=(?,7.1) and draw Bang-5(20)

velocity v(5)=(1,-1) line. (Find v(5) using V-V plot.)

v(5)

Geometric “Integration” (Converting Velocity data to Space-space trajectory)

Fig. 4.11in Unit 1

33Friday, August 28, 2015

Geometric “Integration” (Converting Velocity data to Space-time trajectory)

Start at(1.0,-1.0)

1.0

1.0

-1.0

2.0

3.0

4.0

5.0

6.0

7.0

-2.0

-3.0

-4.0

-5.0

-6.0

Bang-1(01)

Bang-1(01)

Bang-2(12)

Bang-4(12)

Bang-6(12)

m2 Velocity axisVym2

m1 Velocity axis Vym1

81012 14

16

18

7 9 111315 17 19

Bang-2(12)

Bang-3(20)

Bang-4(12)

Bang-6(12)

810

12

5

7

9

5

-5

Bang-3(20)

m1 = 49

m2 = 1

1113

Fig. 5.1in Unit 1

Example with masses: m1=49 and m2=1

34Friday, August 28, 2015

Geometric “Integration” (Converting Velocity data to Space-time trajectory)

Start at(1.0,-1.0)

1.0

1.0

-1.0

2.0

3.0

4.0

5.0

6.0

7.0

-2.0

-3.0

-4.0

-5.0

-6.0

Bang-1(01)

Bang-1(01)

Bang-2(12)

Bang-4(12)

Bang-6(12)

m2 Velocity axisVym2

m1 Velocity axis Vym1

81012 14

16

18

7 9 111315 17 19

Bang-2(12)

Bang-3(20)

Bang-4(12)

Bang-6(12)

810

12

5

7

9

5

-5

Bang-3(20)

m1 = 49

m2 = 1

1113

Fig. 5.1in Unit 1

Example with masses: m1=49 and m2=1Kinetic Energy Ellipse

KE = 12

m1V12 + 1

2m2V2

2 =492

+ 12= 25

1 = V12

2KE /m1

+V2

2

2KE /m2

=x1

2

a12 +

x22

a22

Ellipse radius 1

a1 = 2KE /M1

= 2KE /49

= 50/49 = 1.01

Ellipse radius 2

a2 = 2KE /m2

= 2KE /1

= 50/1 = 7.07

35Friday, August 28, 2015

Multiple collisions calculated by matrix operator products Matrix or tensor algebra of 1-D 2-body collisions“Mass-bang” matrix M, “Floor-bang” matrix F, “Ceiling-bang” matrix C.Geometry and algebra of “ellipse-Rotation” group product: R= C•M

36Friday, August 28, 2015

Multiple Collisions by Matrix Operator Products

v1FIN

v2FIN

⎛

⎝⎜⎜

⎞

⎠⎟⎟=

2VCOM − v1IN

2VCOM − v2IN

⎛

⎝⎜⎜

⎞

⎠⎟⎟=

T-Symmetry & Momentum Axioms give: VCOM = VFIN +V IN

2= m1v1 +m2v2

m1 +m2

Gives vFIN in terms of vIN...

2 m1v1IN +m2v2

IN

m1 +m2

− v1IN

2 m1v1IN +m2v2

IN

m1 +m2

− v2IN

⎛

⎝

⎜⎜⎜⎜⎜

⎞

⎠

⎟⎟⎟⎟⎟

=

m1v1IN −m2v1

IN + 2m2v2IN

2m1v1IN +m2v2

IN −m1v2IN

⎛

⎝⎜⎜

⎞

⎠⎟⎟

m1 +m2

=

m1 −m2 2m2

2m1 m2 −m1

⎛

⎝⎜⎜

⎞

⎠⎟⎟

v1IN

v2IN

⎛

⎝⎜⎜

⎞

⎠⎟⎟

m1 +m2

Finally as a matrix operation: vFIN =M•vIN...

M =

m1 −m2

m1 +m2

2m2

m1 +m2

2m1

m1 +m2

m2 −m1

m1 +m2

⎛

⎝

⎜⎜⎜⎜⎜

⎞

⎠

⎟⎟⎟⎟⎟

Matrix operations include...Floor bounce F of m1: Mass collision M of m1 and m2 : Ceiling bounce C of m2:

F = −1 00 1

⎛⎝⎜

⎞⎠⎟

C = 1 00 −1

⎛⎝⎜

⎞⎠⎟

Let: m1=49 and m2=1 M = 0.96 0.041.96 −0.96

⎛⎝⎜

⎞⎠⎟

Define a “rotation” R as group product: R= C•M = 1 00 −1

⎛⎝⎜

⎞⎠⎟⋅ 0.96 0.041.96 −0.96

⎛⎝⎜

⎞⎠⎟= 0.96 0.04

−1.96 0.96⎛⎝⎜

⎞⎠⎟

37Friday, August 28, 2015

Multiple Collisions by Matrix Operator Products

v1FIN

v2FIN

⎛

⎝⎜⎜

⎞

⎠⎟⎟=

2VCOM − v1IN

2VCOM − v2IN

⎛

⎝⎜⎜

⎞

⎠⎟⎟=

T-Symmetry & Momentum Axioms give: VCOM = VFIN +V IN

2= m1v1 +m2v2

m1 +m2

Gives vFIN in terms of vIN...

2 m1v1IN +m2v2

IN

m1 +m2

− v1IN

2 m1v1IN +m2v2

IN

m1 +m2

− v2IN

⎛

⎝

⎜⎜⎜⎜⎜

⎞

⎠

⎟⎟⎟⎟⎟

=

m1v1IN −m2v1

IN + 2m2v2IN

2m1v1IN +m2v2

IN −m1v2IN

⎛

⎝⎜⎜

⎞

⎠⎟⎟

m1 +m2

=

m1 −m2 2m2

2m1 m2 −m1

⎛

⎝⎜⎜

⎞

⎠⎟⎟

v1IN

v2IN

⎛

⎝⎜⎜

⎞

⎠⎟⎟

m1 +m2

Finally as a matrix operation: vFIN =M•vIN...

Matrix operations include...Floor bounce F of m1: Mass collision M of m1 and m2 : Ceiling bounce C of m2:

Let: m1=49 and m2=1

Define a “rotation” R as group product: R= C•M

38Friday, August 28, 2015

Multiple Collisions by Matrix Operator Products

v1FIN

v2FIN

⎛

⎝⎜⎜

⎞

⎠⎟⎟=

2VCOM − v1IN

2VCOM − v2IN

⎛

⎝⎜⎜

⎞

⎠⎟⎟=

T-Symmetry & Momentum Axioms give: VCOM = VFIN +V IN

2= m1v1 +m2v2

m1 +m2

Gives vFIN in terms of vIN...

2 m1v1IN +m2v2

IN

m1 +m2

− v1IN

2 m1v1IN +m2v2

IN

m1 +m2

− v2IN

⎛

⎝

⎜⎜⎜⎜⎜

⎞

⎠

⎟⎟⎟⎟⎟

=

m1v1IN −m2v1

IN + 2m2v2IN

2m1v1IN +m2v2

IN −m1v2IN

⎛

⎝⎜⎜

⎞

⎠⎟⎟

m1 +m2

=

m1 −m2 2m2

2m1 m2 −m1

⎛

⎝⎜⎜

⎞

⎠⎟⎟

v1IN

v2IN

⎛

⎝⎜⎜

⎞

⎠⎟⎟

m1 +m2

Finally as a matrix operation: vFIN =M•vIN...

Matrix operations include...Floor bounce F of m1: Mass collision M of m1 and m2 : Ceiling bounce C of m2:

Let: m1=49 and m2=1

Define a “rotation” R as group product: R= C•M

39Friday, August 28, 2015

Multiple Collisions by Matrix Operator Products

v1FIN

v2FIN

⎛

⎝⎜⎜

⎞

⎠⎟⎟=

2VCOM − v1IN

2VCOM − v2IN

⎛

⎝⎜⎜

⎞

⎠⎟⎟=

T-Symmetry & Momentum Axioms give: VCOM = VFIN +V IN

2= m1v1 +m2v2

m1 +m2

Gives vFIN in terms of vIN...

2 m1v1IN +m2v2

IN

m1 +m2

− v1IN

2 m1v1IN +m2v2

IN

m1 +m2

− v2IN

⎛

⎝

⎜⎜⎜⎜⎜

⎞

⎠

⎟⎟⎟⎟⎟

=

m1v1IN −m2v1

IN + 2m2v2IN

2m1v1IN +m2v2

IN −m1v2IN

⎛

⎝⎜⎜

⎞

⎠⎟⎟

m1 +m2

=

m1 −m2 2m2

2m1 m2 −m1

⎛

⎝⎜⎜

⎞

⎠⎟⎟

v1IN

v2IN

⎛

⎝⎜⎜

⎞

⎠⎟⎟

m1 +m2

Finally as a matrix operation: vFIN =M•vIN...

M =

m1 −m2

m1 +m2

2m2

m1 +m2

2m1

m1 +m2

m2 −m1

m1 +m2

⎛

⎝

⎜⎜⎜⎜⎜

⎞

⎠

⎟⎟⎟⎟⎟

Matrix operations include...Floor bounce F of m1: Mass collision M of m1 and m2 : Ceiling bounce C of m2:

F = −1 00 1

⎛⎝⎜

⎞⎠⎟

C = 1 00 −1

⎛⎝⎜

⎞⎠⎟

Let: m1=49 and m2=1 M = 0.96 0.041.96 −0.96

⎛⎝⎜

⎞⎠⎟

Define a “rotation” R as group product: R= C•M = 1 00 −1

⎛⎝⎜

⎞⎠⎟⋅ 0.96 0.041.96 −0.96

⎛⎝⎜

⎞⎠⎟= 0.96 0.04

−1.96 0.96⎛⎝⎜

⎞⎠⎟

40Friday, August 28, 2015

Multiple collisions calculated by matrix operator products Matrix or tensor algebra of 1-D 2-body collisions“Mass-bang” matrix M, “Floor-bang” matrix F, “Ceiling-bang” matrix C.Geometry and algebra of “ellipse-Rotation” group product: R= C•M

41Friday, August 28, 2015

Multiple Collisions by Matrix Operator Products

v1FIN

v2FIN

⎛

⎝⎜⎜

⎞

⎠⎟⎟=

2VCOM − v1IN

2VCOM − v2IN

⎛

⎝⎜⎜

⎞

⎠⎟⎟=

T-Symmetry & Momentum Axioms give: VCOM = VFIN +V IN

2= m1v1 +m2v2

m1 +m2

Gives vFIN in terms of vIN...

2 m1v1IN +m2v2

IN

m1 +m2

− v1IN

2 m1v1IN +m2v2

IN

m1 +m2

− v2IN

⎛

⎝

⎜⎜⎜⎜⎜

⎞

⎠

⎟⎟⎟⎟⎟

=

m1v1IN −m2v1

IN + 2m2v2IN

2m1v1IN +m2v2

IN −m1v2IN

⎛

⎝⎜⎜

⎞

⎠⎟⎟

m1 +m2

=

m1 −m2 2m2

2m1 m2 −m1

⎛

⎝⎜⎜

⎞

⎠⎟⎟

v1IN

v2IN

⎛

⎝⎜⎜

⎞

⎠⎟⎟

m1 +m2

Finally as a matrix operation: vFIN =M•vIN...

Matrix operations include...Floor-bang F of m1:

F = −1 00 1

⎛⎝⎜

⎞⎠⎟

Define a “rotation” R as group product: R= C•M

42Friday, August 28, 2015

Multiple Collisions by Matrix Operator Products

v1FIN

v2FIN

⎛

⎝⎜⎜

⎞

⎠⎟⎟=

2VCOM − v1IN

2VCOM − v2IN

⎛

⎝⎜⎜

⎞

⎠⎟⎟=

T-Symmetry & Momentum Axioms give: VCOM = VFIN +V IN

2= m1v1 +m2v2

m1 +m2

Gives vFIN in terms of vIN...

2 m1v1IN +m2v2

IN

m1 +m2

− v1IN

2 m1v1IN +m2v2

IN

m1 +m2

− v2IN

⎛

⎝

⎜⎜⎜⎜⎜

⎞

⎠

⎟⎟⎟⎟⎟

=

m1v1IN −m2v1

IN + 2m2v2IN

2m1v1IN +m2v2

IN −m1v2IN

⎛

⎝⎜⎜

⎞

⎠⎟⎟

m1 +m2

=

m1 −m2 2m2

2m1 m2 −m1

⎛

⎝⎜⎜

⎞

⎠⎟⎟

v1IN

v2IN

⎛

⎝⎜⎜

⎞

⎠⎟⎟

m1 +m2

Finally as a matrix operation: vFIN =M•vIN...

Matrix operations include...Floor-bang F of m1: Ceiling-bang C of m2:

F = −1 00 1

⎛⎝⎜

⎞⎠⎟

C = 1 00 −1

⎛⎝⎜

⎞⎠⎟

Define a “rotation” R as group product: R= C•M

43Friday, August 28, 2015

Multiple Collisions by Matrix Operator Products

v1FIN

v2FIN

⎛

⎝⎜⎜

⎞

⎠⎟⎟=

2VCOM − v1IN

2VCOM − v2IN

⎛

⎝⎜⎜

⎞

⎠⎟⎟=

T-Symmetry & Momentum Axioms give: VCOM = VFIN +V IN

2= m1v1 +m2v2

m1 +m2

Gives vFIN in terms of vIN...

2 m1v1IN +m2v2

IN

m1 +m2

− v1IN

2 m1v1IN +m2v2

IN

m1 +m2

− v2IN

⎛

⎝

⎜⎜⎜⎜⎜

⎞

⎠

⎟⎟⎟⎟⎟

=

m1v1IN −m2v1

IN + 2m2v2IN

2m1v1IN +m2v2

IN −m1v2IN

⎛

⎝⎜⎜

⎞

⎠⎟⎟

m1 +m2

=

m1 −m2 2m2

2m1 m2 −m1

⎛

⎝⎜⎜

⎞

⎠⎟⎟

v1IN

v2IN

⎛

⎝⎜⎜

⎞

⎠⎟⎟

m1 +m2

Finally as a matrix operation: vFIN =M•vIN...

M =

m1 −m2

m1 +m2

2m2

m1 +m2

2m1

m1 +m2

m2 −m1

m1 +m2

⎛

⎝

⎜⎜⎜⎜⎜

⎞

⎠

⎟⎟⎟⎟⎟

Matrix operations include...Floor-bang F of m1: Mass-bang M of m1 and m2 : Ceiling-bang C of m2:

F = −1 00 1

⎛⎝⎜

⎞⎠⎟

C = 1 00 −1

⎛⎝⎜

⎞⎠⎟

Define a “rotation” R as group product: R= C•M

44Friday, August 28, 2015

Multiple Collisions by Matrix Operator Products

v1FIN

v2FIN

⎛

⎝⎜⎜

⎞

⎠⎟⎟=

2VCOM − v1IN

2VCOM − v2IN

⎛

⎝⎜⎜

⎞

⎠⎟⎟=

T-Symmetry & Momentum Axioms give: VCOM = VFIN +V IN

2= m1v1 +m2v2

m1 +m2

Gives vFIN in terms of vIN...

2 m1v1IN +m2v2

IN

m1 +m2

− v1IN

2 m1v1IN +m2v2

IN

m1 +m2

− v2IN

⎛

⎝

⎜⎜⎜⎜⎜

⎞

⎠

⎟⎟⎟⎟⎟

=

m1v1IN −m2v1

IN + 2m2v2IN

2m1v1IN +m2v2

IN −m1v2IN

⎛

⎝⎜⎜

⎞

⎠⎟⎟

m1 +m2

=

m1 −m2 2m2

2m1 m2 −m1

⎛

⎝⎜⎜

⎞

⎠⎟⎟

v1IN

v2IN

⎛

⎝⎜⎜

⎞

⎠⎟⎟

m1 +m2

Finally as a matrix operation: vFIN =M•vIN...

M =

m1 −m2

m1 +m2

2m2

m1 +m2

2m1

m1 +m2

m2 −m1

m1 +m2

⎛

⎝

⎜⎜⎜⎜⎜

⎞

⎠

⎟⎟⎟⎟⎟

Matrix operations include...Floor-bang F of m1: Mass-bang M of m1 and m2 : Ceiling-bang C of m2:

F = −1 00 1

⎛⎝⎜

⎞⎠⎟

C = 1 00 −1

⎛⎝⎜

⎞⎠⎟

Let: m1=49 and m2=1 M = 0.96 0.041.96 −0.96

⎛⎝⎜

⎞⎠⎟

Define a “rotation” R as group product: R= C•M = 1 00 −1

⎛⎝⎜

⎞⎠⎟⋅ 0.96 0.041.96 −0.96

⎛⎝⎜

⎞⎠⎟= 0.96 0.04

−1.96 0.96⎛⎝⎜

⎞⎠⎟

45Friday, August 28, 2015

Multiple collisions calculated by matrix operator products Matrix or tensor algebra of 1-D 2-body collisions“Mass-bang” matrix M, “Floor-bang” matrix F, “Ceiling-bang” matrix C.Geometry and algebra of “ellipse-Rotation” group product: R= C•M

46Friday, August 28, 2015

Multiple Collisions by Matrix Operator Products

v1FIN

v2FIN

⎛

⎝⎜⎜

⎞

⎠⎟⎟=

2VCOM − v1IN

2VCOM − v2IN

⎛

⎝⎜⎜

⎞

⎠⎟⎟=

T-Symmetry & Momentum Axioms give: VCOM = VFIN +V IN

2= m1v1 +m2v2

m1 +m2

Gives vFIN in terms of vIN...

2 m1v1IN +m2v2

IN

m1 +m2

− v1IN

2 m1v1IN +m2v2

IN

m1 +m2

− v2IN

⎛

⎝

⎜⎜⎜⎜⎜

⎞

⎠

⎟⎟⎟⎟⎟

=

m1v1IN −m2v1

IN + 2m2v2IN

2m1v1IN +m2v2

IN −m1v2IN

⎛

⎝⎜⎜

⎞

⎠⎟⎟

m1 +m2

=

m1 −m2 2m2

2m1 m2 −m1

⎛

⎝⎜⎜

⎞

⎠⎟⎟

v1IN

v2IN

⎛

⎝⎜⎜

⎞

⎠⎟⎟

m1 +m2

Finally as a matrix operation: vFIN =M•vIN...

M =

m1 −m2

m1 +m2

2m2

m1 +m2

2m1

m1 +m2

m2 −m1

m1 +m2

⎛

⎝

⎜⎜⎜⎜⎜

⎞

⎠

⎟⎟⎟⎟⎟

Matrix operations include...Floor-bang F of m1: Mass-bang M of m1 and m2 : Ceiling-bang C of m2:

F = −1 00 1

⎛⎝⎜

⎞⎠⎟

C = 1 00 −1

⎛⎝⎜

⎞⎠⎟

Let: m1=49 and m2=1 M = 0.96 0.041.96 −0.96

⎛⎝⎜

⎞⎠⎟

Define “ellipse-Rotation” R as group product: R= C•M= 1 00 −1

⎛⎝⎜

⎞⎠⎟⋅ 0.96 0.041.96 −0.96

⎛⎝⎜

⎞⎠⎟= 0.96 0.04

−1.96 0.96⎛⎝⎜

⎞⎠⎟

47Friday, August 28, 2015

C • M • C • M • C • M • C • M • F IN 0FIN 9 =

v1FIN−9

v2FIN−9

⎛

⎝⎜⎜

⎞

⎠⎟⎟= 1 0

0 −1⎛⎝⎜

⎞⎠⎟⋅ 0.96 0.04

1.96 −0.96⎛⎝⎜

⎞⎠⎟⋅ 1 0

0 −1⎛⎝⎜

⎞⎠⎟⋅ 0.96 0.04

1.96 −0.96⎛⎝⎜

⎞⎠⎟⋅ 1 0

0 −1⎛⎝⎜

⎞⎠⎟⋅ 0.96 0.04

1.96 −0.96⎛⎝⎜

⎞⎠⎟⋅ 1 0

0 −1⎛⎝⎜

⎞⎠⎟⋅ 0.96 0.04

1.96 −0.96⎛⎝⎜

⎞⎠⎟⋅ −1 0

0 +1⎛⎝⎜

⎞⎠⎟

v1IN = −1

v2IN = −1

⎛

⎝⎜⎜

⎞

⎠⎟⎟

INITIAL (0)( )

48Friday, August 28, 2015

C • M • C • M • C • M • C • M • F

v1FIN−9

v2FIN−9

⎛

⎝⎜⎜

⎞

⎠⎟⎟= 0.96 0.04

−1.96 0.96⎛⎝⎜

⎞⎠⎟

⋅ 0.96 0.04−1.96 0.96

⎛⎝⎜

⎞⎠⎟

⋅ 0.96 0.04−1.96 0.96

⎛⎝⎜

⎞⎠⎟

⋅ 0.96 0.04−1.96 0.96

⎛⎝⎜

⎞⎠⎟

⋅v1 = 1v2 = −1

⎛

⎝⎜⎜

⎞

⎠⎟⎟

after Bang-1( )

R • R • R • R • F

IN 0

IN 0FIN 9 =

FIN 9 =

v1FIN−9

v2FIN−9

⎛

⎝⎜⎜

⎞

⎠⎟⎟= 1 0

0 −1⎛⎝⎜

⎞⎠⎟⋅ 0.96 0.04

1.96 −0.96⎛⎝⎜

⎞⎠⎟⋅ 1 0

0 −1⎛⎝⎜

⎞⎠⎟⋅ 0.96 0.04

1.96 −0.96⎛⎝⎜

⎞⎠⎟⋅ 1 0

0 −1⎛⎝⎜

⎞⎠⎟⋅ 0.96 0.04

1.96 −0.96⎛⎝⎜

⎞⎠⎟⋅ 1 0

0 −1⎛⎝⎜

⎞⎠⎟⋅ 0.96 0.04

1.96 −0.96⎛⎝⎜

⎞⎠⎟⋅ −1 0

0 +1⎛⎝⎜

⎞⎠⎟

v1IN = −1

v2IN = −1

⎛

⎝⎜⎜

⎞

⎠⎟⎟

INITIAL (0)( )

“ellipse-Rotation” group product: R= C•M

49Friday, August 28, 2015

C • M • C • M • C • M • C • M • F

v1FIN−9

v2FIN−9

⎛

⎝⎜⎜

⎞

⎠⎟⎟= 0.96 0.04

−1.96 0.96⎛⎝⎜

⎞⎠⎟

⋅ 0.96 0.04−1.96 0.96

⎛⎝⎜

⎞⎠⎟

⋅ 0.96 0.04−1.96 0.96

⎛⎝⎜

⎞⎠⎟

⋅ 0.96 0.04−1.96 0.96

⎛⎝⎜

⎞⎠⎟

⋅v1 = 1v2 = −1

⎛

⎝⎜⎜

⎞

⎠⎟⎟

after Bang-1( )

=v1 = 0.2925v2 = −6.768

⎛

⎝⎜⎜

⎞

⎠⎟⎟

after Bang-9( )

R • R • R • R • F

IN 0

IN 0FIN 9 =

FIN 9 =

v1FIN−9

v2FIN−9

⎛

⎝⎜⎜

⎞

⎠⎟⎟= 1 0

0 −1⎛⎝⎜

⎞⎠⎟⋅ 0.96 0.04

1.96 −0.96⎛⎝⎜

⎞⎠⎟⋅ 1 0

0 −1⎛⎝⎜

⎞⎠⎟⋅ 0.96 0.04

1.96 −0.96⎛⎝⎜

⎞⎠⎟⋅ 1 0

0 −1⎛⎝⎜

⎞⎠⎟⋅ 0.96 0.04

1.96 −0.96⎛⎝⎜

⎞⎠⎟⋅ 1 0

0 −1⎛⎝⎜

⎞⎠⎟⋅ 0.96 0.04

1.96 −0.96⎛⎝⎜

⎞⎠⎟⋅ −1 0

0 +1⎛⎝⎜

⎞⎠⎟

v1IN = −1

v2IN = −1

⎛

⎝⎜⎜

⎞

⎠⎟⎟

INITIAL (0)( )

“ellipse-Rotation” group product: R= C•M

50Friday, August 28, 2015

Start at(1.0,-1.0)

1.0

1.0

-1.0

2.0

3.0

4.0

5.0

6.0

7.0

-2.0

-3.0

-4.0

-5.0

-6.0

Bang-1(01)

Bang-1(01)

Bang-2(12)

Bang-4(12)

Bang-6(12)

m2 Velocity axisVym2

m1 Velocity axis Vym1

81012 14

16

18

7 9 111315 17 19

Bang-2(12)

Bang-3(20)

Bang-4(12)

Bang-6(12)

810

12

5

7

9

5

-5

Bang-3(20)

m1 = 49

m2 = 1

1113

C • M • C • M • C • M • C • M • F

v1FIN−9

v2FIN−9

⎛

⎝⎜⎜

⎞

⎠⎟⎟= 0.96 0.04

−1.96 0.96⎛⎝⎜

⎞⎠⎟

⋅ 0.96 0.04−1.96 0.96

⎛⎝⎜

⎞⎠⎟

⋅ 0.96 0.04−1.96 0.96

⎛⎝⎜

⎞⎠⎟

⋅ 0.96 0.04−1.96 0.96

⎛⎝⎜

⎞⎠⎟

⋅v1 = 1v2 = −1

⎛

⎝⎜⎜

⎞

⎠⎟⎟

after Bang-1( )

=v1 = 0.2925v2 = −6.768

⎛

⎝⎜⎜

⎞

⎠⎟⎟

after Bang-9( )

R • R • R • R • F

IN 0

IN 0FIN 9 =

FIN 9 =

v1FIN−9

v2FIN−9

⎛

⎝⎜⎜

⎞

⎠⎟⎟= 1 0

0 −1⎛⎝⎜

⎞⎠⎟⋅ 0.96 0.04

1.96 −0.96⎛⎝⎜

⎞⎠⎟⋅ 1 0

0 −1⎛⎝⎜

⎞⎠⎟⋅ 0.96 0.04

1.96 −0.96⎛⎝⎜

⎞⎠⎟⋅ 1 0

0 −1⎛⎝⎜

⎞⎠⎟⋅ 0.96 0.04

1.96 −0.96⎛⎝⎜

⎞⎠⎟⋅ 1 0

0 −1⎛⎝⎜

⎞⎠⎟⋅ 0.96 0.04

1.96 −0.96⎛⎝⎜

⎞⎠⎟⋅ −1 0

0 +1⎛⎝⎜

⎞⎠⎟

v1IN = −1

v2IN = −1

⎛

⎝⎜⎜

⎞

⎠⎟⎟

INITIAL (0)( )

R

R

R

R

F

“ellipse-Rotation” group product: R= C•M

51Friday, August 28, 2015

Start at(1.0,-1.0)

1.0

1.0

-1.0

2.0

3.0

4.0

5.0

6.0

7.0

-2.0

-3.0

-4.0

-5.0

-6.0

Bang-1(01)

Bang-1(01)

Bang-2(12)

Bang-4(12)

Bang-6(12)

m2 Velocity axisVym2

m1 Velocity axis Vym1

81012 14

16

18

7 9 111315 17 19

Bang-2(12)

Bang-3(20)

Bang-4(12)

Bang-6(12)

810

12

5

7

9

5

-5

Bang-3(20)

m1 = 49

m2 = 1

1113

C • M • C • M • C • M • C • M • F

v1FIN−9

v2FIN−9

⎛

⎝⎜⎜

⎞

⎠⎟⎟= 0.96 0.04

−1.96 0.96⎛⎝⎜

⎞⎠⎟

⋅ 0.96 0.04−1.96 0.96

⎛⎝⎜

⎞⎠⎟

⋅ 0.96 0.04−1.96 0.96

⎛⎝⎜

⎞⎠⎟

⋅ 0.96 0.04−1.96 0.96

⎛⎝⎜

⎞⎠⎟

⋅v1 = 1v2 = −1

⎛

⎝⎜⎜

⎞

⎠⎟⎟

after Bang-1( )

=v1 = 0.2925v2 = −6.768

⎛

⎝⎜⎜

⎞

⎠⎟⎟

after Bang-9( )

R • R • R • R • F

IN 0

IN 0FIN 9 =

FIN 9 =

v1FIN−9

v2FIN−9

⎛

⎝⎜⎜

⎞

⎠⎟⎟= 1 0

0 −1⎛⎝⎜

⎞⎠⎟⋅ 0.96 0.04

1.96 −0.96⎛⎝⎜

⎞⎠⎟⋅ 1 0

0 −1⎛⎝⎜

⎞⎠⎟⋅ 0.96 0.04

1.96 −0.96⎛⎝⎜

⎞⎠⎟⋅ 1 0

0 −1⎛⎝⎜

⎞⎠⎟⋅ 0.96 0.04

1.96 −0.96⎛⎝⎜

⎞⎠⎟⋅ 1 0

0 −1⎛⎝⎜

⎞⎠⎟⋅ 0.96 0.04

1.96 −0.96⎛⎝⎜

⎞⎠⎟⋅ −1 0

0 +1⎛⎝⎜

⎞⎠⎟

v1IN = −1

v2IN = −1

⎛

⎝⎜⎜

⎞

⎠⎟⎟

INITIAL (0)( )

R

R

R

R

F

v1FIN−11

v2FIN−11

⎛

⎝⎜⎜

⎞

⎠⎟⎟= 0.96 0.04

−1.96 0.96⎛⎝⎜

⎞⎠⎟⋅

v1FIN−9

v2FIN−9

⎛

⎝⎜⎜

⎞

⎠⎟⎟

=v1 = 0.0100v2 = −7.071

⎛

⎝⎜⎜

⎞

⎠⎟⎟

after Bang-11( )

“ellipse-Rotation” group product: R= C•M

52Friday, August 28, 2015

Ellipse rescaling-geometry and reflection-symmetry analysisRescaling KE ellipse to circle

How this relates to Lagrangian, l’Etrangian, and Hamiltonian mechanics in Ch. 12

53Friday, August 28, 2015

Ellipse rescaling geometry and reflection symmetry analysis Convert to rescaled velocity: symmetrize: V1 = v1 ⋅ m1 , V2 = v2 ⋅ m1 , KE =2

1m1v12+2

1m2v22 =2

1 V12 +21 V22

11

33

77

1177

1199

11331155

55

991111

2211

v1

v2

slope1/1

slope-1/1

[11]-tangentslope

-m1/m2= -49

22

44

54Friday, August 28, 2015

Ellipse rescaling geometry and reflection symmetry analysis Convert to rescaled velocity: symmetrize: V1 = v1 ⋅ m1 , V2 = v2 ⋅ m1 , KE =2

1m1v12+2

1m2v22 =2

1 V12 +21 V22

v1FIN1

v2FIN1

⎛

⎝

⎜⎜

⎞

⎠

⎟⎟= 1M

m1 −m2 2m2

2m1 m2 −m1

⎛

⎝⎜⎜

⎞

⎠⎟⎟

v1v2

⎛

⎝⎜⎜

⎞

⎠⎟⎟

becomes: V1FIN1 / m1

V2FIN1 / m2

⎛

⎝

⎜⎜

⎞

⎠

⎟⎟ =

1M

m1 −m2 2m2

2m1 m2 −m1

⎛

⎝⎜⎜

⎞

⎠⎟⎟

V1 / m1

V2 / m2

⎛

⎝⎜⎜

⎞

⎠⎟⎟

11

33

77

1177

1199

11331155

55

991111

2211

v1

v2

slope1/1

slope-1/1

[11]-tangentslope

-m1/m2= -49

22

44

55Friday, August 28, 2015

Ellipse rescaling geometry and reflection symmetry analysis Convert to rescaled velocity: symmetrize: V1 = v1 ⋅ m1 , V2 = v2 ⋅ m1 , KE =2

1m1v12+2

1m2v22 =2

1 V12 +21 V22

v1FIN1

v2FIN1

⎛

⎝

⎜⎜

⎞

⎠

⎟⎟= 1M

m1 −m2 2m2

2m1 m2 −m1

⎛

⎝⎜⎜

⎞

⎠⎟⎟

v1v2

⎛

⎝⎜⎜

⎞

⎠⎟⎟

becomes: V1FIN1 / m1

V2FIN1 / m2

⎛

⎝

⎜⎜

⎞

⎠

⎟⎟ =

1M

m1 −m2 2m2

2m1 m2 −m1

⎛

⎝⎜⎜

⎞

⎠⎟⎟

V1 / m1

V2 / m2

⎛

⎝⎜⎜

⎞

⎠⎟⎟

or: V1FIN1

V2FIN1

⎛

⎝

⎜⎜

⎞

⎠

⎟⎟= 1M

m1 −m2 2 m1m2

2 m1m2 m2 −m1

⎛

⎝⎜⎜

⎞

⎠⎟⎟

V1

V2

⎛

⎝⎜⎜

⎞

⎠⎟⎟=M i

V , or:

V1FIN2

V2FIN2

⎛

⎝

⎜⎜

⎞

⎠

⎟⎟= 1M

m1 −m2 2 m1m2

−2 m1m2 m1 −m2

⎛

⎝⎜⎜

⎞

⎠⎟⎟

V1

V2

⎛

⎝⎜⎜

⎞

⎠⎟⎟= C iM i

V

11

33

77

1177

1199

11331155

55

991111

2211

v1

v2

slope1/1

slope-1/1

[11]-tangentslope

-m1/m2= -49

22

44

56Friday, August 28, 2015

Ellipse rescaling geometry and reflection symmetry analysis Convert to rescaled velocity: symmetrize: V1 = v1 ⋅ m1 , V2 = v2 ⋅ m1 , KE =2

1m1v12+2

1m2v22 =2

1 V12 +21 V22

v1FIN1

v2FIN1

⎛

⎝

⎜⎜

⎞

⎠

⎟⎟= 1M

m1 −m2 2m2

2m1 m2 −m1

⎛

⎝⎜⎜

⎞

⎠⎟⎟

v1v2

⎛

⎝⎜⎜

⎞

⎠⎟⎟

becomes: V1FIN1 / m1

V2FIN1 / m2

⎛

⎝

⎜⎜

⎞

⎠

⎟⎟ =

1M

m1 −m2 2m2

2m1 m2 −m1

⎛

⎝⎜⎜

⎞

⎠⎟⎟

V1 / m1

V2 / m2

⎛

⎝⎜⎜

⎞

⎠⎟⎟

or: V1FIN1

V2FIN1

⎛

⎝

⎜⎜

⎞

⎠

⎟⎟= 1M

m1 −m2 2 m1m2

2 m1m2 m2 −m1

⎛

⎝⎜⎜

⎞

⎠⎟⎟

V1

V2

⎛

⎝⎜⎜

⎞

⎠⎟⎟=M i

V , or:

V1FIN2

V2FIN2

⎛

⎝

⎜⎜

⎞

⎠

⎟⎟= 1M

m1 −m2 2 m1m2

−2 m1m2 m1 −m2

⎛

⎝⎜⎜

⎞

⎠⎟⎟

V1

V2

⎛

⎝⎜⎜

⎞

⎠⎟⎟= C iM i

V

Then collisions become reflections and double-collisions become rotations

where:

cosθ sinθsinθ −cosθ

⎛

⎝⎜⎞

⎠⎟cosθ sinθ−sinθ cosθ

⎛

⎝⎜⎞

⎠⎟

cosθ ≡ m1 −m2

m1 +m2

⎛⎝⎜

⎞⎠⎟

and: sinθ ≡2 m1m2

m1 +m2

⎛

⎝⎜⎞

⎠⎟ with: m1 −m2

m1 +m2

⎛⎝⎜

⎞⎠⎟

2

+2 m1m2

m1 +m2

⎛

⎝⎜⎞

⎠⎟

2

= 1

11

33

77

1177

1199

11331155

55

991111

2211

v1

v2

slope1/1

slope-1/1

[11]-tangentslope

-m1/m2= -49

22

44

57Friday, August 28, 2015

Ellipse rescaling geometry and reflection symmetry analysis Convert to rescaled velocity: symmetrize: V1 = v1 ⋅ m1 , V2 = v2 ⋅ m1 , KE =2

1m1v12+2

1m2v22 =2

1 V12 +21 V22

v1FIN1

v2FIN1

⎛

⎝

⎜⎜

⎞

⎠

⎟⎟= 1M

m1 −m2 2m2

2m1 m2 −m1

⎛

⎝⎜⎜

⎞

⎠⎟⎟

v1v2

⎛

⎝⎜⎜

⎞

⎠⎟⎟

becomes: V1FIN1 / m1

V2FIN1 / m2

⎛

⎝

⎜⎜

⎞

⎠

⎟⎟ =

1M

m1 −m2 2m2

2m1 m2 −m1

⎛

⎝⎜⎜

⎞

⎠⎟⎟

V1 / m1

V2 / m2

⎛

⎝⎜⎜

⎞

⎠⎟⎟

or: V1FIN1

V2FIN1

⎛

⎝

⎜⎜

⎞

⎠

⎟⎟= 1M

m1 −m2 2 m1m2

2 m1m2 m2 −m1

⎛

⎝⎜⎜

⎞

⎠⎟⎟

V1

V2

⎛

⎝⎜⎜

⎞

⎠⎟⎟=M i

V , or:

V1FIN2

V2FIN2

⎛

⎝

⎜⎜

⎞

⎠

⎟⎟= 1M

m1 −m2 2 m1m2

−2 m1m2 m1 −m2

⎛

⎝⎜⎜

⎞

⎠⎟⎟

V1

V2

⎛

⎝⎜⎜

⎞

⎠⎟⎟= C iM i

V

Then collisions become reflections and double-collisions become rotations

where:

cosθ sinθsinθ −cosθ

⎛

⎝⎜⎞

⎠⎟cosθ sinθ−sinθ cosθ

⎛

⎝⎜⎞

⎠⎟

cosθ ≡ m1 −m2

m1 +m2

⎛⎝⎜

⎞⎠⎟

and: sinθ ≡2 m1m2

m1 +m2

⎛

⎝⎜⎞

⎠⎟ with: m1 −m2

m1 +m2

⎛⎝⎜

⎞⎠⎟

2

+2 m1m2

m1 +m2

⎛

⎝⎜⎞

⎠⎟

2

= 1

Fig. 5.2a-c(revised)

θ=16.26°1

m1 - m2m1+m2 2√m1 m2

m1+m2

4850=

1450=

slope :

−m2

m1

= −17

slope :

+m2

m1

= +17

11

33

77

1177

1199

11331155

55

991111

2211

θ=16.26°V1

V2

[11]-tangent slope-√m1/√m2= -7

22

44

11

33

77

1177

1199

11331155

55

991111

2211

v1

v2

slope1/1

slope-1/1

[11]-tangentslope

-m1/m2= -49

22

44

58Friday, August 28, 2015

Ellipse rescaling geometry and reflection symmetry analysis Convert to rescaled velocity: symmetrize: V1 = v1 ⋅ m1 , V2 = v2 ⋅ m1 , KE =2

1m1v12+2

1m2v22 =2

1 V12 +21 V22

v1FIN1

v2FIN1

⎛

⎝

⎜⎜

⎞

⎠

⎟⎟= 1M

m1 −m2 2m2

2m1 m2 −m1

⎛

⎝⎜⎜

⎞

⎠⎟⎟

v1v2

⎛

⎝⎜⎜

⎞

⎠⎟⎟

becomes: V1FIN1 / m1

V2FIN1 / m2

⎛

⎝

⎜⎜

⎞

⎠

⎟⎟ =

1M

m1 −m2 2m2

2m1 m2 −m1

⎛

⎝⎜⎜

⎞

⎠⎟⎟

V1 / m1

V2 / m2

⎛

⎝⎜⎜

⎞

⎠⎟⎟

or: V1FIN1

V2FIN1

⎛

⎝

⎜⎜

⎞

⎠

⎟⎟= 1M

m1 −m2 2 m1m2

2 m1m2 m2 −m1

⎛

⎝⎜⎜

⎞

⎠⎟⎟

V1

V2

⎛

⎝⎜⎜

⎞

⎠⎟⎟=M i

V , or:

V1FIN2

V2FIN2

⎛

⎝

⎜⎜

⎞

⎠

⎟⎟= 1M

m1 −m2 2 m1m2

−2 m1m2 m1 −m2

⎛

⎝⎜⎜

⎞

⎠⎟⎟

V1

V2

⎛

⎝⎜⎜

⎞

⎠⎟⎟= C iM i

V

Then collisions become reflections and double-collisions become rotations

where:

cosθ sinθsinθ −cosθ

⎛

⎝⎜⎞

⎠⎟cosθ sinθ−sinθ cosθ

⎛

⎝⎜⎞

⎠⎟

cosθ ≡ m1 −m2

m1 +m2

⎛⎝⎜

⎞⎠⎟

and: sinθ ≡2 m1m2

m1 +m2

⎛

⎝⎜⎞

⎠⎟ with: m1 −m2

m1 +m2

⎛⎝⎜

⎞⎠⎟

2

+2 m1m2

m1 +m2

⎛

⎝⎜⎞

⎠⎟

2

= 1

Fig. 5.2a-c(revised)

θ=16.26°1

m1 - m2m1+m2 2√m1 m2

m1+m2

4850=

1450=

slope :

−m2

m1

= −17

slope :

+m2

m1

= +17

11

33

77

1177

1199

11331155

55

991111

2211

θ=16.26°V1

V2

[11]-tangent slope-√m1/√m2= -7

22

44

11

33

77

1177

1199

11331155

55

991111

2211

v1

v2

slope1/1

slope-1/1

[11]-tangentslope

-m1/m2= -49

22

44

Note: If m1·m2 is perfect-square, then θ-triangle is rational (32+42=52, etc.)59Friday, August 28, 2015

Ellipse rescaling geometry and reflection symmetry analysis Convert to rescaled velocity: symmetrize: V1 = v1 ⋅ m1 , V2 = v2 ⋅ m1 , KE =2

1m1v12+2

1m2v22 =2

1 V12 +21 V22Then collisions become reflections and double-collisions become rotations

where:

cosθ sinθsinθ −cosθ

⎛

⎝⎜⎞

⎠⎟

cosθ ≡ m1 − m2

m1 + m2

⎛⎝⎜

⎞⎠⎟

and: sinθ ≡2 m1m2

m1 + m2

⎛

⎝⎜

⎞

⎠⎟ with: m1 − m2

m1 + m2

⎛⎝⎜

⎞⎠⎟

2

+2 m1m2

m1 + m2

⎛

⎝⎜

⎞

⎠⎟

2

= 1

Fig. 5.2a-c(revised)

θ=16.26°1

m1 - m2m1+m2 2√m1 m2

m1+m2

4850=

1450=

slope :

−m2

m1

= −17

slope :

+m2

m1

= +17

11

33

77

1177

1199

11331155

55

991111

2211

θ=16.26°V1

V2

[11]-tangent slope-√m1/√m2= -7

22

44

11

33

77

1177

1199

11331155

55

991111

2211

v1

v2

slope1/1

slope-1/1

[11]-tangentslope

-m1/m2= -49

22

44

60Friday, August 28, 2015

Fig. 5.2a-c(revised)

θ=16.26°1

m1 - m2m1+m2 2√m1 m2

m1+m2

4850=

1450=

slope :

−m2

m1

=−17

slope :

+m2

m1

=+17

11

33

77

1177

1199

11331155

55

991111

2211

v1

v2

slope1/1

slope-1/1

[11]-tangentslope

-m1/m2= -49

22

44

11

33

77

1177

1199

11331155

55

991111

2211

θ=16.26°V1

V2

[11]-tangent slope-√m1/√m2= -7

22

44

61Friday, August 28, 2015

Ellipse rescaling-geometry and reflection-symmetry analysisRescaling KE ellipse to circle

How this relates to Lagrangian, l’Etrangian, and Hamiltonian mechanics in Ch. 12

62Friday, August 28, 2015

What ellipse rescaling leads to...(in Ch. 9-12)

(a) Lagrangian L = L(v1,v2)

v1

v2(b) Estrangian E = E(V1,V2)

V1=√m1v1

(c) Hamiltonian H = H(p1,p2)

p1=m1v1

p2=m2v2

V2=√m2v2

COM Bisectorslope = 1/1

Collision line andCOM tangent slope= -m1/m2 =-16

Collision line andCOM tangent slope=-√m1/√m2=-4

COM Bisector slope= √m2/√m1 =1/4

Collision line andCOM tangent slope

= -1/1

COM Bisector slope= m2/m1 =1/16

slope√m1√m2

=4

slope=1

Fig. 12.1

How this relates to Lagrangian, l’Etrangian, and Hamiltonian mechanics in Ch. 12

velocity v1 rescaled to momentum: p1 = m1v1

velocity v2 rescaled to momentum: p2 = m2v2

63Friday, August 28, 2015

What ellipse rescaling leads to...(in Ch. 9-12)

(a) Lagrangian L = L(v1,v2)

v1

v2(b) Estrangian E = E(V1,V2)

V1=√m1v1

(c) Hamiltonian H = H(p1,p2)

p1=m1v1

p2=m2v2

V2=√m2v2

COM Bisectorslope = 1/1

Collision line andCOM tangent slope= -m1/m2 =-16

Collision line andCOM tangent slope=-√m1/√m2=-4

COM Bisector slope= √m2/√m1 =1/4

Collision line andCOM tangent slope

= -1/1

COM Bisector slope= m2/m1 =1/16

slope√m1√m2

=4

slope=1

Fig. 12.1

How this relates to Lagrangian, l’Etrangian, and Hamiltonian mechanics in Ch. 12

velocity v1 rescaled to momentum: p1 = m1v1

velocity v2 rescaled to momentum: p2 = m2v2

Lagrangian L(v1,v2 ) = KE =12m1v1

2 +12m2v2

2

rescaled to

Hamiltonian H (p1, p2 ) = KE =p1

2

2m1

+p2

2

2m2

64Friday, August 28, 2015

What ellipse rescaling leads to...(in Ch. 9-12)

(a) Lagrangian L = L(v1,v2)

v1

v2(b) Estrangian E = E(V1,V2)

V1=√m1v1

(c) Hamiltonian H = H(p1,p2)

p1=m1v1

p2=m2v2

V2=√m2v2

COM Bisectorslope = 1/1

Collision line andCOM tangent slope= -m1/m2 =-16

Collision line andCOM tangent slope=-√m1/√m2=-4

COM Bisector slope= √m2/√m1 =1/4

Collision line andCOM tangent slope

= -1/1

COM Bisector slope= m2/m1 =1/16

slope√m1√m2

=4

slope=1

Fig. 12.1

(Now we call this one “l’Etrangian”)

How this relates to Lagrangian, l’Etrangian, and Hamiltonian mechanics in Ch. 12

65Friday, August 28, 2015

Ellipse rescaling-geometry and reflection-symmetry analysisRescaling KE ellipse to circle

How this relates to Lagrangian, l’Etrangian, and Hamiltonian mechanics in Ch. 12(Preview of Lecture 3)

Reflections in the clothing store: “It’s all done with mirrors!” Introducing hexagonal symmetry D6~C6v (Resulting for m1/m2=3)

Group multiplication and product tableClassical collision paths with D6~C6v (Resulting from m1/m2=3)

66Friday, August 28, 2015

Reflections in clothing store mirrors

Fig. 5.4a-b

67Friday, August 28, 2015

σφacts1st

σφacts2nd

(a)Reflections σA=( ),

x=σA·x=( )10

y=( )01

-y=( )=σA·y

-10

1 00 -1 -σA=( )-1 0

0 1

y=( ) = -σA·y01

10

-x=( )= -σA·x

-10

x=( )

(c) σφ reflection ( )of x-vector: ...of y-vector:

σφ·x=( )ycosφsinφ

cosφ sinφsinφ -cosφ

xφ/2

φ/2

σφ·y=( )sinφ-cosφ

y

xφ/2

φ/2φ

φ -cosφsinφ

cosφ

sinφ

(b)Reflections σB=( ),

x=( )=σB·y

10

y=( ) =σB·x01

0 11 0 -σB=( )

-y=( )= -σB·x

0-1

y=( )01

0 -1-1 0

-x=( )= -σB·y

-10

Mirror plane(edge-on)

(d)Rotation:R+φ=σφσA=( ) (e)Rotation:R−φ=σAσφ=( )cosφ -sinφsinφ cosφ

cosφ sinφ- sinφ cosφy

xφ/2

φ/2

y

xφ/2

φ/2

σA·y

σA·xσφ·y

σφ·x

φ/2

φ/2

σA acts1st

σA acts2nd

+φ y

x

−φ

−φ

Symmetry: It’s all done with mirrors!

Fig. 5.3a-e

68Friday, August 28, 2015

σφacts1st

σφacts2nd

(a)Reflections σA=( ),

x=σA·x=( )10

y=( )01

-y=( )=σA·y

-10

1 00 -1 -σA=( )-1 0

0 1

y=( ) = -σA·y01

10

-x=( )= -σA·x

-10

x=( )

(c) σφ reflection ( )of x-vector: ...of y-vector:

σφ·x=( )ycosφsinφ

cosφ sinφsinφ -cosφ

xφ/2

φ/2

σφ·y=( )sinφ-cosφ

y

xφ/2

φ/2φ

φ -cosφsinφ

cosφ

sinφ

(b)Reflections σB=( ),

x=( )=σB·y

10

y=( ) =σB·x01

0 11 0 -σB=( )

-y=( )= -σB·x

0-1

y=( )01

0 -1-1 0

-x=( )= -σB·y

-10

Mirror plane(edge-on)

(d)Rotation:R+φ=σφσA=( ) (e)Rotation:R−φ=σAσφ=( )cosφ -sinφsinφ cosφ

cosφ sinφ- sinφ cosφy

xφ/2

φ/2

y

xφ/2

φ/2

σA·y

σA·xσφ·y

σφ·x

φ/2

φ/2

σA acts1st

σA acts2nd

+φ y

x

−φ

−φ

Symmetry: It’s all done with mirrors!

Fig. 5.3a-e

69Friday, August 28, 2015

σφacts1st

σφacts2nd

(a)Reflections σA=( ),

x=σA·x=( )10

y=( )01

-y=( )=σA·y

-10

1 00 -1 -σA=( )-1 0

0 1

y=( ) = -σA·y01

10

-x=( )= -σA·x

-10

x=( )

(c) σφ reflection ( )of x-vector: ...of y-vector:

σφ·x=( )ycosφsinφ

cosφ sinφsinφ -cosφ

xφ/2

φ/2

σφ·y=( )sinφ-cosφ

y

xφ/2

φ/2φ

φ -cosφsinφ

cosφ

sinφ

(b)Reflections σB=( ),

x=( )=σB·y

10

y=( ) =σB·x01

0 11 0 -σB=( )

-y=( )= -σB·x

0-1

y=( )01

0 -1-1 0

-x=( )= -σB·y

-10

Mirror plane(edge-on)

(d)Rotation:R+φ=σφσA=( ) (e)Rotation:R−φ=σAσφ=( )cosφ -sinφsinφ cosφ

cosφ sinφ- sinφ cosφy

xφ/2

φ/2

y

xφ/2

φ/2

σA·y

σA·xσφ·y

σφ·x

φ/2

φ/2

σA acts1st

σA acts2nd

+φ y

x

−φ

−φ

Symmetry: It’s all done with mirrors!

Fig. 5.3a-e

70Friday, August 28, 2015