Physics 2D Lecture Slides Lecture 7 : Jan 12th 2005 Vivek Sharma - - PDF document

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Physics 2D Lecture Slides Lecture 7 : Jan 12th 2005

Vivek Sharma UCSD Physics

First Quiz This Friday !

• Bring a Blue Book, calculator;

check battery

– Make sure you remember the code number for this couse given to you (record it some place safe!)

• No “cheat Sheet” please, I will give

you equations and constants that I think you need

• When you come for the quiz, pl.
• ccupy seats in the front first.
• Pl. observe one seat distance in the

back rows (there is plenty of space)

• Academic Honesty is for you to
• bserve and for me to enforce:

– Be a good citizen, in this course and forever !

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Lorentz Transformation Between Ref Frames

2

' ' ' ' ( ) y y z z v t t x t c x v x

• =
• =
• =
• =

Lorentz Transformation

2

' ' ' ') ' ' ( v t x x vt t c y z x y z

• =

+

• =

+

• =
• =

Inverse Lorentz Transformation As v→0 , Galilean Transformation is recovered, as per requirement

Notice : SPACE and TIME Coordinates mixed up !!!

Lorentz Velocity Transformation Rule

' ' ' 2 1 x' ' ' ' 2 1 x' 2 x' 2 x' 2 '

In S' frame, u , u , u 1 For v << c, u (Gali divide by dt' ' lean Trans. Restor ( ) ( ed) )

x x x

x x dx t t dt dx vdt v dt dx c v dt dt dx dx v dx c u u u d v v t c v

• =

=

• =
• =

= = =

• S

S’ v

u

S and S’ are measuring ant’s speed u along x, y, z axes

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2 ' 2 ' 2

divide by dt on (1 ) There is a change in velocity in the direction to S-S' motion ' , ' ' ( H ) ' S ! R ( )

x y y y

u u dy dy dy v dt dt dx dy c u u v dy dt dx c v c

• =

= =

• =

=

• Velocity Transformation Perpendicular to S-S’ motion

' 2

Similarly Z component of Ant' s velocity transforms (1 ) as

z z x

u u v c u

• =
• Inverse Lorentz Velocity Transformation

' x ' ' ' 2 ' 2 ' 2

Inverse Velocity Transform: (1 u ) 1 1 ( )

y y z x z x x x

u v vu u u v c u v c u c u u

• =

+ = + + = +

As usual, replace

v ⇒ - v

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Does Lorentz Transform “work” For Topgun ?

Consistent with Special Theory of Relativity

Two rockets A &B travel in

• pposite directions

An observer on earth (S) measures speeds = 0.75c And 0.85c for A & B respectively What does A measure as B’s speed? 0.75c

• 0.85c

A B Place an imaginary S’ frame on Rocket A ⇒ v = 0.75c relative to Earth Observer S y’ y x

S

O (Earth guy) x’

S’

O’

Example of Inverse Velocity Transformation

Biker moves with speed = 0.8c past stationary observer Throws a ball forward with speed = 0.7c What does stationary

• bserver see as velocity
• f ball ?

Place S’ frame on biker Biker sees ball speed uX’ =0.7c Speed of ball relative to stationary observer

uX ?

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Velocity Transformation Perpendicular to S-S’ motion

2 bike gang leaders racing at relativistic speeds along perpendicular paths How fast does BETA recede over right shoulder of ALPHA as seen by ALPHA To get BETA’s speed of recession as seen by ALPHA, associate S’ frame to be ALPHA’s make Policeman’s frame to be S (measuring ux and uy) and calculate u’x and u’y

Policeman measures : ALPHA : ux = 0.75c uy = 0 BETA: ux = 0 uy = 0.90c

x y

Velocity Transformation Perpendicular to S-S’ motion

ux

' = ux V

1 uxV c2 = 0 0.75c 1 0.0c 0.75c c2 = 0.75c uy

' =

uy 1 uxV c2

• =

0.90c 1 0.75c c

• 2

1 0 = 0.595c

V S Make Alpha the S’ frame at rest, Policeman is frame S and we know what he (S) measures as the x,y components of BETA’s velocity in his frame.

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Hollywood Yarns Of Time Travel !

Terminator : Can you be seen to be born before your mother? A frame of Ref where sequence of events is REVERSED ?!!

S S’

1 1 ' ' 1 1

( , ) ( , ) x t x t u

2 2 ' ' 2 2

( , ) ( , ) x t x t

I take off from SD I arrive in SF ' ' 2 1 2

' Reversing sequence of even ' ts v x t t t c t t

• =
• =
• <

7 I Can’t be seen to arrive in SF before I take off from SD

S S’

1 1 ' ' 1 1

( , ) ( , ) x t x t

u

2 2 ' ' 2 2

( , ) ( , ) x t x t

t' = t2

' t1 ' = t

v x c2

• For what value of v can t' < 0

t' < 0 t < v x c2 1 < v x c2t = v u c2 v c > c u v > c : Not allowed !! Relativistic Momentum and Revised Newton’s Laws

Need to generalize the laws of Mechanics & Newton to confirm to Lorentz Transform and the Special theory of relativity: Example : p

mu =

• 1

2 Before v1’=0 v2’ 2 1 After V’ S’ S 1 2 Before v v 2 1 After V=0 P = mv –mv = 0 P = 0

v1

' = v1 v

1 v1v c2 = 0, v2

' = v2 v

1 v1v c2 = 2v 1+ v2 c2 , V ' = V v 1 Vv c2 = v pbefore

'

= mv1

' + mv2 ' = 2mv

1+ v2 c2 , pafter

'

= 2mV ' = 2mv

pbefore

'

pafter

'

Need to re-examine definition of relativistic momemtum

Watching an Inelastic Collision between two putty balls

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Definition (without proof) of Relativistic Momentum

2

1 ( / ) mu p mu u c

• =

=

• With the new definition relativistic

momentum is conserved in all frames of references : Do the exercise

New Concepts

Rest mass = mass of object measured In a frame of ref. where object is at rest

2

is velocity of the object NOT of a referen 1 1 ( / ) ! ce frame u u c =

• Nature of Relativistic Momentum

2

1 ( / ) mu p mu u c

• =

=

• With the new definition of

Relativistic momentum Momentum is conserved in all frames of references

m

u

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Relativistic Force & Acceleration Relativistic Force And Acceleration

2

1 ( / ) mu p mu u c

• =

=

• (

) ( ) ( )

3/2 2 2 2 2 2 2 3/2 2 3/2 2 2 2

1 ( / ) : Relativistic For 1 2 ce ( )( ) 1 ( / ) Since A 2 1 ( / ) 1 ( ccel / ) 1 ( e ) a / r d du d use dt dt du m mu u du F c dt u c u c mc mu mu du F du dt dp d mu F dt dt u c m F u c dt c u c =

• =

+

• +
• =
• =

=

• =
• 3/2

2

tion a = Note: As / 1, a 0 !!!! It [rate of change of v s harder to accelerate when you get closer to s elocity , F ] peed of l a = 1 ( / ) m ight du u c dt u c

• Reason why you cant

quite get up to the speed

• f light no matter how

hard you try!

A Linear Particle Accelerator

V

+

• F
• E

E= V/d F= eE

3/2 3/2 2 2 2 2

Charged particle q moves in straight line in a uniform electric field E with speed u accelarates under f F=qE a 1 =

• rce

larger 1 the potential difference V a du F u qE u dt m c m c

• =

=

• cross

plates, larger the force on particle d

q

Under force, work is done

• n the particle, it gains

Kinetic energy New Unit of Energy

1 eV = 1.6x10-19 Joules 1 MeV = 1.6x10-13 Joules 1 GeV = 1.6x10-10 Joules

Parallel Plates

10 Your Television (the CRT type) is a Small Particle Accelerator !

PEP-II accelerator schematic and tunnel view PEP-II accelerator schematic and tunnel view

Linear Particle Accelerator : 50 GigaVolts Accelating Potential 3/ 2 2

eE a= 1 ( / ) m u c

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Hollywood Yarns !