Physics 2D Lecture Slides Lecture 6: Jan 13th 2004 Vivek Sharma - - PDF document

Quiz 1 is This Friday Bring Blue Book, check calculator battery

Physics 2D Lecture Slides Lecture 6: Jan 13th 2004

Vivek Sharma UCSD Physics

Lorentz Transformation Between Ref Frames

2

' ' ' ' ( ) y y z z v t t x t c x v x γ γ = ⎛ ⎞ = − ⎜ ⎟ ⎝ = − ⎠ =

Lorentz Transformation

2

' ' ' ') ' ' ( v t x x vt t c y z x y z γ γ = + ⎛ ⎞ = + ⎜ ⎟ ⎝ = ⎠ =

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' x' ' 2 2

In S' frame, u , u , divide by dt u 1 For v << c, u (Galilean Trans. Resto ' ( red) ( ) )

x x x

x x dx t t dt dx vdt v v dt dt dx c dt dx c v v c dx dx v u u u v dt γ γ − = = − − = − − = − = − = − = −

S S’ v

u

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

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 dt dt dx c v c γ γ γ = = = ⊥ − − = = −

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 γ = −

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 γ γ = + = + + = +

As usual, replace

v ⇒ - v

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 Transform

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 ?

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 t a k e

• f

f f r

• m

S D I arrive in SF ' ' 2 1 2

' Reversing sequence of even ' ts v x t t t c t t γ ⎡ ∆ ⎤ ⎛ ⎞ ∆ = − = ∆ −⎜ ⎟ ⎢ ⎥ ⎝ ⎠ ⎣ ⎦ ⇒ ∆ <

I Cant ‘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

' 2 2 2 ' 2 1 2

' ' For what value of v v can : Not al lowe u 1 < ' d c v x t t c c v c u v x v c t c v x t t t t c t γ ⎡ ∆ ⎤ ⎛ ⎞ ∆ = − = ∆ −⎜ ⎟ ⎢ ⎥ ⎝ ⎠ ⎣ ⎦ ∆ < ∆ ∆ ∆ = ∆ ⇒ < ⇒ > ⇒ < ⇒ ∆ >

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

' ' ' 1 2 ' ' 1 2 1 2 2 1 1 2 2 2 2 2 2 '

' ' before after

2 0, , ' 2 1 1 1 1 1 , 2 2 '

p p

after before

mv p mv m v v v v v V v v v V v v v v v V v v c v v c c c c p mV mv − − − − = = = = = − = + = − − − − + = ≠ + = = −

Watching an Inelastic Collision between two putty balls

Definition (without proof) of Relativistic Momentum

2

1 ( / ) mu p mu u c γ = = −

• With the new definition relativistic

momentum is conserved in all frames

• f 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 γ = −

Nature of Relativistic Momentum

2

1 ( / ) mu p mu u c γ = = −

• With the new definition of

Relativistic momentum Momentum is conserved in all frames of references

m

u

Relativistic Force & Acceleration Relativistic Force And Acceleration

2

1 ( / ) mu p mu u c γ = = −

• (

) ( ) ( )

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

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

PEP PEP-

• II accelerator schematic and tunnel view

II accelerator schematic and tunnel view

Linear Particle Accelerator : 50 GigaVolts Accelating Potential

3/2 2

eE a= 1 ( / ) m u c ⎡ ⎤ − ⎣ ⎦