Physics 2D Lecture Slides Jan 14 Vivek Sharma UCSD Physics - - PowerPoint PPT Presentation

Physics 2D Lecture Slides Jan 14

Vivek Sharma UCSD Physics

“Fixing” (upgrading) Newtonian mechanics

• To confirm with fast velocities
• Re-examine

– Spacetime : X,Y,Z,t – Velocity : Vx , Vy , Vz – Momentum : Px , Py , Pz – (Proper) Rest Mass – Acceleration: ax , ay , az – Force – Work Done & Energy Change – Kinetic Energy – Mass IS energy

• Nuclear Fission
• Nuclear Fusion
• Making baby universes Learning about the first three minutes

since the beginning of the universe

Lorentz Transformation Between Ref Frames

2

' ' ' ' ( ) y y z z v t t x t c x v x γ γ =   = −    = −  =

Lorentz Transformation

2

' ' ' ) ' ' ( y y z v t x x v t c t x z γ γ   = +    = = +  =

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

Notice : SPACE and TIME Coordinates mixed up !!!

Lorentz Transform for Pair of Events

Can understand Simultaneity, Length contraction & Time dilation formulae from this Time dilation: Bulb in S frame turned on at t1 & off at t2 : What ∆t’ did S’ measure ? two events occur at same place in S frame => ∆x = 0

∆t’ = γ ∆t (∆t = proper time)

S

x

S’

X’

Length Contraction: Ruler measured in S between x1 & x2 : What ∆x’ did S’ measure ? two ends measured at same time in S’ frame => ∆t’ = 0

∆x = γ (∆x’ + 0 ) => ∆x’ = ∆x / γ

(∆x = proper length)

x1 x2 ruler

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 γ γ − = = − − = − − = = = = − − − −

S S’ v

u

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

Does Lorentz Transform “work” ?

Two rockets 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? Place an imaginary S’ frame on Rocket A ⇒ v = 0.75c relative to Earth Observer S Consistent with Special Theory of Relativity y x

S

x’

S’

y’ 0.7c

• 0.85c

A B O O’

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 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

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 ?

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

' For what value of v can ' v x t t t t c 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 = −