Physics 2D Lecture SlidesJan 15

Vivek SharmaUCSD Physics

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 2Before

v1’=0

v2’21

After V’

S’

S

1 2

Before v v 21

After V=0

P = mv –mv = 0 P = 0

' ' '1 2

' '1 21 2 2

1 122 2

2

2

2

'

' 'before after

20, , '

2

11 1

1

1

, 2 2

'

p p

afterbeforemvp mv m

v v v v v V vv v V vv v v v V

vvc

vvcc c c

p mV mv

− − − −= = = = =

−= +

= −−− − +

=

≠

+= = −

Wat

chin

g an

Inel

astic

Col

lisio

n be

twee

n tw

o pu

tty b

alls

Definition (without proof) of Relativistic Momentum

21 ( / )mup muu c

γ= =−

With the new definition relativistic momentum is conserved in all frames of references : Do the exercise

New ConceptsRest mass = mass of object measuredIn a frame of ref. where object is at rest

2

is velocity of the objectNOT of a referen

11 ( / )

!ce frame u

u cγ =

−

Nature of Relativistic Momentum

21 ( / )mup muu c

γ= =−

With the new definition of Relativistic momentum

Momentum is conserved in all frames of references

mu

Good old Newton

Relativistic Force & Acceleration

Relativistic Force And

Acceleration

21 ( / )mup muu c

γ= =−

( )

( )

( )

3 / 2 22 2

2 2 2

3 / 2

2

3 / 2

2

2

2

1 ( / )

: Relativistic For

1 2

ce

( )( )

1 ( / )

Since A

21 ( / ) 1 (

ccel

/ )

1 (

e

)

a

/

r

d du dusedt dt du

m mu u duFc dtu c u c

mc mu mu duF

dudt

dp d muFdt dt u c

mFu c

dtc u c

=

− − = + × − −

− + = −

= = −

= −

3 / 22

tion a =

Note: As / 1, a 0 !!!!Its harder to accelerate when

,

Fa =

you get closer to speed of light

1 ( / )m

d

c

ut

u c

u

d

⇒

→

−

→

Reason why you cant quite get up to the speed of light no matter how hard you try!

A Linear Particle Accelerator

V

+- F

E

E= V/dF=eE

3/ 2 3/ 22 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 =

orce

larger

1

the potential difference V a

du F u qE udt m c m c

= = − −

cross

plates, larger the force on particle

d

q

Under force, work is done on the particle, it gains

Kinetic energy

New Unit of Energy

1 eV = 1.6x 10-19 Joules

PEPPEP--II accelerator schematic and tunnel viewII accelerator schematic and tunnel view

A Linear Particle Accelerator

3/ 22eEa= 1 ( / )m

u c −

Magnetic Confinement & Circular Particle Accelerator V

2

2

ClassicallyvF mrvqvB mr

=

=BF

B

r

2

2

( )

(Centripetal accelaration)

dp d mu duF m quBdt dt dt

du udt rum quB mu qBr p qBr

r

γ γ

γ γ

= = = =

=

⇒ == ⇒ =

Charged Form of Matter & Anti-Matter in a B Field

Circular Particle Accelerator: LEP @ CERN, Geneve

Magnets Keep Circular Orbit of Particles

Inside A Circular Particle Accelerator @ CERN

Accelerating Electrons Thru RF Cavities

Test of Relativistic Momentum In Circular Accelerator

2

1 ( / )mup muu c

qBrm

mu qB

u

rγ

γ

γ

=

=

=

=−

Relativistic Work Done & Change in Energy

x1 , u=0

X2 , u=u

2 2

1 1

3 / 22 2

2 2

3 / 220

2

22

3 / 2 1 / 22 20

2 2

2 2

substitute i

. .

, , 1 1

n W

(change in var x u 1

1 1

W rk d

)

o

x x

x x

u

u

dpW F dx dxdt

dummu dp dtpdtu u

c cdumdtWuc

mudu m

udt

mccW

c

c

c

mmcu u

γ

= =

= ∴ = − −

∴ =

−

= = − =

− −

→

−

∫ ∫

∫

∫

2

2

2

2

one is change in Kinetic energy KK = or Total Ener E= gy

mc mcKmc mcγ

γ −

= +

Why Can’s Anything go faster than light ?

( )

2

2 222 21/ 2 1/ 22 2

2 2

2 22 4 2

22 2

2

2

(Parabolic in Vs )

1 2Non-relativistic case: K =

1 ( 1)

1 1

2

1

uK Ku c

mc mcK mc K mcu uc c

u m c K mcc

mc mc

Kmu um

−

−

= − ⇒ + = − −

⇒ − = +

⇒

−

+

=

=

⇒

Lets accelerate a particle from rest, particle gains velocity & kinetic energy

Relativistic Kinetic Energy

When Electron Goes Fast it Gets “Fat”

2E mcγ=vAs 1, c

Apparent Mass approaches

γ→ → ∞

∞

Relativistic Kinetic Energy & Newtonian Physics2

12 22

2

2

2

22

2

22

Relativistic KE =

1When , 1- 1 ...smaller terms2

1so [1 ] (classical form recovered)1

22

u uu cc c

uK mc

mc

mcc

mc

mu

γ−

−

<< ≅ − +

≅ − − =

2 2

2

For a particle

Total Energy of a Pa

at rest, u = 0

Total Energy E=

r

m

ticle

c

E mc KE mcγ= = +

⇒

Relationship between P and E2 2 2 4

2 2 2 2 2 2

2 2 2 2 4 2 2 2 2 2 2 2 2

2 2 2 42 2

2

2 2

2

2 2 2 2

2

22

2

2

2 4

( )

= ( ) ( )

........important relati( on

For

)

1

p

u

E mc

p mu

E p c mc

E m c

p c m u c

E p c m c m u c m c um c m cc u c m

c uu

c

c

γ

γ

γ

γ

γ

γ γ

=

=

⇒ =

⇒ =

⇒ − = − = −

+

−−

=

− =−

=

2 2 2 2 4

EE= pc or p = (light has momentum!)c

Re

articles with zero rest mass like photon (EM

lativistic Invariance :

waves)

: In all Ref Frames

Rest M

E p c m c− =

ass is a "finger print" of the particle

Mass Can “Morph” into Energy & Vice Verca

• Unlike in Newtonian mechanics• In relativistic physics : Mass and Energy are the same

thing• New word/concept : Mass-Energy• It is the mass-energy that is always conserved in every

reaction : Before & After a reaction has happened• Like squeezing a balloon :

– If you squeeze mass, it becomes (kinetic) energy & vice verca !• CONVERSION FACTOR = C2

Mass is Energy, Energy is Mass : Mass-Energy Conservation

be

2

f

2

ore after

2 22

2

2

2

2

2

2

2

2 2 1

Kinetic energy has been transformed

E E

into mass increase

2 2 - 21

1 1

mc mc Mc

K m

u uc

cM M

mM m

m

ucc

c c u

=

+ = ⇒

− −

∆ = = =

= >

−

−

2

2

2

mc

c

−

Examine Kinetic energy Before and After Inelastic Collision: Conserved?

S

1 2

Before v v 21

After V=0

K = mu2 K=0

Mass-Energy Conservation: sum of mass-energy of a system of particles before interaction must equal sum of mass-energy after interaction

Kinetic energy is not lost, its transformed into more mass in final state

Conservation of Mass-Energy: Nuclear Fission

22 22 31 2

1 2 32 2 21 2 32 2 21 1 1

M cM c M cMcu u uc c

M M

c

M M= +

−

> ++

−

+⇒

−

M M1 M2M3+ + Nuclear Fission

< 1 < 1 < 1

Loss of mass shows up as kinetic energy of final state particlesDisintegration energy per fission Q=(M – (M1+M2+M3))c2 =∆Mc2

1423692

10

-28

355

9092+ +3 n

m=0.177537u=2.947

U Cs R

1 10 165.4 MeV

b

kg∆ × =

→

What makes it explosive is 1 mole U = 6.023 x 1023 Nuclei !!

Relativistic Kinematics of Subatomic ParticlesReconstructing Decay of a π Meson