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The Theory of Special Relativity. Ch 26. Two Theories of Relativity. Special Relativity (1905) Inertial Reference frames only Time dilation Length Contraction Momentum and mass (E=mc 2 ) General Relativity Noninertial reference frames (accelerating frames too)

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two theories of relativity
Two Theories of Relativity

Special Relativity (1905)

  • Inertial Reference frames only
  • Time dilation
  • Length Contraction
  • Momentum and mass (E=mc2)

General Relativity

  • Noninertial reference frames (accelerating frames too)
  • Explains gravity and the curvature of space time
classical and modern physics
Classical and Modern Physics

Classical Physics – Larger, slow moving

  • Newtonian Mechanics
  • EM and Waves
  • Thermodynamics

Modern Physics

  • Relativity – Fast moving objects
  • Quantum Mechanics – very small
slide4

10% c

Speed

Atomic/molecular size

Size

correspondence principle
Correspondence Principle
  • Below 10% c, classical mechanics holds (relativistic effects are minimal)
  • Above 10%, relativistic mechanics holds (more general theory)
inertial reference frames
Inertial Reference Frames
  • Reference frames in which the law of inertia holds
  • Constant velocity situations
    • Standing Still
    • Moving at constant velocity (earth is mostly inertial, though it does rotate)
slide7

Basic laws of physics are the same in all inertial reference frames

  • All inertial reference frames are equally valid
speed of light problem
Speed of Light Problem
  • According to Maxwell’s Equations, c did not vary
  • Light has no medium
  • Some postulated “ether” that light moved through
  • No experimental confirmation of ether (Michelson-Morley experiment)
two postulates of special relativity
Two Postulates of Special Relativity

Einstein (1905)

  • The laws of physics are the same in all inertial reference frames
  • Light travels through empty space at c, independent of speed of source or observer

There is no absolute reference frame

of time and space

simultaneity
Simultaneity
  • Time always moves forward
  • Time measured between things can vary

Lightning strikes point A and B at the same time

O will see both at the same time and call them simultaneous

moving observers 1
Moving Observers 1

On train O2 - train O1 moves to the right

On train O1 – train O2 moves to the left

moving observers 2
Moving Observers 2
  • Lightning strikes A and B at same time as both trains are opposite one another
slide18

Train O2 will observe the strikes as simultaneous

  • Train O1 will observe strike B first (not simultaneous

Neither reference frame is “correct.”

Time is NOT absolute

time dilation
Time Dilation
  • Consider light beam reflected and observed on a moving spaceship and from the ground
slide21

Distance is shorter from the ship

  • Distance is longer from the ground
  • c = D/t
  • Since D is longer from the ground, so t must be too.
slide22

On Spaceship:

c = 2D/Dto

Dto = 2D/c

On Earth:

c = 2 D2 + L2

Dt

v = 2L/Dt

L = vDt

2

slide23

c = 2 D2 + v2 (Dt)2/4

Dt

c2 = 4D2 + v2

Dt2

Dt = 2D

c 1 –v2/c2

Dt = Dto

1 - v2/c2

slide24

Dt = Dto

√ 1 - v2/c2

Dto

  • Proper time
  • time interval when the 2 events are at the same point in space
  • In this example, on the spaceship
is this real experimental proof
Is this real? Experimental Proof
  • Jet planes (clocks accurate to nanoseconds)
  • Elementary Particles – muon
    • Lifetime is 2.2 ms at rest
    • Much longer lifetime when travelling at high speeds
time dilation ex 1
Time Dilation: Ex 1

What is the lifetime of a muon travelling at 0.60 c (1.8 X 108 m/s) if its rest lifetime is 2.2 ms?

Dt = Dto

√ 1 - v2/c2

Dt = (2.2 X 10-6 s) = 2.8 X 10-6 s

1- (0.60c)2 1/2

c2

time dilation ex 2
Time Dilation: Ex 2

If our apatosaurus aged 10 years, calculate how many years will have passed for his twin brother if he travels at:

  • ¼ light speed
  • ½ light speed
  • ¾ light speed
time dilation ex 234
Time Dilation: Ex 2
  • 10.3 y
  • 11.5 y
  • 10.5 y
time dilation ex 3
Time Dilation: Ex 3

How long will a 100 year trip (as observed from earth) seem to the astronaut who is travelling at 0.99 c?

Dt = Dto

1 - v2/c2

Dto = Dt 1 - v2/c2

Dto = 4.5 y

time dilation ex 336
Time Dilation: Ex 3

If our apatosaurus aged 10 years, and his brother aged 70 years, calculate the apatosaurus’ average speed for his trip. (Express your answer in terms of c).

ANS: 0.99 c

length contraction
Length Contraction
  • Observers from earth would see a spaceship shorten in the length of travel
slide38

Only shortens in direction of travel

  • The length of an object is measured to be shorter when it is moving relative to an observer than when it is at rest.
slide39

Dto = Dt √ 1 - v2/c2

v = L Dto = L/v (L is from spacecraft)

Dto

Dt = Lo/v

Lo = L

v v √ 1 - v2/c2

L = Lo √ 1 - v2/c2

slide40

L = Lo √ 1 - v2/c2

Lo = Proper Length (at rest)

L = Length in motion (from stationary observer)

length contraction ex 1
Length Contraction: Ex 1

A painting is 1.00 m tall and 1.50 m wide. What are its dimensions inside a spaceship moving at 0.90 c?

length contraction ex 2
Length Contraction: Ex 2

What are its dimensions to a stationary observer?

Still 1.00 m tall

L = Lo √ 1 - v2/c2

L = (1.50 m)(√ 1 - (0.90 c)2/c2)

L = 0.65 m

length contraction ex 3
Length Contraction: Ex 3

The apatosaurus had a length of about 25 m. Calculate the dinosaur’s length if it was running at:

  • ½ lightspeed
  • ¾ lightspeed
  • 95% lightspeed
slide46
21.7 m
  • 15.5 n
  • 7.8 m
four dimensional space time
Four-Dimensional Space-Time

Consider a meal on a train (stationary observer)

  • Meal seems to take longer to observer
  • Meal plate is more narrow to observer
slide48

Move faster – Time is longer but length is shorter

  • Move slower – Time is shorter but length is longer
  • Time is the fourth dimension
momentum and the mass increase
Momentum and the Mass Increase

p = mov

1 - v2/c2

Mass increases with speed

mo = proper (rest) mass

m = mo

1 - v2/c2

mass increase ex 1
Mass Increase: Ex 1

Calculate the mass of an electron moving at 4.00 X 107 m/s in the CRT of a television tube.

m = mo

1 - v2/c2

m = 9.11 X 10-31 kg = 9.19 X 10-31 kg

1 - (4.00 X 107 m/s)2/c2

mass increase ex 2
Mass Increase: Ex 2

Calculate the mass of an electron moving at 0.98 c in an accelerator for cancer therapy.

m = mo

√ 1 - v2/c2

m = 9.11 X 10-31 kg = 4.58 X 10-30 kg (5mo)

√ 1 - (0.98c)2/c2

the speed limit
The Speed Limit
  • Nothing below the speed of light can be accelerated to the speed of light
  • Would require infinite energy
    • Mass becomes infinite
    • Length goes to zero
    • Time becomes infinite
mass increase ex 256
Mass Increase: Ex 2

The apatosaurus had a mass of about 35,000 kg. Calculate the dinosaur’s mass if it was running at:

  • ½ lightspeed
  • ¾ lightspeed
  • 95% lightspeed
slide57
40,415 kg
  • 53,915 kg
  • 112,090 kg
relativistic momentum
Relativistic Momentum

p = mov

1 - v2/c2

e mc 2
E = mc2

Particle at Rest

E = moc2

E = Total Energy

mo = Rest mass

c = speed of light

Moving Particles

E2 = mo2c4 + p2c2

slide60

E = moc2 + KE

rest kinetic KE does not equal ½ mv2 at

energy energy relativistic speeds

e mc 2 ex 1
E=mc2: Ex 1

How much energy would be released if a p0 meson (mo=2.4 X 10-28 kg) decays at rest.

E = mc2

E = moc2 (particle is at rest)

E = (2.4 X 10-28 kg)(3.0 X 108 m/s)2

E = 2.16 X 10-11 J

e mc 2 ex 2
E=mc2: Ex 2

A p0 meson (mo=2.4 X 10-28 kg) travels at 0.80 c.

  • Calculate the new mass [4 X 10-28 kg]
  • Calculate the relativistic momentum [9.6 X 10-20 kg m/s]
  • Calculate the energy of the particle (E2 = mo2c4 + p2c2 ) [ANS: 3.6 X 10-11 J]
e mc 2 ke ex 3
E=mc2 + KE: Ex 3

What is the kinetic energy of the p0 meson in the former example.

E = moc2 + KE

KE = E – moc2

KE = 3.6 X 10-11J - (2.4 X 10-28 kg)(3.0X108 m/s)2

KE = 1.4 X 10-11 J

e mc 2 ke
E=mc2 + KE

An electron is moving at 0.999c in the CERN accelerator.

  • Calculate the rest energy
  • Calculate the relativistic momentum
  • Calculate the relativistic energy
  • Calculate the Kinetic energy
electron volts
Electron Volts

1 eV= 1.6 X 10-19 J

1 MeV = 106 eV = 1.60 X 10-13 J

What is the rest mass of an electron in MeV?

E = mc2

E = moc2 (particle is at rest)

E = (9.11 X 10-31 kg)(3.0 X 108 m/s)2

E = 8.20 X 10-14 J

relativistic addition of velocity
Relativistic Addition of Velocity

Classical Addition

  • Bus moves at 40 mph
  • You walk to the front at 5 mph
  • Overall speed = 45 mph
slide68

Relativistic Addition

  • Cannot simply add velocities above ~0.10 c
  • Length and time are in different reference frames
  • Formula

u = v + u’

1 + vu’/c2

u = overall speed with respect to stationary observer

v = speed of moving object with respect to st. observer

u’ =speed of 2nd object with respect to moving observer

relativistic addition ex 1
Relativistic Addition: Ex 1

What is the speed of the second stage of the rocket shown with respect to the earth?

slide70

u = v + u’

1 + vu’/c2

u = 0.60c + 0.60c

1 + [(0.60c)(0.60c)/c2 ]

u = 0.88 c

(classical addition would give you 1.20c, over the speed of light)

relativistic addition ex 2
Relativistic Addition: Ex 2

Suppose a car travelling at 0.60c turns on its headlights. What is the speed of the light travelling out from the car?

u = v + u’

1 + vu’/c2

u = 0.60c + c = 1.60c

1 + [(0.60c)(c)/c2 ] 1.60

u = c

relativistic addition ex 3
Relativistic Addition: Ex 3

Now the car is travelling at c and turns on its headlights.

u = v + u’

1 + vu’/c2

u = c + c = 2c

1 + [(c)(c)/c2 ] 2

u = c