Classical RelativityGalilean Transformations

x’

y’

P

v

x

y

x = x’ + vt x’ = x - vt

Divide equations by t

u = u’ + v u’ = u - v

u’

Example:A train travels through a station at a constant speed of 8.0ms-1 . One observer sits on the train and another sits on the platform. as they pass each other, they start their stopwatches and take measurements of a dog on the train who is running in the same direction the train is moving.

(a) The train observer measures the velocity of the dog to be 2.0ms-1 . What is the velocity relative to the platform observer?

(b) After 5s how far has the dog moved according to the observer on the train?(c) After 5s how far has the dog moved according to the observer on the

platform?

8.0ms-1

50m

50m

Calculate the time for each boat to make a round trip 50 meters out and back along the path shown.

5.0ms-1

5.0ms

-1

3.0 ms-1

50m

50m

Calculate the time for each boat to make a round trip 50 meters out and back along the path shown.

5.0ms-1

5.0ms

-1

Michelson-Morley Experiment An experiment using an interferometer to detect the motion of the Earth through the ether.

When the entire apparatus is rotated a shift in the interference pattern should occur.

Ether Wind

Michelson-Morley Experiment There was no shift in the interference pattern showing the ether did not exist, that light could travel through a vacuum, and its speed is independent of the source motion.

Special Relativity

Inertial Reference Frame

• Reference frames in which Newton’s law are valid.

• Reference frames with constant velocity.• Special Relativity only deals with events in

inertial reference frames.• General relativity deals with non-inertial

reference frames.

Maxwell’s electromagnetic equations

• Unified electricity and magnetism.• Predicted the existence of electromagnetic waves.• Gave the speed of electromagnetic waves as a

constant regardless of reference frame.

s

mc

oo

8100.31

μo = Vacuum permeability

εo = Vacuum permittivity

Einstein Postulates of Special Relativity

• The laws of Physics have the same form in all inertial reference frames.

• Light propagates through empty space with a definite speed c independent of the speed of the source or the observer.

Classical Relativity

1,000,000 ms-1 1,000,000 ms-1

■ How fast is Spaceship A approaching Spaceship B?

■ Both Spaceships see the other approaching at 2,000,000 ms-1.

■ This is Classical Relativity.

Einstein’s Special Relativity

1,000,000 ms-1

0 ms-1

300,000,000 ms-1

n Both spacemen measure the speed of the approaching ray of light.n How fast do they measure the speed of light to be?

Special Relativity

• Stationary man– 300,000,000 ms-1

• Man travelling at 1,000,000 ms-1

– 301,000,000 ms-1?– Wrong!

• The Speed of Light is the same for all observers

Time Travel!

• Time between ‘ticks’ = distance / speed of light

• Light in the moving clock covers more distance…– …but the speed of light is constant…– …so the clock ticks slower!

• Moving clocks run more slowly!V

The Lorentz factor

Time dilation Δ 𝑡=𝛾 Δ𝑡𝑜Δto = Proper time – The time measured by a clock at rest relative to the event.

An observer sets up an experiment to measure the time of oscillation of a mass suspended from a vertical spring. He measures the time period as 2.0s. To another observer this time period is measured as 2.66s. Calculate the relative velocity between the two observers.

Length Contraction L=𝐿𝑜

𝛾The length measured by an observer who is at rest relative to the object.

Mary is traveling in a space ship, which is not accelerating. To her the space ship has a length of 100m. To Paul who is traveling in another space ship, which is also not accelerating. Mary’s space ship has a length of 98m. Calculate the relative velocity of Paul and Mary.

A spaceship is traveling away from the Earth with a speed of 0.6c as measured by an observer on the Earth. The spaceship sends a light pulse back to Earth every 10 minutes as measured by a clock on the space ship.

(a) Calculate the distance that the spaceship travels between light pulses as measured by

i. the observer on Earth.ii. somebody on the space ship.

(b) If the Earth observer measures the length of the spaceship as 60m, determine the proper length of the spaceship.

Muon Decay Half life of 3.1E-6s as measured in a reference frame at which they are at rest. Muons are created in the upper atmosphere (10km) of the Earth from cosmic ray bombardment. These muons have very high velocities (0.98c)

Special Relativity

v = 0.7c ux = -0.8c

■ How fast is Spaceship A approaching Spaceship B?

𝑢𝑥′ =

𝑢𝑥−𝑣

1−𝑢𝑥𝑣

𝑐2

impossible

𝑢𝑥′ =

− .8𝑐− .7𝑐

1−(− .8 𝑐 ) ( .7𝑐 )

𝑐2

𝑢𝑥′ =0.96 𝑐

Relativistic Mass, Energy, and Momentum

Relativistic MassThe mass of a moving object is greater than than the rest mass.

m mom = relativistic mass - the mass measured moving relative to the object.

mo = rest mass – the mass measured at rest relative to the object

= the Lorentz factor

Classical EnergyIn classical physics if a constant force is applied to an object it experiences a constant acceleration. The work done by the force is transferred to kinetic energy.

EK 1

2mov

2

v

t

c

Relativistic EnergyAccording to special relativity no object can exceed the speed of light (c). So the acceleration of the object must decrease, but where does the work done by the force transfer to…….MASS

E K mc2 moc2

EK m mo c 2

EK mo mo c 2

EK 1 moc 2

t

v

c

EK = total energy – rest energy

Let’s look at the relativistic kinetic energy equation at low velocities.

EK 1 moc 2

EK moc

2

1 v 2

c 2

moc2

Binomial expansion when x is small

1 x n 1 nx

1v 2

c 2

1

2

1v 2

2c 2

EK 1v 2

2c 2

moc

2 moc2

EK moc2 moc

2v 2

2c 2 moc2

EK 1

2mov

2

Example: An electron is accelerated through a potential difference of 1.0MV. Calculate its velocity

Classical Calculation:

Eo W E f

UE EK

qV 1

2mv2

v 2eV

m

2 1.6E 19C 1.0E6J

C

9.11E 315.9E8

m

s

IMPOSSIBLE

Example: An electron is accelerated through a potential difference of 1.0MV. Calculate its velocity

Relativistic Calculation

Eo W E f

UE EKqV 1 moc 2

(1.6E 19C)(1.0E6J

C) ( 1)(9.11E 31kg)(3.0E8

m

s)2

1.6E 13J ( 1)(8.2E 14J)

1.95 ( 1)

2.95

1

1 v 2

c 2

2.95 1

1 v 2

c 2

.34 1 v 2

c 2

.12 1 v 2

c 2

v 2

c 2 .88

v 2 .88c 2

v .94c

Lets work the same example using different units and working in terms of c

UE EKqV 1 moc 2

e(1.0MV ) 1 .511MeVc 2 c 2

1.0MeV 1 (.511MeV )

( 1) 1.96

2.96

1

1 v 2

c 2

1

2 1 v 2

c 2

v 2

c 2 1 1

2

v c 1 1

2 c 1 1

2.962 .94c

Example: Calculate the pd necessary to accelerate an electron to a velocity of 0.8c

The classical relation between energy and momentum

EK 1

2mv2 p mv

EK p2

2m

EK mv 2

2mm2v 2

2m

1

2mv2

The relativistic relation between energy and momentum:

m mo

m 1

1 v 2

c 2

mo

m2 mo

2

1 v 2

c 2

m2 mv2

c 2 mo2

m2c 2 mv2 moc2

m2c 2 m2v 2 moc2

m2c 2 m2v 2 moc2

c 2 m2c 2 m2v 2 moc2

m2c 4 (m2v 2)c 2 moc4

E 2 p2c 2 moc4

p mv

p2 m2v 2

E mc 2

E 2 m2c 4

Example: What is the momentum of an electron with a kinetic energy of 1.0MeV?

Total energy = rest energy + kinetic energy

E 2 p2c 2 mo2c 4

1.5MeV 2 p2c 2 .511MeVc 2 2c 4

2.25MeV 2 p2c 2 .26MeV 2

1.99MeV 2 p2c 2

p2 1.99MeV 2

c 2

p 1.41MeVc 1

E .511MeV 1.0MeV 1.5MeV

Example: What is the speed of an electron with a momentum of 2.0MeVc-1?