Chapter 27 Relativity – Lecture 22

27.5 Length Contraction

27.6 Addition of Velocities

27.7 Relativistic Momentum

27.8 What is “Mass” ?

27.9 Mass and Energy

Length Contraction

• Alice marks two points (A and B) on the ground and measures length L0 between them

• Ted travels in the x-direction at constant velocity v and reads his clock as he passes point A and point B

• This is the proper time interval of the motion

Section 27.5

Length Contraction, cont.

• Distance measured by Alice = L0 = v Δt

• Distance measured by Ted = L = v Δt0

• Since Δt ≠ Δt0, L ≠ L0

• The difference is due to time

dilation and

• The length measured

by Ted is shorter than

Alice’s length

Section 27.5

t t v c

t t v c

2 2

0

2 2

0

/ 1 /

1 /

2 2

0 1 /L L v c

Proper Length

• Ted is at rest

• Alice moves on the

meterstick with speed v

relative to Ted

• Ted measures a length

shorter than Alice

• Moving metersticks are

shortened

• The proper length, Lo,

is the length measured

by the observer (Alice)

at rest relative to the

meterstick

Section 27.5

2 2

0 1 /L L v c

Length Contraction Equation

• Length contraction is

described by

• When the speed is very

small, the contraction

factor is very close to 1

• This is the case for

typical terrestrial

speeds

Section 27.5

2

2

0

1L v

L c

Proper Length and Time

Review:

• Proper time is measured by an observer who is at

rest relative to the clock used for the measurement

• Proper length is measured by an observer who is at

rest relative to the object whose length is being

measured

Section 27.5

Experimental Support:

• A large number of experiments have shown that time

dilation and length contraction actually do occur

• At ordinary terrestrial speeds the effects are

negligibly small

• For objects moving at speeds approaching the

speed of light, the effects become significant

Addition of Velocities

• Ted is traveling on a railroad car at constant speed vTA with respect to Alice

• He throws an object with a speed relative to himself of vOT

• What is the velocity vOA of the ball relative to Alice? • Alice is at rest on the

ground

Section 27.6

• Newton would predict that vOA

= vOT + vTA

Newton’s Addition of Velocities

• Newton would predict that vOA = vOT + vTA

• The velocity of the object relative to Alice = the velocity

of the object relative to Ted + the velocity of Ted

relative to Alice

• This result is inconsistent with the postulates of

special relativity when the speeds are very high

• For example, if the object’s speed relative to Ted is

0.9 c and the railroad car is moving at 0.9 c, then the

object would be traveling at 1.8 c relative to Alice

• Newton’s theory gives a speed faster than the speed

of light

Section 27.6

Relativistic Addition of Velocities

• The result of special relativity for the addition of velocities is

• The velocities are:

• vOT – the velocity of an object

relative to an observer

• vTA – the velocity of one observer

relative to a second observer

• vOA – the velocity of the object

relative to the second observer

OT TAOA

OT TA

v vv

v v

c21

+=

+

Section 27.6

Relativistic Addition of Velocities, cont.

• When the velocities vOT and vTA are much less than

the speed of light, the relativistic addition of

velocities equations gives nearly the same result as

the Newtonian equation

• For speeds less than approximately 10% the speed

of light, the Newtonian velocity equation works well

• For the example, with each speed being 0.9 c, the

relativistic result is 0.994 c

• Compared to 1.8 c from Newton’s prediction

• Experiments with particles moving

at very high speeds show that

the relativistic result is correct

Section 27.6

OT TAOA

OT TA

v vv

v v

c21

+=

+

Relativistic Velocities and the Speed of Light

• A slightly different result occurs when the velocities

are perpendicular to each other

• Again when vOT and vTA are both less than c, then

vOA is also less than c

• In general, if an object has a speed less than c for

one observer, its speed is less than c for all other

observers

• Since no experiment has ever observed an object

with a speed greater than the speed of light, c is a

universal “speed limit”

Section 27.6

Relativistic Velocities, final

• Assume the object leaving Ted’s hand is a pulse of

light Then vOT = c

• From the relativistic velocity equation, Alice observes

the pulse is vOA = c

• Alice sees the pulse traveling at the speed of light

regardless of Ted’s speed

• If an object moves at the speed of light for one

observer, it moves at the speed of light for all

observers Section 27.6

OT TAOA

OT TA

v v c v c v c vv c

v v cv v c v c

c cc 22

( ) /1 11

Momentum

• According to Newton’s mechanics, a particle of

mass m0 moving with speed v has a momentum

given by

p = m0 v

• Conservation of momentum is one of the

fundamental conservation rules in physics and is

believed to be satisfied by all the laws of physics,

including the theory of special relativity

• The momentum of a single particle can also be

written as

Section 27.7

0

xp m

t

Relativistic Momentum

• From time dilation and length contraction,

measurements of both Δx and Δt can be different for

observers in different inertial reference frames

• Should proper time or proper length be used?

• Einstein showed that you should use the proper time

to calculate momentum

• Uses a clock traveling along with the particle

• The result from special relativity is

Section 27.7

00 0

2 2 2 20 1 / 1 /

m vx xp m m

t t v c v c

Relativistic Momentum, cont.

• Einstein showed that when the momentum is

calculated by using the special relativity equation,

the principle of conservation of momentum is obeyed

exactly

• This is the correct expression for momentum and

applies even for a particle moving at high speed,

close to the speed of light

• When a particle’s speed is small compared to the

speed of light, the relativistic momentum becomes p

= m0 v which is Newton’s momentum

Section 27.7

m vx xp m m m v if v c

t t v c v c

00 0 0

2 2 2 20

,1 1

Newton’s vs. Relativistic Momentum

• As v approaches the

speed of light, the

relativistic result is very

different than Newton’s

• There is no limit to how

large the momentum

can be

• However, even when

the momentum is very

large, the particle’s

speed never quite

reaches the speed of

light Section 27.7

m vp

v c

0

2 21

Mass

• Newton’s second law gives mass, m0, as the

constant of proportionality that relates acceleration

and force

• This works well as long as the object’s speed is

small compared with the speed of light

• At high speeds, though, Newton’s second law

breaks down

Section 27.8

Rest Mass

• When the speed of the mass is close to the speed of

light, the particle responds to a force as if it had a

mass larger than m0

• The same result happens with momentum where at

high speeds the particle responds to impulses and

forces as if its mass were larger than m0

• Rest mass is denoted by m0

• This is the mass measured by an observer who is

moving very slowly relative to the particle

• The best way to describe the mass of a particle is

through its rest mass

Section 27.8

Relativistic Mass-Energy

Section 27.8

0 0

0 0

2 2 3/22 2

0 0

2 2 3/2 2 2 3/2

0 0

2 220 0

02 2 2 2

0

220

02 2

,

(1 / )1 /

(1 / ) (1 / )

,1 / 1 /

1 /

x x

x x

t v

v

dp dpW Fdx dx F ma W work

dt dt

m v mdp d dv

dt dt v c dtv c

m m vdvdvW vdt

v c dt v c

m c m cW m c W KE

v c v c

m cKE m c

v c

Mass and Energy

• Relativistic effects need to be taken into account

when dealing with energy at high speeds

• From special relativity and work-energy,

• For v << c, this gives KE ≈ ½ m0 v2 which is the

expression for kinetic energy from Newton’s results

Section 27.9

220

02 21 /

m cKE m c

v c

Kinetic Energy and Speed

• For small velocities, KE

is given by Newton’s

results

• As v approaches c, the

relativistic result has a

different behavior than

does Newton

• Although the KE can be

made very large, the

particle’s speed never

quite reaches the speed

of light

Section 27.9

220

02 21 /

m cKE m c

v c

Rest and Total Energies

• The kinetic energy can also be thought of as the

difference between the final and initial energies of

the particle

• The initial energy, m0c2, is a constant called the rest

energy of the particle

• A particle will have this much energy even when it is at

rest

• The total relativistic energy of the particle is the sum

of the kinetic energies and the rest energy

Section 27.9

total

total

m cKE m c E TE KE m c

v c

m cE TE Eq

v c

22 20

0 02 2

2

0

2 2

,1

.(27.33)1

Mass as Energy

• The rest energy equation implies that mass is a form

of energy

• It is possible to convert an amount of energy (m0c2)

into a particle of mass m0

• It is possible to convert a particle of mass m0 into an

amount of energy (m0c2)

• The principle of conservation of energy must be

extended to include this type of energy

• When v 0, (famous “Einstein Equation”)

Section 27.9

E m c2

0

total

m cE TE Eq

v c

2

0

2 2.(27.33)

1

Speed of Light as a Speed Limit

• Several results of special relativity suggest that

speeds greater than the speed of light are not

possible

• The factor that appears in time dilation and

length contraction is imaginary if v > c

• The relativistic momentum of a particle becomes

infinite as v → c

• This suggests that an infinite force or impulse is

needed for a particle to reach the speed of light

Section 27.9

v c 2 21

Speed Limit, cont.

• The total energy of a particle becomes infinite as v → c

• This suggests that an infinite amount of mechanical work

is required to accelerate a particle to the speed of light

• The idea that c is a “speed limit” is not one of the

postulates of special relativity

• Combining the two postulates of special relativity leads

to the conclusion that it is not possible for a particle to

travel faster than the speed of light

Section 27.9

Two Postulates of Special Relativity

1. All the laws of physics are the same in all inertial

reference frames

2. The speed of light in a vacuum is a constant, independent

of the motion of the light source and all observers

Mass-Energy Conversions

• Conversion of mass into energy is important in

nuclear reactions, but also occurs in other cases

• A chemical reaction occurs when a hydrogen atom is

dissociated

• The mass of a hydrogen atom must be less than the

sum of the masses of an electron and proton

• The energy is lower by 13.6 eV when bound in the

atom

• Mass is not conserved when a hydrogen atom

dissociates

• Δm0 = 2.4 x 10-35 kg

• This is much less than the mass of a proton and can

be ignored (mp = 1.67 x 10-27 kg)

Section 27.9

Conservation Principles

• Conservation of mass

• Mass is a conserved quantity in Newton’s mechanics

• The total mass of a closed system cannot change

• Special relativity indicates that mass is not conserved

• The principle of conservation of energy must be extended to

include mass

• Momentum is conserved in collisions

• Use the relativistic expression for momentum

• Electric charge is conserved

• It is possible to create or annihilate charges as long as

the total charge does not change

Section 27.9