Fisica Generale - Alan Giambattista, Betty McCarty Richardson
Chapter 26: Relativity
•The Postulates of Special Relativity
•Simultaneity
•Time Dilation
•Length Contraction
•Relativistic Velocity Addition
•Relativistic Momentum
•Relativistic Rest Mass Energy, Kinetic Energy, and Total Energy
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Fisica Generale - Alan Giambattista, Betty McCarty Richardson
§26.1 Postulates of Relativity
Postulate 1: The laws of physics are the same in all
inertial reference frames (the principle of relativity).
An inertial reference frame is one in which no
accelerations are observed in the absence of external
forces. (Recall Newton’s first law).
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Fisica Generale - Alan Giambattista, Betty McCarty Richardson
Scientists in the 19th century believed that light was a
vibration in an invisible medium called the ether.
The Michelson-Morley experiment found that there was no
observable change in the speed of light due to the Earth’s
motion around the Sun. The conclusion was that the ether
did not exist.
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Fisica Generale - Alan Giambattista, Betty McCarty Richardson
Postulate 2: The speed of light in vacuum is the same in
all inertial reference frames. The value is independent of
the motion of the source of light or of the observer.
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Fisica Generale - Alan Giambattista, Betty McCarty Richardson
What follows does not show that Newtonian physics is
wrong; it is just not complete. Newtonian physics breaks
down at speeds near c; this is where special relativity is
needed. All of the familiar relationships from Newtonian
physics are contained within special relativity in the limit of
low speeds. This is known as the correspondence
principle.
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§26.2 Simultaneity and Ideal
Observers
The location of an event can be specified by four
coordinates: the three spatial coordinates (x,y,z) and a time
coordinate t. Taken together (x,y,z,t) are the four
coordinates of space-time.
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Fisica Generale - Alan Giambattista, Betty McCarty Richardson
Consider a high speed train with AC=BC & A’C’=B’C’. The
marks ACB are on a stationary train platform and the marks
A’C’B’ are on the moving train.
v
A’
C’
B’
A
C
B
At t=0: AA’, BB’, and CC’ are lined up. At this instant two
lightning bolts strike at AA’ and BB’.
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Fisica Generale - Alan Giambattista, Betty McCarty Richardson
An observer standing at point C (on the platform) will see
both strikes simultaneously.
The observer at C’ (on the train) will see the strike at AA’ first
followed by the strike at BB’.
The observers do not agree on what
happened, but both are correct.
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Fisica Generale - Alan Giambattista, Betty McCarty Richardson
This experiment shows that events that are simultaneous in
one frame (the platform) are not simultaneous in another
frame (the train).
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Fisica Generale - Alan Giambattista, Betty McCarty Richardson
§26.3 Time Dilation
Consider a light clock. A light pulse reflects back and forth
between two mirrors. One complete trip can be considered
one “tick” on the clock.
Mirror
L
Light pulse
Mirror
The time interval for a round
trip by the light pulse is
d 2L
t0  
.
v
c
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Fisica Generale - Alan Giambattista, Betty McCarty Richardson
As long as an observer is stationary with respect to the
clock, he will measure a time interval of t0 between clock
ticks. Now put the light clock on a moving train. What does
a stationary observer outside the train see?
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Fisica Generale - Alan Giambattista, Betty McCarty Richardson
The motion of the train is to the right with speed v.
The light pulse appears to have the path shown below. How
long does it take the light pulse to return to the bottom
mirror?
1

2 L   vt 
d
2

t  
c
c
2
2
L
vt
d 2L
Know t 0  
v
c
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Fisica Generale - Alan Giambattista, Betty McCarty Richardson
Eliminating L gives t 
t0
v
1  
c
v
1
Let,   and  
2
c
1 
2
 t0 .
(Lorentz Gamma factor)
The person outside the train will measure a longer
time interval between ticks compared to the observer
in the train.
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Fisica Generale - Alan Giambattista, Betty McCarty Richardson
The relative motion between the observers (train and
outside the train) changes the rate at which time passes.
The effect is known as time dilation.
Moving clocks run slower.
The quantity t0 is known as the proper time. This is time
interval between two events that occur in the same place.
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Fisica Generale - Alan Giambattista, Betty McCarty Richardson
Example (text problem 26.6): An unstable particle called
the pion has a mean lifetime of 25 ns in its own frame. A
beam of pions travels through the laboratory at a speed of
0.60c. (a) What is the mean lifetime of the pion as
measured in the laboratory?
Given: t0 = 25 ns and v = 0.60c (=0.60)

1
1 
2

1
1  0.60
2
 1.250
t  t0  1.25025 ns   31 ns
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Fisica Generale - Alan Giambattista, Betty McCarty Richardson
Example continued:
(b) How far does a pion travel (as measured by laboratory
observers) during this time?
d  vt  0.6c31 ns   5.6 m
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§26.4 Length Contraction
To measure the length of an object, its ends must be located
simultaneously. The proper length is measured when an
object is at rest relative to you.
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Fisica Generale - Alan Giambattista, Betty McCarty Richardson
A person on a train platform will measure the length of the
platform to be L0. A moving object will pass by the platform
in a time t so L0 = vt (note: t is not the proper time).
A person riding on a train will measure the length of the
platform to be L = vt0 (this person measures the proper
time).
L0 vt
t



L vt0 t0
or L 
L0

where L0 is the proper length.
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Fisica Generale - Alan Giambattista, Betty McCarty Richardson
The two observers measure different lengths for the platform.
Moving meter sticks are shorter.
This effect is known as length contraction and only applies
to lengths parallel to the direction of motion.
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Fisica Generale - Alan Giambattista, Betty McCarty Richardson
Example (text problem 26.15): A cosmic ray particle travels
directly over a football field from one goal line to the other, at
a speed of 0.50c. (a) If the length of the field between goal
lines is 91.5 m, what is the length measured in the rest
frame of the particle?
Given: L0 = 91.5 m and v = 0.50c (=0.50)

1
1 
2

1
1  0.50
2
 1.155
L0
91.5 m
L

 79 m.

1.155
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Fisica Generale - Alan Giambattista, Betty McCarty Richardson
Example continued:
(b) How long does it take the particle to go from one goal
line to the other according to earth observers?
L0
91.5
t 

 6.110 7 s
v 0.50c
(c) How long does it take in the rest frame of the particle?
L 79 m
t0  
 5.3 10 7 s
v 0.50c
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Fisica Generale - Alan Giambattista, Betty McCarty Richardson
Example (text problem 26.17): Two spaceships are moving
directly toward each other with a relative velocity of 0.90c. If
an astronaut measures the length of his own spaceship to
be 30.0 m, how long is the spaceship as measured by an
astronaut in the other ship?
Given: L0 = 30.0 m and v = 0.90c (=0.90)

1
1 
2

1
1  0.90
2
 2.294
L0
30.0m
L

 13 m.

2.294
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Fisica Generale - Alan Giambattista, Betty McCarty Richardson
§26.5 Velocities in Different
Reference Frames
What is the velocity of the probe as measured by Abe?
According to Galilean relativity, it is vPA= vPB+vBA.
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Fisica Generale - Alan Giambattista, Betty McCarty Richardson
Once the speeds get large enough, the relativistic velocity
transformation formula must be used.
vPA
vPB  vBA

vPBvBA
1
c2
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Fisica Generale - Alan Giambattista, Betty McCarty Richardson
Example (text problem 26.25): The rogue starship, Galaxa,
is being chased by the battlecruiser, Millenia. The Millenia is
catching up to the Galaxa at a rate of 0.55c when the
captain of the Millenia decides it is time to fire a missile.
First the captain shines a laser range finder to determine the
distance to the Galaxa, and then he fires a missile that is
moving at a speed of 0.45c with respect to Millenia. What
speed does the Galaxa measure for (a) the laser beam and
(b) the missile as they both approach the starship?
The laser beam will be measured to have speed c.
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Fisica Generale - Alan Giambattista, Betty McCarty Richardson
Example continued:
The velocity of Millenia relative to Galaxa is vMG=0.55c.
The velocity of the missile relative to Millenia is vmG = 0.45c.
The velocity of the missile relative to Galaxa is
vmG
vmM  vMG
0.45c  0.55c


 0.802c.

vmM vMG
0.45c 0.55c 
1
1
2
c2
c
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Fisica Generale - Alan Giambattista, Betty McCarty Richardson
§26.6 Relativistic Momentum
p  mv
Note: it is true that p =
Ft, but Fma.
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Fisica Generale - Alan Giambattista, Betty McCarty Richardson
Example: Show that the expression for relativistic momentum
reduces to its classical form when v<<c.
When v<<c, <<1
 1
p  mv  
 1  2


mv


  1 2
1 2
 1      mv  mv   mv
2
  2 
v 1 v
 mc   mc 
c 2 c
 mv
3
To first order in v/c
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Fisica Generale - Alan Giambattista, Betty McCarty Richardson
Example (text problem 26.36): A body has a mass of
12.6 kg and a speed of 0.87c. (a) What is the magnitude
of the momentum?
Given: m = 12.6 kg and v = 0.87c (=0.87)

1
1 
2

1
1  0.87
2
 2.028
p  mv
 2.02812.6 kg 0.87c   6.7 109 kg m/s.
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Example continued:
(b) If a constant force of 424.6 N acts in the direction
opposite to the body’s motion, how long must the force act
to bring the body to rest?
p
6.7 109 kg m/s
t 

424.6 N
F
 6.58 10 s  0.50 years
7
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Fisica Generale - Alan Giambattista, Betty McCarty Richardson
§26.7 Mass and Energy
The rest mass energy of a particle is its energy measured
in its rest frame.
E0  mc 2
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Fisica Generale - Alan Giambattista, Betty McCarty Richardson
1 eV (electron volt) is the change in energy that a charge
e (the fundamental unit of charge) experiences when
accelerated through a 1 volt potential difference.
W  qV  e 1 Volt  1 eV
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A proton has m=1.6710-27 kg; its rest mass energy is
E0  mc
2
 938 MeV
 0.938 GeV
The mass of the proton can be written as 938 MeV/c2.
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Fisica Generale - Alan Giambattista, Betty McCarty Richardson
A quantity is conserved if its value is unchanged in a given
reference frame.
A quantity is invariant if its value is the same in all reference
frames.
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Fisica Generale - Alan Giambattista, Betty McCarty Richardson
§26.8 Relativistic Kinetic Energy
2


K    1 mc
Kinetic energy
E  E0  K
 mc    1mc
2
2
Total energy
 mc 2
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Fisica Generale - Alan Giambattista, Betty McCarty Richardson
Example:
Show that the expression for relativistic kinetic energy
reduces to its classical form when v<<c.
When v<<c, <<1
 1
 2
KE    1mc  
 1mc
 1  2



  1 2  2 1 2 2 1 2
 1       1mc   mc  mv .
2
2
  2

2
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The relationship between relativistic momentum and total
energy is
 .
E   pc   mc
2
2
2 2
 pc
2
Other useful relationships are
 K 2  2KE0
v pc
 .
c E
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Fisica Generale - Alan Giambattista, Betty McCarty Richardson
Example (text problem 26.48): When an electron travels at
0.60c, what is its total energy in MeV?
Given: v = 0.60c (=0.60)

1
1 
2

E  mc2
1
1  0.60

2
 1.150

 1.250 0.511 MeV/ c 2 c 2  0.65 MeV.
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Fisica Generale - Alan Giambattista, Betty McCarty Richardson
Example (text problem 26.53): An electron has a total
energy of 6.5 MeV. (a) What is its momentum (in MeV/c)?
 .
The energy-momentum relationship is E   pc   mc
2
2
2 2
Solving for p:
 
pc  E  mc
2

2 2
6.5 MeV   0.511 MeV 
2
2
 6.5 MeV
p  6.5 MeV/ c
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Example continued:
(b) What is the value of  for this electron?
E
6.5 MeV
 2
 12.7
2 2
mc
0.511 MeV/ c c


(c) What is the kinetic energy of this electron?
K    1mc2


 11.7  0.511 MeV/ c 2 c 2  6.0 MeV
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Fisica Generale - Alan Giambattista, Betty McCarty Richardson
Summary
•The Postulates of Relativity
•Simultaneity
•Time Dilation (moving clocks run slower)
•Length Contraction (moving meter sticks are shorter)
•Addition of Relativistic Velocities
•Relativistic Momentum
•Rest Mass Energy
•Relativistic Kinetic Energy
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