Do Now (5/6/14):
• Turn in your homeworks!!!
• You are standing on a bus that is moving at 30
m/s. You toss a baseball up and down.
– What is the horizontal speed of the ball that you see?
– What is the speed of the ball as seen by someone
standing on the street?
• You are on the same bus, but now you toss the
ball forward at 20 m/s.
– What is the horizontal speed of the ball that you see?
– What is the speed of the ball as seen by someone
standing on the street?
Do Now (5/16/14):
• Pick up a paper in the back on your way in
• You are standing on a bus that is moving at 30
m/s. You hold a baseball in your hand.
– What is the horizontal speed of the ball that you see?
– What is the speed of the ball as seen by someone
standing on the street?
• You are on the same bus, but now you toss the
ball forward at 20 m/s.
– What is the horizontal speed of the ball that you see?
– What is the speed of the ball as seen by someone
standing on the street?
Relativity
Michelson-Morley Experiment
• The ether was a hypothetical medium in which
it was believed that electromagnetic waves
(visible light, infrared radiation, ultraviolet
radiation, radio waves, X-rays, -rays, ...) would
propagate.
Michelson-Morley Experiment
•
In 1887, Albert A.
Michelson and Edward W.
Morley tried to measure the
speed of the ether.
• The concept of the ether was
made in analogy with other
types of media in which
different types of waves are
able to propagate; sound
waves can, for example,
propagate in air or other
materials.
Michelson-Morley Experiment
• The result of the MichelsonMorley experiment was that the
speed of the Earth through the
ether (or the speed of the ether
wind) was zero. Therefore, this
experiment also showed that
there is no need for any ether
at all, and it appeared that the
speed of light in vacuum was
independent of the speed of
the observer!
Michelson-Morley Experiment
• Michelson and Morley repeated
their experiment many times up
until 1929, but always with the
same results and conclusions.
Michelson won the Nobel Prize in
Physics in 1907.
Relativity: Two Parts:
•General
•Special
The Postulates of Special Relativity
• On June 30, 1905 Einstein formulated
the two postulates of special relativity:
1. The Principle of Relativity
The laws of physics are the same in
all inertial frames of reference.
2. The Constancy of Speed of Light in
Vacuum
The speed of light in vacuum has the
same value c in all inertial frames of
reference.
The Postulates of Special Relativity
• The speed of light in vacuum c (299792458 m/s)
is so enormous that we do not notice a delay
between the transmission and reception of
electromagnetic waves under normal
circumstances.
• The speed of light in vacuum is actually the only
speed that is absolute and the same for all
observers as was stated in the second postulate.
Postulates of Special Relativity
• Nothing can travel faster than the speed of
light!
Classical Relativity
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.
1,000,000 ms-1
Einstein’s Special Relativity
0 ms-1
300,000,000 ms-1
1,000,000 ms-1


Both spacemen measure the speed of the approaching ray of light.
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!
V
• Moving clocks run more slowly!
Three effects
• 3 strange effects of special relativity
– Length contraction
– Time dilation
– Relativistic mass
Constant Velocity of Light
Platform v = 30 m/s to right relative to ground.
c
c
10 m/s
10 m/s
Velocities observed inside car
c
c
20 m/s
40 m/s
Velocities observed from outside
car
The light from two flashlights and the two balls travel in opposite
directions. The observed velocities of the ball differ, but the speed of light is
independent of direction.
Velocity of Light (Cont.)
Platform moves 30 m/s to right relative to boy.
c
10 m/s
c
10 m/s
30
m/s
Each observer sees c = 3 x 108
m/s
Outside observer sees very different velocities for
balls.
The velocity of light is unaffected by relative motion and is exactly equal to:
c = 2.99792458 x 108 m/s
Time Dilation Equation
Einstein’s Time dilation
Equation:
t 
t0
1  v2 c2
t = Relative time (Time measured in frame moving relative to actual event).
to= Proper time (Time measured in the same frame as the event itself).
v = Relative velocity of two frames.
c = Free space velocity of light (c = 3 x 108 m/s).
Example 1: Ship A passes ship B with a relative velocity of 0.8c
(eighty percent of the velocity of light). A woman aboard Ship B
takes 4 s to walk the length of her ship. What time is recorded by
the man in Ship A?
Proper time to = 4 s
A
Find relative time t
t 
t 
B
t0
v = 0.8c
1  v2 c2
4.00 s
1- (0.8c) 2 / c 2

4.00 s
1- 0.64
t = 6.67 s
Length Contraction
Since time is affected by relative motion,
length will also be different:
0.9c
Lo
L  L0 1  v 2 c 2
Lo is proper length
L is relative length
Moving objects are foreshortened due to relativity.
L
Example 2: A meter stick moves at 0.9c relative to an
observer. What is the relative length as seen by the
observer?
Lo
L  L0 1  v 2 c 2
L  (1 m) 1  (0.9c) / c
2
1m
0.9c
2
L=?
L  (1 m) 1  0.81  0.436 m
Length recorded by observer:
L = 43.6 cm
If the ground observer held a meter stick, the same contraction would be seen
from the ship.
Relativistic Mass
If momentum is to be conserved, the relativistic mass m must be consistent with the
following equation:
Relativistic mass:
m
m0
1  v2 c2
Note that as an object is accelerated by a resultant force, its mass increases,
which requires even more force. This means that:
The speed of light is an ultimate speed!
Example 3: The rest mass of an electron is 9.1 x 10-31 kg.
What is the relativistic mass if its velocity is 0.8c ?
m
-
m0
1 v c
2
0.8c
mo = 9.1 x 10-31 kg
2
-31
9.1 x 10 kg
-31
9.1 x 10 kg
m

0.36
1  (0.6c) 2 c 2
The mass has increased by 67% !
m = 15.2 x 10-31 kg
Practice:
Use the rest of class to complete
the Relativity sheet
Do Now (5/7/14):
The rest mass of an electron is 9.1
x 10-31 kg. What is the relativistic
mass if its velocity is 0.8c
Do Now (5/8/14):
Draw me a funny/cute picture on
your Do Now sheet
Review:
if you are exempt, complete your
Relativity sheet
if you are not, complete your
review packet!!!
Mass and Energy
Prior to the theory of relativity, scientists considered mass and energy as
separate quantities, each of which must be conserved.
Now mass and energy must be considered as
the same quantity. We may express the mass
of a baseball in joules or its energy in
kilograms! The motion adds to the massenergy.
Total Relativistic Energy
The general formula for the relativistic total energy involves the rest mass
mo and the relativistic momentum p = mv.
Total Energy, E
E  (m0c )  p c
2
For a particle with zero momentum p =
0:
E = moc2
For an EM wave, m0 = 0, and E simplifies
to:
E = pc
2 2
Mass and Energy (Cont.)
The conversion factor between mass m and energy E is:
Eo = moc2
The zero subscript refers to proper or rest values.
A 1-kg block on a table has an energy Eo and mass mo relative to
table:
Eo = (1 kg)(3 x 108 m/s)2
1 kg
Eo = 9 x 1016 J
If the 1-kg block is in relative motion, its kinetic energy adds to the total energy.
Total Energy
According to Einstein's theory, the total energy E of a particle of is given by:
Total Energy: E = mc2
(moc2 + K)
Total energy includes rest energy and energy of motion. If we are interested in
just the energy of motion, we must subtract moc2.
Kinetic Energy: K = mc2 – moc2
Kinetic Energy: K = (m – mo)c2
Example 4: What is the kinetic energy of a proton
(mo = 1.67 x 10-27 kg) traveling at 0.8c?
m
+
m0
1 v c
2
2
0.7c
mo = 1.67 x 10-27 kg
1.67 x 10-27 kg
1.67 x 10-27 kg; m = 2.34 x 10-27 kg
m

0.51
1  (0.7c) 2 c 2
K = (m – mo)c2 = (2.34 x 10-27 kg – 1.67 x 10-17 kg)c2
Relativistic Kinetic Energy K = 6.02 x 10-11 J
Summary
Einstein’s Special Theory of Relativity, published in 1905, was based on two
postulates:
I. The laws of physics are the same for all frames of reference moving at a
constant velocity with respect to each other.
II. The free space velocity of light c is constant for all observers, independent
of their state of motion. (c = 3 x 108 m/s)
Summary (Cont.)
Relativistic time:
Relativistic length:
Relativistic mass:
t 
t0
1  v2 c2
L  L0 1  v 2 c 2
m
m0
1  v2 c2
Summary(Cont.)
Relativistic
momentum:
p
Total energy: E = mc2
Kinetic energy: K = (m – mo)c2
m0 v
1  v2 c2
CONCLUSION: Chapter 38A
Relativity
Chapter 38A - Relativity
A PowerPoint Presentation by
Paul E. Tippens, Professor of Physics
Southern Polytechnic State University
© 2007
Objectives: After completing this
module, you should be able to:
• State and discuss Einstein’s two postulates
regarding special relativity.
• Demonstrate your understanding of time
dilation and apply it to physical problems.
• Demonstrate and apply equations for
relativistic length, momentum, mass, and
energy.
Special Relativity
Einstein’s Special Theory of Relativity, published in 1905, was based on two
postulates:
I. The laws of physics are the same for all frames of reference moving at a
constant velocity with respect to each other.
II. The free space velocity of light c is constant for all observers, independent
of their state of motion. (c = 3 x 108 m/s)
Rest and Motion
What do we mean when we say that an object is at rest . . . or in motion? Is
anything at rest?
We sometimes say that man, computer,
phone, and desk are at rest.
We forget that the Earth is also in motion.
What we really mean is that all are moving with the same velocity. We can only
detect motion in reference to something else.
No Preferred Frame of Reference
What is the velocity of the bicyclist?
25 m/s
West
We cannot say without a frame of
reference.
East
10 m/s
Earth
Assume bike moves at 25 m/s,W relative to Earth and that platform moves 10 m/s, E
relative to Earth.
What is the velocity of the bike relative to platform?
Assume that the platform is the reference, then look at relative motion of Earth
and bike.
Reference for Motion (Cont.)
To find the velocity of the bike relative to platform, we must imagine that we
are sitting on the platform at rest (0 m/s) relative to it.
We would see the Earth moving westward at 10 m/s and the bike moving west
at 35 m/s.
Earth as Reference
25 m/s
0 m/s
35 m/s
East
10 m/s
West
Earth
Platform as Reference
West
10 m/s
East
0 m/s
Frame of
Reference
Consider the velocities for three
different frames of reference.
Earth as Reference
25 m/s
10 m/s
West
Earth
Platform as Reference
35 m/s
West
10 m/s
East
0
Bicycle as Reference
East
0 m/s
0 m/s
West
25 m/s
East
35 m/s
Constant Velocity of Light
Platform v = 30 m/s to right relative to ground.
c
c
10 m/s
10 m/s
Velocities observed inside car
c
c
20 m/s
40 m/s
Velocities observed from outside
car
The light from two flashlights and the two balls travel in opposite
directions. The observed velocities of the ball differ, but the speed of light is
independent of direction.
Velocity of Light (Cont.)
Platform moves 30 m/s to right relative to boy.
c
10 m/s
c
10 m/s
30
m/s
Each observer sees c = 3 x 108
m/s
Outside observer sees very different velocities for
balls.
The velocity of light is unaffected by relative motion and is exactly equal to:
c = 2.99792458 x 108 m/s
Simultaneous Events
The judgment of simultaneous events is also a matter of relativity. Consider observer OT
sitting on top of moving train while observer OE is on ground.
At t = 0, lightning strikes train and
ground at A and B.
Not simultaneous
OT
B
A
BT
AT
AE
OE
BE
Observer OE sees lightning events
AE & BE as simultaneous.
Simultaneous
Observer OT says event BT occurs before event AT due to motion of train. Each
observer is correct!
Time Measurements
Since our measurement of time involves
judg-ments about simul-taneous events,
we can see that time may also be affected
by relative motion of observers.
In fact, Einstein's theory shows that observers in relative motion will judge
times differently - furthermore, each is correct.
Relative Time
Consider cart moving with velocity v under a mirrored ceiling. A light pulse travels to
ceiling and back in time to for rider and in time t for watcher.
2d
c
t 0
d
to
Light path for
rider
2R
c
t
R
R
d
x
t
Light path for
watcher
ct
vt
R
; x
2
2
Relative Time (Cont.)
2d
c
t 0
d
to
Light path for
rider
Substitution of:
ct0
d
2
c
2 t
t
R
d
v 2t
2
2
 c   v 
2


d

 

 2 t   2 t 
2
2

t
0
 c t   v   ct0 

 
 2  2

 2t   21t v c 2 
2
Time Dilation Equation
Einstein’s Time dilation
Equation:
t 
t0
1  v2 c2
t = Relative time (Time measured in frame moving relative to actual event).
to= Proper time (Time measured in the same frame as the event itself).
v = Relative velocity of two frames.
c = Free space velocity of light (c = 3 x 108 m/s).
Proper Time
The key to applying the time dilation equation is to distinguish clearly between
proper time to and relative time t. Look at our example:
Event
Frame
d
to
Proper Time
Relative Frame
t
Relative Time
t > to
Example 1: Ship A passes ship B with a relative velocity of 0.8c
(eighty percent of the velocity of light). A woman aboard Ship B
takes 4 s to walk the length of her ship. What time is recorded by
the man in Ship A?
Proper time to = 4 s
A
Find relative time t
t 
t 
B
t0
v = 0.8c
1  v2 c2
4.00 s
1- (0.8c) 2 / c 2

4.00 s
1- 0.64
t = 6.67 s
The Twin Paradox
Two twins are on Earth.
One leaves and travels for
10 years at 0.9c.
Traveling twin ages more!
When traveler returns, he is
23 years older due to time
dilation!
t 
t0
1  v2 c2
Paradox: Since motion is relative, isn’t it just as true
that the man who stayed on earth should be 23 years
older?
The Twin Paradox Explained
The traveling twin’s motion was not
uniform. Acceleration and forces
were needed to go to and return
from space.
Traveling twin ages more!
The traveler ages more and
not the one who stayed
home.
This is NOT science fiction. Atomic clocks placed
aboard aircraft sent around Earth and back have
verified the time dilation.
Length Contraction
Since time is affected by relative motion,
length will also be different:
0.9c
Lo
L  L0 1  v 2 c 2
Lo is proper length
L is relative length
Moving objects are foreshortened due to relativity.
L
Example 2: A meter stick moves at 0.9c relative to an
observer. What is the relative length as seen by the
observer?
Lo
L  L0 1  v 2 c 2
L  (1 m) 1  (0.9c) / c
2
1m
0.9c
2
L=?
L  (1 m) 1  0.81  0.436 m
Length recorded by observer:
L = 43.6 cm
If the ground observer held a meter stick, the same contraction would be seen
from the ship.
Foreshortening of Objects
Note that it is the length in the direction of relative motion that contracts
and not the dimensions perpendicular to the motion.
Assume each holds a meter stick, in
example.
Wo
If meter stick is 2 cm wide, each will say
the other is only 0.87 cm wide, but
they will agree on the length.
0.9c
1 m=1 m
W<Wo
Relativistic Momentum
The basic conservation laws for momentum and energy can not be violated due to
relativity.
Newton’s equation for momentum (mv) must be changed as follows to account for
relativity:
Relativistic momentum:
p
m0 v
1  v2 c2
mo is the proper mass, often called the rest mass. Note that for large values of
v, this equation reduces to Newton’s equation.
Relativistic Mass
If momentum is to be conserved, the relativistic mass m must be consistent with the
following equation:
Relativistic mass:
m
m0
1  v2 c2
Note that as an object is accelerated by a resultant force, its mass increases,
which requires even more force. This means that:
The speed of light is an ultimate speed!
Example 3: The rest mass of an electron is 9.1 x 10-31 kg.
What is the relativistic mass if its velocity is 0.8c ?
m
-
m0
1 v c
2
0.8c
mo = 9.1 x 10-31 kg
2
-31
9.1 x 10 kg
-31
9.1 x 10 kg
m

0.36
1  (0.6c) 2 c 2
The mass has increased by 67% !
m = 15.2 x 10-31 kg
Mass and Energy
Prior to the theory of relativity, scientists considered mass and energy as
separate quantities, each of which must be conserved.
Now mass and energy must be considered as
the same quantity. We may express the mass
of a baseball in joules or its energy in
kilograms! The motion adds to the massenergy.
Total Relativistic Energy
The general formula for the relativistic total energy involves the rest mass
mo and the relativistic momentum p = mv.
Total Energy, E
E  (m0c )  p c
2
For a particle with zero momentum p =
0:
E = moc2
For an EM wave, m0 = 0, and E simplifies
to:
E = pc
2 2
Mass and Energy (Cont.)
The conversion factor between mass m and energy E is:
Eo = moc2
The zero subscript refers to proper or rest values.
A 1-kg block on a table has an energy Eo and mass mo relative to
table:
Eo = (1 kg)(3 x 108 m/s)2
1 kg
Eo = 9 x 1016 J
If the 1-kg block is in relative motion, its kinetic energy adds to the total energy.
Total Energy
According to Einstein's theory, the total energy E of a particle of is given by:
Total Energy: E = mc2
(moc2 + K)
Total energy includes rest energy and energy of motion. If we are interested in
just the energy of motion, we must subtract moc2.
Kinetic Energy: K = mc2 – moc2
Kinetic Energy: K = (m – mo)c2
Example 4: What is the kinetic energy of a proton
(mo = 1.67 x 10-27 kg) traveling at 0.8c?
m
+
m0
1 v c
2
2
0.7c
mo = 1.67 x 10-27 kg
1.67 x 10-27 kg
1.67 x 10-27 kg; m = 2.34 x 10-27 kg
m

0.51
1  (0.7c) 2 c 2
K = (m – mo)c2 = (2.34 x 10-27 kg – 1.67 x 10-17 kg)c2
Relativistic Kinetic Energy K = 6.02 x 10-11 J
Summary
Einstein’s Special Theory of Relativity, published in 1905, was based on two
postulates:
I. The laws of physics are the same for all frames of reference moving at a
constant velocity with respect to each other.
II. The free space velocity of light c is constant for all observers, independent
of their state of motion. (c = 3 x 108 m/s)
Summary (Cont.)
Relativistic time:
Relativistic length:
Relativistic mass:
t 
t0
1  v2 c2
L  L0 1  v 2 c 2
m
m0
1  v2 c2
Summary(Cont.)
Relativistic
momentum:
p
Total energy: E = mc2
Kinetic energy: K = (m – mo)c2
m0 v
1  v2 c2
CONCLUSION: Chapter 38A
Relativity