Download A8 Teaching Einstein`s Relativity in Unit 3

Einstein’s Theory of
Special Relativity
VCE Physics Unit 3 2017
Jill Detez
[email protected]
An overview of how I will teach Einstein’s Relativity in Unit 3. This session is aimed at
teachers who are new to teaching this topic. An outline of my approach and sharing of
resources, activities and assessment ideas will be provided.
Study Design References: Pages 39 & 40
Teaching Special Relativity in Unit 3
• Time allocation: 4 to 5 x 50 minute periods
• Text reference: Heinemann Physics 12 4th
Edition
• Assessment: Informal topic test.
• My SAC on Unit 3 Area of Study 3 will consist
of annotations of practical activities (circular
motion, strain potential energy, etc.) and may
include a reference to the modelling γ and
muon analyses.
Frames of reference
A reference frame allows us to refer to the
position of a particle. Reference frames consist
of an origin together with a set of axes.
• In Physics, we generally use the Cartesian
reference frame. (x, y, z and t)
• Others can be used:
– Sailors use latitude and longitude
– Astronomers use angles to define the position of a
star (altitude and azimuth, or right ascension
Inertial reference frames
We could roughly define an inertial reference
frame as a frame that is at rest or in constant
motion (i.e. not accelerating).
But then the question arises: at rest with respect
to what?
Michelson-Morley experiment
• Discuss the Michelson-Morley experiment at
this point: determining the relative motion of
the Earth with respect to the aether.
http://scienceworld.wolfram.com/physics/MichelsonMorleyExperiment.html
Laws of Physics and inertial reference
frames
We can more easily define an inertial reference frame as
a frame in which an object obeys Newton’s first law.
There are some good videos to introduce this concept.
Try
• Introduction to Special Relativity (Art Doing Physics, 4.5
min)
• https://www.youtube.com/watch?v=PpJ-LPOuvPc
• Also UNSW Physclips:
http://www.animations.physics.unsw.edu.au/jw/Newt
on.htm#frames
Galilean relativity and Newton’s
postulates
Galileo’s principle of relativity states that all motion
is relative to some particular frame of reference,
but there can be no frame of reference that has an
absolute zero velocity.
Newton assumed that space and time are constant,
uniform and straight (i.e. a metre ruler has the
same length whether vertical or horizontal, at
ground level or in space, and time flows at the same
rate no matter the location of the observer)
Einstein
Michelson and Morley demonstrated that the speed of
light (determined by Maxell’s equations) in vacuum is
constant in all directions, the accepted value is
c = 299 792 458 ms-1 (≈ 3.0 × 108 ms-1)
In trying to explain this fixed upper limit for the speed of
light, Einstein realised that Newton’s assumptions of
space-time may not hold true for all motion, especially
motion on scales involving huge distances and at speeds
approaching the speed of light.
If the speed of light is considered to be
absolute, then Newtonian physics
cannot be used to explain this
phenomenon.
The two postulates of special relativity
Einstein’s two postulates:
• The laws of Physics are the same in all inertial
(non-accelerated) frames of reference
• The speed of light has a constant value for all
observers regardless of their motion or the
motion of the source.
http://www.pbslearningmedia.org/resource/phy03.sci.phys.energy.sprelativit
y/einsteins-special-theory-of-relativity/
(7 min)
Heinemann Physics 12 Questions
• 6.1 Review page 211
• Questions 1 - 10
Some of the most dramatic differences between
our everyday perceptions of space and time and
the predictions made by special relativity
concern time and length differences that arise
between frames of reference moving relative to
each other.
Simultaneity
Simultaneity is the relation between two events
assumed to be happening at the same time in a
frame of reference. According to Einstein's
theory of relativity, simultaneity is not an
absolute relation between events; what is
simultaneous in one frame of reference will not
necessarily be simultaneous in another.
https://www.youtube.com/watch?v=fqVN2XmH_48 (2 min)
Proper Time
Einstein’s thought experiments regarding the
concept of simultaneity lead him to question
Newton’s concept of time being a constant in all
reference frames.
Watch this video: 60 Second adventure in
Special Relativity (Nice for light clock)
• www.youtube.com/watch?v=mU04-vJB6gc
Evidence for time difference in
different reference frames
The difference in time for the events discussed
by observers in a train would be so
infinitesimally small as to be negligible.
• However, aircraft flying at supersonic speeds
recording time using atomic clocks can detect
the difference.
• To particles travelling at speeds approaching
the speed of light, such as those in particle
accelerators, the differences in time between
the frames of reference become significant.
Proper time
• Proper time, Δto, is the term given for the time
interval in the frame of reference that is
‘stationary’.
• In the VCE Study Design, we will define
proper time, to, as the the time interval
between two events in a reference frame
where the two events occur at the same point
in space.
• Alternatively, it is the shortest possible
measured time interval between two events.
Proper time (continued)
• Consider the idea of the light clock. A beam of
light is reflected between two mirrors, and
ticks each time the light is reflected off the
bottom mirror.
• An observer sits in the room watching the light
reflected between the mirrors from a distance,
the time taken for the light to return to a mirror is
t = 2L/c (diagram 1)
• Now a second observer moves sideways at a
constant velocity v relative to the mirrors. In the
frame of this observer, the light appears to travel
at an angle to the direction of motion, but of
course at the same speed of light.
• Hence the moving person sees the light
travelling a greater distance at the same
speed, so the measured time for this must be
greater.
Time dilation and the Lorentz Factor
We can use the Pythagorean theorem to show
that the relationship between the time
measured in the reference frame, to, is related to
the time measured in the moving frame, t, is:
Note that the
t = toγ
derivation of the
where
time dilation
relationship and
the Lorentz
Factor are not
required for the
VCE course.
Time dilation and the Lorentz Factor
(cont.)
H. A. Lorentz (shared
Nobel Prize in Physics
1902) first introduced
the γ factor while
attempting to explain
the results of the
Michelson-Morley
experiment, hence it is
known as the Lorentz
factor.
• v is the speed of the
moving frame of
reference
• c is the speed of light in
vacuum
• t is the time observed in
the stationary frame
• to is the time observed in
the moving frame
Using time dilation - example
• Suppose a news report stated that the starship
Enterprise has just returned from a 5.0 year voyage
(as measured by Earth time) while travelling at 0.89c.
How much time has elapsed on the ship?
Time dilation
• Note that the observer on Earth measures the
time for the same event as longer. In other
words, the time is ‘dilated’ (expanded).
Log-book Exercise:
Using Excel to model the effect of
speed on the value of γ
• Investigate the effect of changing speed to obtain
values for γ for a range of values of the ratio v/c.
• Plot a graph of
γ vs. v/c for values of 0 ≤ v/c ≤ 1
Observations: Describe the trend of your graph. At
what speeds do relativistic effects become
noticeable?
Model the effect of speed on the value
of γ
Velocity
(ms-1)
3.00E+02
3.00E+05
3.00E+06
3.00E+07
5.70E+07
8.40E+07
1.11E+08
1.38E+08
1.65E+08
1.92E+08
2.19E+08
2.46E+08
2.73E+08
2.83E+08
2.93E+08
2.96E+08
2.99E+08
3.00E+08
v/c
γ
1.00E-06
1.00E-03
1.00E-02
1.00E-01
1.90E-01
2.80E-01
3.70E-01
4.60E-01
5.50E-01
6.40E-01
7.30E-01
8.20E-01
9.10E-01
9.43E-01
9.77E-01
9.87E-01
9.97E-01
1.00E+00
1.0000000000
1.0000005000
1.0000500038
1.0050378153
1.0185538856
1.0416666667
1.0763894729
1.1262289640
1.1973686802
1.3014480157
1.4631709377
1.7471413945
2.4119153510
3.0134397773
4.6563421829
6.1442394039
12.2576676967
#DIV/0!
10
9
8
7
6
γ
5
4
3
2
1
0
0.0
0.2
0.4
0.6
v/c
0.8
1.0
Useful videos
Time Dilation with Stephen Hawking Ph.D: A
video that describes what it might be like to
travel on a spaceship at relativistic speeds.
• https://www.youtube.com/watch?v=RSeomJx
_f5E
Explanation of time dilation.wmv (Mechanical
Universe series explains time dilation)
• https://www.youtube.com/watch?v=uIGegibP
V-c
Muons
• Muons are high energy sub-atomic particles that are
created in the Earth’s atmosphere when cosmic rays
strike oxygen atoms at about 15 km above the Earth.
Muons are unstable, with a mean lifetime of 2.196 μs.
• These muons typically travel at 99.97% of the speed of
light, and Newtonian physics predicts that a muon
would travel about 659 m in one half life.
• After 10 lifetimes there would be essentially no muons
remaining, and so after 6.59 km ( a height of 8.42 km
above the Earth) we would expect that no muons
would be detected.
http://hyperphysics.phyastr.gsu.edu/hbase/Relativ/muon.html
Explaining
high-altitude muons
• Time dilation explains the observation that
muons are, in fact, detected at the Earth’s
surface.
• If we apply the time dilation equation to the
muons, we see that the mean life (in the muon’s
reference frame) becomes 49.21 μs, which is
more than 22 times longer than in the stationary
frame.
• This is plenty of time for the muons to cascade
down to the Earth’s surface and hence be
detected.
Muon Investigation (Log Book)
Cosmic radiation from distant parts of the universe interacts with
particles in the atmosphere and produces muons.
• Muons are unstable with a half life of approximately 1.5 µs (1 µs = 1
x 10 -6 s). Muons travel at approximately 0.995 c.
• If the number of muons at time t = 0 is N0 and the muons enter the
atmosphere at a height of 2000 m, calculate the percentage of
muons which reach sea level.
• The value you have calculated does not agree with empirical
research, which detects 70 % of the original muons reaching the
ground. Use the theory of relativity to find how many of the initial
muons survive and calculate the half life of the muons in the Earth’s
frame of reference.
Heinemann Physics 12 Questions
• 6.2 Review Page 219
• Questions 1 - 10
Proper Length
In addition to time dilation, the length of an
object also changes when observed in frames
moving relative to the object.
• We define proper length, Lo, as the length of
the object in the frame which is at rest (the
frame which contains both the measuring
device and the object).
Length Contraction
The length of an object moving at relativistic speeds
can be found by again applying the Lorentz factor.
• The measured length in the moving frame
becomes
This equation shows that an object with a proper
length Lo will have a shorter length, L, parallel to
the motion of its moving frame of reference when
measured by an observer in a stationary frame of
reference.
https://www.nobelprize.org/educational/physi
cs/relativity/transformations-2.html
http://www.staff.science.uu.nl/~hooft
101/animations.html
Length contraction: Example
(Giancoli p 809)
• A rectangular painting measures 1.00 m tall
and 1.50 m wide. It is hung on the side wall of
a spaceship which is moving past the Earth at
a speed of 0.90c. (a) What are the dimensions
of the painting as seen by the captain of the
spaceship? (b) What are the dimensions as
seen by an observer on the Earth?
Using the Lorentz transformation
Sometimes it is useful to make v the subject in
the Lorentz factor. This produces:
• Students are encouraged to include this on
their notes sheet for use during the November
examination.
Heinemann Physics 12 Questions
• 6.3 Review page 224
• Questions 1 – 10
• Also Chapter Review Question pages 225 - 226
Relativistic mass increase
According to Newton’s Second Law, an object of mass m subjected to a
force F will experience an acceleration a, as given by F = m×a.
The Special Theory predicts that:
• As the speed of the body increases, then to an observer at rest
relative to the body the mass of the body will increase.
If mo is the mass of the body when it is at rest relative to the observer,
then this is the rest mass. When it is moving at speed v relative to the
observer, then its mass m is given by
m =  mo
Newton’s Second Law does not put any upper limit on the velocity an
object can attain.
• If, for example, a force of 1000 N were applied to a mass of 1.0 g for
1000 s, then it would reach a velocity……….
This value is much greater than the speed of light!
However, the Lorentz transformation equations
show us that if the relative velocity between two
objects is c, then the time dilation is infinite and the
length contraction is zero. In other words, time
stays still for objects travelling at the speed of light,
and the object will have no length in the direction
of travel. The mass equation shows that the mass
would be infinite. Hence the Special Theory tells us
that the maximum velocity an object can attain is c,
and also that this velocity is unattainable. It would
seem that the only thing that can travel at the
speed of light is light itself!
Mass-Energy
As a consequence of the Special Theory, the mass of a moving
object increases. A net force is needed to accelerate any
object and this force does work on the object.
Newtonian mechanics indicates that the conservation of
energy laws imply that the work done in accelerating an
object equals the increase in kinetic energy of the object.
In other words:
Work done = Force x displacement = gain in kinetic energy
So
F x s = (EK) = ½ m (v2)
This equation does not apply in Special Relativity. Since the
Laws of Physics hold in all reference frames, we need to find a
new relationship between work done and energy transferred.
In exploring these ideas, Einstein came to the idea of mass
and energy being interchangeable, such that the gain in mass
of an accelerated body could be equal to a gain in energy.
Hence he established his famous equation:
E = mc2
where E is the total energy of the object and m is its
relativistic mass. If the object is at rest then it has rest mass
energy given by
Eo = moc2
These two equations can be combined in terms
of the work done in accelerating and object
from rest to a velocity v, then the object will
have a total mass-energy mc2. Its mass energy
will have changed by an amount EK, where
EK = E - moc2 = mc2 – moc2
• This is the gain in kinetic energy, and is equal
to the work done on the object.
The key to understanding these equations is
Energy and mass are entirely equal!
Mass Energy Equivalence in the VCE
Physics Study Design
Etot = Ek + E0 = γmc2
where
E0 = mc2
and where kinetic energy can be calculated by:
Ek = (γ – 1)mc2
Example:
Example: Calculate the mass of uranium that is
converted into energy in one year (3.15 x 107 s )
in a nuclear power station that has an annual
power output of 100 MW.
Conservation of Mass-Energy
We can no longer consider the laws of conservation of
energy and mass as two separate laws of Physics, but
rather we need to combine them into one single law.
Consider the following example:
An electron is accelerated through a potential difference
of 2 x 106 volts. Calculate the final velocity of the electron
applying:
• classical Newtonian mechanics
• relativistic mechanics
Nuclear Fusion
This is the process of joining together two
smaller nuclei to form a larger, more stable
nucleus. Extremely high temperatures are
needed in order for this to occur, for example, in
the Sun or other stars.
A typical reaction in smaller stars (like our Sun)
is:
http://hyperphysics.phyastr.gsu.edu/hbase/Astro/
procyc.html
Mass defect: The binding energy of the
nucleus appears as a loss in mass, Δm,
which can be calculated using E = mc2
http://slideplayer.com
/slide/8701359/
Heinemann Physics 12 Questions
• 7.5 Review page 261
• Questions 4 – 10 (questions 1 -3 relate to
relativistic momentum, which is not
mentioned in the new Study Design)
Data Sheet References
Sources of Sample Questions
• Text book (Heinemann Physics 12 4th Edition Chapters 6
& 7 have a good selection)
• Heinemann Physics 12 3rd Edition has a good selection
of multiple choice questions in Chapter 6 (from the old
Relativity Option)
• Sample and Trial exam papers (2005 -2016 Option,
2017 core)
• Other textbooks (Such as Giancoli)
• Exam papers from alternative courses (IB, GCSE, etc.)
Other aspects to discuss but not
specified for the VCE course
• Twin paradox: (Minute Physics)
https://www.youtube.com/watch?v=Bg9MVRQY
mBQ
• Electricity and magnetism: (Veratisium)
https://www.youtube.com/watch?v=1TKSfAkW
WN0&t=14s
Contact me….
Email: [email protected]