Senior High School
NOT
General Physics 2
Quarter 4 – Module 5
Relativity
E = mc2
𝛾∆𝑡0
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General Physics 2 - Grade 12
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Quarter 4 - Module 5: Relativity
First Edition, 2020
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Senior High School
General Physics 2
Quarter 4 – Module 5
Relativity
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Table of Contents
What This Module is About ........................................................................................................... i
What I Need to Know ..................................................................................................................... i
How to Learn from this Module ................................................................................................... ii
Icons of this Module ...................................................................................................................... ii
What I Know.................................................................................................................................. iii
FOURTH QUARTER
Lesson 1: Postulates of Relativity
What Is It: Einstein’s Postulates ....................................................................... 1
What I Have Learned: ........................................................................................ 2
What’s More: ....................................................................................................... 2
Lesson 2: Relativity of Time, Length, and Mass
What Is It: Simultaneity and Time Dilation ..................................................... .. 3
What I Have Learned: ...................................................................................... 10
What Is It: Relativity of Length ........................................................................ 11
What Is It: Relativity of Mass ........................................................................... 12
Lesson 3: Relativistic Dynamics
What Is It: Relativistic Velocity ........................................................................ 14
What Is It: Relativistic Momentum ................................................................... 15
What Is It: Relativistic Energy and Rest Energy ............................................ 16
What I Have Learned: ...................................................................................... 16
Summary ……………………………………………………………………………………….. 17
Key to Answers
References
What This Module is About
This module demonstrates your understanding on the concepts of Relativity. It
specifically discusses about Postulates of Relativity, Relativity of Time, Length, and
Mass, and Relativistic Dynamics.
This module will help you explore the key concepts on topics that will help you
answer the questions pertaining to the special theory of relativity.
This module has three (3) lessons:
•
•
•
Lesson 1 – Postulates of Relativity
Lesson 2 – Relativity of Time, Length, and Mass
Lesson 3 – Relativistic Dynamics
What I Need to Know
At the end of this module, you should be able to:
1. State the postulates of Special Relativity and their consequences
STEM_GP12MP-IVg-39;
2. Apply the time dilation, length contraction, and relativistic velocity addition to worded
problems
STEM_GP12MP-IVg-40-41; and
3. Calculate kinetic energy, rest energy, momentum, and speed of objects moving with
speeds comparable to the speed of light
STEM_GP12MP-IVg-42
i
How to Learn from this Module
To achieve the objectives cited above, you are to do the following:
•
Take your time reading the lessons carefully.
•
Follow the directions and/or instructions in the activities and exercises diligently.
•
Answer all the given tests and exercises.
Icons of this Module
What I Need to
This part contains learning objectives that
Know
are set for you to learn as you go along the
module.
What I know
This is an assessment as to your level of
knowledge to the subject matter at hand,
meant specifically to gauge prior related
knowledge
This part connects previous lesson with that
What’s In
of the current one.
What’s New
An introduction of the new lesson through
various activities, before it will be presented
to you
What is It
These are discussions of the activities as a
way to deepen your discovery and understanding of the concept.
What’s More
These are follow-up activities that are intended for you to practice further in order to
master the competencies.
What I Have
Activities designed to process what you
Learned
have learned from the lesson
What I can do
These are tasks that are designed to showcase your skills and knowledge gained, and
applied into real-life concerns and situations.
ii
What I Know
.
A. Matching Type
Match column A with column B. Write only the letter of the best answer on the space before
each number.
Column A
1. It is the study of the interactions of matter
and energy in the universe.
2. It is the remnant of a supernova.
3. It is a theory of relativity.
4. Its energy output is hundred times that of
the solar system.
5. It is an exploding supergiant.
6. He formulated the theory of relativity.
7. It is a huge cloud of gas and dust in space.
8. It determines the nature of stars.
9. It is an extremely dense object from which
no light could escape.
10. It is a large cool star which emits red light.
Column B
A. nebula
B. black hole
C. Einstein
D. pulsar
E. quasar
F. supernova
G. E = mc2
H. spectroscope
I. red giant
J. neutron star
K. astrophysics
B. Choose the letter of the best answer.
1. Which of the following is NOT a postulate of the special theory of relativity?
A. Relativity Postulate
B. speed of light postulate
C. Energy Postulate
2. Which of the following physical properties does NOT change when speed of objects
approaches the speed of light?
A. length
B. mass
C. time
D. none of the above because all will change
3. Who were the two American scientists who tried to detect the existence of ether
experimentally?
A. Michelson and Newton B. Michelson and Einstein
C. Michelson and Morley D. Michelson and Graham
4. When an object moves at a much greater speed (closer to the speed of light), its mass
increases.
A. True
B. False
5. When an object moves at a much greater speed (closer to the speed of light), its length
parallel to its direction of motion will be observed to be shorter.
A. True
B. False
iii
Lesson
1
Postulates of Relativity
What is it
When the year 1905 began, Albert Einstein was an unknown 25-year-old clerk in the
Swiss patent office. By the end of that amazing year he had published three papers of
extraordinary importance. One was an analysis of Brownian motion; a second (for which he
was awarded the Nobel Prize) was on the photoelectric effect. In the third, Einstein introduced
his special theory of relativity, proposing drastic revisions in the Newtonian concepts of
space and time.
The special theory of relativity has made wide-ranging changes in our understanding
of nature, but Einstein based it on just two simple postulates. One states that the laws of
physics are the same in all inertial frames of reference; the other states that the speed of light
in vacuum is the same in all inertial frames. These innocent-sounding propositions have farreaching implications. Here are three: (1) Events that are simultaneous for one observer may
not be simultaneous for another. (2) When two observers moving relative to each other
measure a time interval or length, they may not get the same results. (3) For the conservation
principles for momentum and energy to be valid in all inertial systems, Newton’s second law
and the equations for momentum and kinetic energy have to be revised.
Relativity has important consequences in all areas of physics, including
electromagnetism, atomic and nuclear physics, and high-energy physics.
1.1 Einstein’s First Postulate
Einstein’s first postulate, called the Principle of Relativity, states that “The laws of
physics are the same in every inertial frames of reference.” If the laws differed, that
difference could distinguish one inertial frame from the others or make one frame somehow
more “correct” than the other. Here is an example. Suppose you watch two children playing
catch with a ball while the three of you are aboard a train moving with constant velocity. Your
observations of the motion of the ball, no matter how carefully done, can’t tell you how fast (or
whether) the train is moving. This is because Newton’s laws of motion are the same in every
inertial frame.
Another example is if you are in a bus moving with constant velocity and you throw a
ball up, it will simply fall down on your lap in free fall motion. But an observer outside the bus,
say, on the street across the moving bus, will observe the ball as a projectile which was thrown
at an angle from the horizontal.
1
1.2 Einstein’s Second Postulate
During the 19th century, most physicists believed that light traveled through a
hypothetical medium called the ether (a hypothetical medium pervading the universe in which
light waves were supposed to travel), just as sound waves travel through air. If so, the speed
of light measured by observers would depend on their motion relative to the ether and would
therefore be different in different directions. The Michelson-Morley experiment (using light
beams and half-silvered mirror) was an effort to detect motion of the earth relative to the ether.
The results of the experiment show that no matter which direction the beams of light were
aimed, they always bounce back at exactly the same instant. Therefore, there is no ether at
all and so there is no such thing as “absolute motion” relative to the ether. The result also
shows that the speed of light is the same for all observers, which is not true of waves (such
as sound waves and water waves) that need a material medium in which to occur.
Einstein’s second postulate states that “The speed of light in vacuum is the same
in all inertial frames of reference and is independent of the motion of the source”. This
postulate implies that “It is impossible for an inertial observer to travel at c, the speed of
light in vacuum.”
To illustrate this statement, consider a rocket which is launched from a space station.
Light is emitted from the station at 300,000 km/s, or c. Regardless of the velocity of the rocket,
an observer in the rocket sees the flash of light pass her at the same speed c. If a flash is sent
to the station from the rocket, observers in the station will measure the speed of light to be c
also. Thus, it could be inferred that all observers who measure the velocity of light will find it
to have the same value c. When you look at the stars, you are actually looking backward in
time. The farthest stars that you see in the sky are actually the stars you may have seen long
ago.
Using the two postulates, Einstein was able to prove mathematically that Newtonian
laws are for objects at rest or moving at very low speeds. But when speeds involved are
comparable to that of light, as in the case of atomic particles, there are corresponding changes
in the physical properties.
What I Have Learned
Answer the following questions thoroughly.
1. What is an inertial frame of reference?
2. An accelerated frame is a noninertial frame. Is this statement true? Explain.
What’s More
Make your own example to each of the two postulates of relativity and make a
justification/discussion on the examples that you have formulated. Write this on a one whole
sheet of paper.
2
Lesson
2
Relativistic Time, Length, and
Mass
What is it
2.1 Simultaneity and Time Dilation
Do time intervals depend on who observes them? Intuitively, we expect the time for a
process, such as the elapsed time for a foot race, to be the same for all observers. Our
experience has been that disagreements over elapsed time have to do with the accuracy of
measuring time. When we carefully consider just how time is measured, however, we will find
that elapsed time depends on the relative motion of an observer with respect to the process
being measured.
Measuring times and time intervals involve the concept of simultaneity. In a given
frame of reference, and event is an occurrence that has a definite position and time. When
you say that you arrived school at 7:15 of the clock, you mean that the two events (your arriving
and your clock showing 7:15) occurred simultaneously. The fundamental problem in
measuring time intervals is this: In general, two events that are simultaneous in one frame of
reference are not simultaneous in a second frame of reference that is moving relative to the
first, even if both are inertial frames.
2.1.1 Simultaneity
Consider how we measure elapsed time. If we use a stopwatch, for example, how do
we know when to start and stop the watch? One method is to use the arrival of light from the
event, such as observing a light turning green to start a drag race. The timing will be more
accurate if some sort of electronic detection is used, avoiding human reaction times and other
complications.
Now suppose we use this method to measure the time interval between two flashes of
light produced by flash lamps. (See Figure 2.1.) Two flash lamps with observer A midway
between them are on a rail car that moves to the right relative to observer B. The light flashes
are emitted just as A passes B, so that both A and B are equidistant from the lamps when the
light is emitted. Observer B measures the time interval between the arrival of the light flashes.
According to postulate 2, the speed of light is not affected by the motion of the lamps relative
to B. Therefore, light travels equal distances to him at equal speeds. Thus, observer B
measures the flashes to be simultaneous.
3
Figure 2.1. Observer B measures the elapsed time between the arrival of light flashes as described in the text.
Observer A moves with the lamps on a rail car. Observer B receives the light flashes simultaneously, but he notes
that observer A receives the flash from the right first. B observes the flashes to be simultaneous to him but not to
A. Simultaneity is not absolute.
Now consider what observer B sees happen to observer A. She receives the light from
the right first, because she has moved towards that flash lamp, lessening the distance the light
must travel and reducing the time it takes to get to her. Light travels at speed c relative to both
observers, but observer B remains equidistant between the points where the flashes were
emitted, while A gets closer to the emission point on the right. From observer B’s point of view,
then, there is a time interval between the arrival of the flashes to observer A. Observer B
measures the flashes to be simultaneous relative to him but not relative to A. Here a relative
velocity between observers affects whether two events are observed to be
simultaneous. Simultaneity is not absolute.
This illustrates the power of clear thinking. We might have guessed incorrectly that if
light is emitted simultaneously, then two observers halfway between the sources would see
the flashes simultaneously. But careful analysis shows this not to be the case. Einstein was
brilliant at this type of thought experiment (in German, “Gedankenexperiment”). He very
carefully considered how an observation is made and disregarded what might seem obvious.
The validity of thought experiments, of course, is determined by actual observation. The
genius of Einstein is evidenced by the fact that experiments have repeatedly confirmed his
theory of relativity.
4
In summary: Two events are defined to be simultaneous if an observer measures them as
occurring at the same time (such as by receiving light from the events). Two events are not
necessarily simultaneous to all observers.
2.1.2 Time Dilation
The consideration of the measurement of elapsed time and simultaneity leads to an important
relativistic effect, which is the time dilation.
Time dilation is the phenomenon of time passing slower for an observer who is moving relative
to another observer.
Suppose, for example, an astronaut measures the time it takes for light to cross her
ship, bounce off a mirror, and return. (See Figure 2.2.) How does the elapsed time the
astronaut measures compare with the elapsed time measured for the same event by a person
on the Earth? Asking this question (another thought experiment) produces a profound result.
We find that the elapsed time for a process depends on who is measuring it. In this case, the
time measured by the astronaut is smaller than the time measured by the Earth-bound
observer. The passage of time is different for the observers because the distance the light
travels in the astronaut’s frame is smaller than in the Earth-bound frame. Light travels at the
same speed in each frame, and so it will take longer to travel the greater distance in the Earthbound frame.
Figure 2.2. (a) An astronaut measures the time Δt0 for light to cross her ship using an electronic timer. Light travels
a distance 2D in the astronaut’s frame. (b) A person on the Earth sees the light follow the longer path 2s and take
a longer time Δt. (c) These triangles are used to find the relationship between the two distances 2D and 2s.
5
To quantitatively verify that time depends on the observer, consider the paths followed
by light as seen by each observer. (See Figure 2.2c.) The astronaut sees the light travel
straight across and back for a total distance of 2D, twice the width of her ship. The Earthbound observer sees the light travel a total distance 2s. Since the ship is moving at speed v to
the right relative to the Earth, light moving to the right hits the mirror in this frame. Light travels
at a speed c in both frames, and because time is the distance divided by speed, the time
measured by the astronaut is
∆𝑡0 =
2𝐷
Eqn. 2.1
𝑐
This time has a separate name to distinguish it from the time measured by the Earth-bound
observer.
2.1.3 Proper Time
Proper time Δt0 is the time measured by an observer at rest relative to the event being
observed.
In the case of the astronaut observe the reflecting light, the astronaut measures proper time.
The time measured by the Earth-bound observer is
2𝑠
∆𝑡 =
Eqn. 2.2
𝑐
To find the relationship between Δt0 and Δt, consider the triangles formed by D and s. (See
Figure 2c.) The third side of these similar triangles is L, the distance the astronaut moves as
the light goes across her ship. In the frame of the Earth-bound observer,
𝐿=
𝑣∆𝑡
Eqn. 2.3
2
Using the Pythagorean Theorem, the distance s is found to be
s = √𝐷 2 + (
𝑣∆𝑡 2
)
2
Substituting s into the expression for the time interval Δt gives
𝑣∆𝑡 2
2
2𝑠 2√𝐷 + ( 2 )
∆𝑡 =
=
𝑐
𝑐
We square this equation, which yields
(∆𝑡)2 =
𝑣 2 (∆𝑡)2
) 4𝐷 2 𝑣 2
4
= 2 + 2 (∆𝑡)2
𝑐2
𝑐
𝑐
4(𝐷 2 +
6
Note that if we square the first expression we had for Δt0, we get
4𝐷 2
𝑐2
(∆𝑡0 )2 =
This term appears in the preceding equation, giving us a means to relate the two time intervals.
Thus,
(∆𝑡)2 = (∆𝑡0 )2 +
𝑣2
(∆𝑡)2
𝑐2
Gathering terms, we solve for Δt:
(∆𝑡)2 (1 −
𝑣2
) = (∆𝑡0 )2
𝑐2
Thus,
(∆𝑡)2 =
(∆𝑡0 )2
𝑣2
1− 2
𝑐
Taking the square root yields an important relationship between elapsed times:
∆𝑡 =
∆𝑡0
2
√1−𝑣2
= 𝛾∆𝑡0
Eqn. 2.4
𝑐
where
𝛾=
1
2
√1−𝑣2
𝑐
Eqn. 2.5
This equation for Δt is truly remarkable. First, as contended, elapsed time is not the
same for different observers moving relative to one another, even though both are in inertial
frames. Proper time Δt0 measured by an observer, like the astronaut moving with the
apparatus, is smaller than time measured by other observers. Since those other observers
measure a longer time Δt, the effect is called time dilation. The Earth-bound observer sees
time dilate (get longer) for a system moving relative to the Earth. Alternatively, according to
the Earth-bound observer, time slows in the moving frame, since less time passes there. All
clocks moving relative to an observer, including biological clocks such as aging, are observed
to run slow compared with a clock stationary relative to the observer.
Note that if the relative velocity is much less than the speed of light (v << c),
then v2/c2 is extremely small, and the elapsed times Δt and Δt0 are nearly equal. At low
velocities, modern relativity approaches classical physics—our everyday experiences have
very small relativistic effects.
7
The equation Δt = γΔt0 also implies that relative velocity cannot exceed the speed of
light. As v approaches c, Δt approaches infinity. This would imply that time in the astronaut’s
frame stops at the speed of light. If v exceeded c, then we would be taking the square root of
a negative number, producing an imaginary value for Δt.
Example
Suppose a cosmic ray colliding with a nucleus in the Earth’s upper atmosphere
produces a muon that has a velocity v = 0.950c. The muon then travels at constant velocity
and lives 1.52 μs as measured in the muon’s frame of reference. (You can imagine this as the
muon’s internal clock.) How long does the muon live as measured by an Earth-bound
observer? (See Figure 3.)
Figure 3. A muon in the Earth’s atmosphere lives longer as
measured by an Earth-bound observer than measured by the
muon’s internal clock.
Analysis
A clock moving with the system being measured
observes the proper time, so the time we are given
is Δt0 = 1.52 μs. The Earth-bound observer
measures Δt as given by the equation Δt = γΔt0.
Since we know the velocity, the calculation is
straightforward.
Solution
Given: v = 0.950c, Δt0 = 1.52 μs
Unknown: Δt
To solve for the unknown, Δt, let us use Eqn. 2.4.
Using Eqn. 2.5 to solve for 𝛾, it will give us 𝛾 = 3.20.
Use the calculated value of γ to determine Δt.
Δt = γΔt0 = (3.20)(1.52μs) = 4.87μs
Discussion:
One implication of this example is that since γ = 3.20 at 95.0% of the speed of light
(v = 0.950c), the relativistic effects are significant. The two time intervals differ by this factor
of 3.20, where classically they would be the same. Something moving at 0.950c is said to be
highly relativistic.
8
Another implication of the preceding example is that everything an astronaut does
when moving at 95.0% of the speed of light relative to the Earth takes 3.20 times longer when
observed from the Earth. Does the astronaut sense this? Only if she looks outside her
spaceship. All methods of measuring time in her frame will be affected by the same factor of
3.20. This includes her wristwatch, heart rate, cell metabolism rate, nerve impulse rate, and
so on. She will have no way of telling, since all of her clocks will agree with one another
because their relative velocities are zero. Motion is relative, not absolute. But what if she does
look out the window?
REAL WORLD CONNECTIONS
It may seem that special relativity has little effect on your life, but it is probably
more important than you realize. One of the most common effects is through the Global
Positioning System (GPS). Emergency vehicles, package delivery services, electronic
maps, and communications devices are just a few of the common uses of GPS, and
the GPS system could not work without taking into account relativistic effects. GPS
satellites rely on precise time measurements to communicate. The signals travel at
relativistic speeds. Without corrections for time dilation, the satellites could not
communicate, and the GPS system would fail within minutes.
2.1.4 The Twin Paradox
An intriguing consequence of time dilation is that a space traveler moving at a high
velocity relative to the Earth would age less than her Earth-bound twin. Imagine the astronaut
moving at such a velocity that γ = 30.0, as in Figure 5. A trip that takes 2.00 years in her frame
would take 60.0 years in her Earth-bound twin’s frame. Suppose the astronaut traveled 1.00
year to another star system. She briefly explored the area, and then traveled 1.00 year back.
If the astronaut was 40 years old when she left, she would be 42 upon her return. Everything
on the Earth, however, would have aged 60.0 years. Her twin, if still alive, would be 100 years
old.
The situation would seem different to the astronaut. Because motion is relative, the
spaceship would seem to be stationary and the Earth would appear to move. (This is the
sensation you have when flying in a jet.) If the astronaut looks out the window of the spaceship,
she will see time slow down on the Earth by a factor of γ = 30.0. To her, the Earth-bound sister
will have aged only 2/30 (1/15) of a year, while she aged 2.00 years. The two sisters cannot
both be correct.
9
Figure 4. The twin paradox asks why the traveling twin ages
less than the Earth-bound twin. That is the prediction we
obtain if we consider the Earth-bound twin’s frame. In the
astronaut’s frame, however, the Earth is moving and time
runs slower there. Who is correct?
As with all paradoxes, the premise is faulty and
leads to contradictory conclusions. In fact, the
astronaut’s motion is significantly different from
that of the Earth-bound twin. The astronaut
accelerates to a high velocity and then
decelerates to view the star system. To return to
the Earth, she again accelerates and decelerates.
The Earth-bound twin does not experience these
accelerations. So the situation is not symmetric,
and it is not correct to claim that the astronaut will observe the same effects as her Earthbound twin. If you use special relativity to examine the twin paradox, you must keep in mind
that the theory is expressly based on inertial frames, which by definition are not accelerated
or rotating. Einstein developed general relativity to deal with accelerated frames and with
gravity, a prime source of acceleration. You can also use general relativity to address the twin
paradox and, according to general relativity, the astronaut will age less. Some important
conceptual aspects of general relativity are discussed in General Relativity and Quantum
Gravity of this course.
In 1971, American physicists Joseph Hafele and Richard Keating verified time dilation
at low relative velocities by flying extremely accurate atomic clocks around the Earth on
commercial aircraft. They measured elapsed time to an accuracy of a few nanoseconds and
compared it with the time measured by clocks left behind. Hafele and Keating’s results were
within experimental uncertainties of the predictions of relativity. Both special and general
relativity had to be taken into account, since gravity and accelerations were involved as well
as relative motion.
What I Have Learned
Solve the following problems clearly and completely. Write your solution on a one-whole piece
of paper.
1. What is γ if v = 0.650c?
2. Particles called π-mesons are produced by accelerator beams. If these particles travel at
2.70 × 108 m/s and live 2.60 × 10−8 s when at rest relative to an observer, how long do they
live as viewed in the laboratory?
3. Suppose a particle called a kaon is created by cosmic radiation striking the atmosphere. It
moves by you at 0.980c, and it lives 1.24 × 10−8 s when at rest relative to an observer. How
long does it live as you observe it?
4. A neutral π-meson is a particle that can be created by accelerator beams. If one such
particle lives 1.40 × 10−16 s as measured in the laboratory, and 0.840 × 10−16 s when at rest
relative to an observer, what is its velocity relative to the laboratory?
5. If relativistic effects are to be less than 1%, then γ must be less than 1.01. At what relative
velocity is γ = 1.01?
6. (a) At what relative velocity is γ = 1.50? (b) At what relative velocity is γ = 100?
10
2.2 Relativity of Length
We have discussed that because of time dilation, observers moving at a constant
velocity relative to each other measure different time intervals. The question now is whether
the observers measure different distances between the Earth and a distant galaxy, say, Alpha
Centauri. According to the special theory of relativity, the answer is yes. The distances
measured by the observers from Earth and those at Alpha Centauri can be calculated using
the following equation:
2
𝐿
𝐿 = 𝐿0 √1 − 𝑣 ⁄𝑐 2 = 0
𝛾
Eqn. 2.6
where: L0 = proper length and L = contracted length.
The proper length L0 is the length (or distance) between two points as measured by observers
at rest with respect to them. Since v is smaller than c, the value of the radicand is less than 1
so is the value of the square root and L is less than L0. Note that length contraction occurs
only along the direction of motion. Distances perpendicular to the motion is not shortened.
Example:
A spaceship flies past earth at a speed of 0.990c. A crew member on board the
spaceship measures its length, obtaining the value 400 m. What length do observers
measure on earth?
Solution:
Given: L0 = 400 m; v = 0.990c
Using Eqn. 2.6 we can solve for the length of the spaceship as measured by the observer on
earth.
2
𝐿 = 𝐿0 √1 − 𝑣 ⁄𝑐 2 = (400 𝑚)√1 −
(0.990𝑐 )2⁄
𝑐 2 = 56.43 𝑚
This answer makes sense: The spaceship is shorter in a frame of reference in which it is in
motion than in a frame in which it is at rest.
11
2.3 Relativity of Mass
You have learned that the following physical quantities are classified as the
fundamental quantities of measure: time, length, and mass. The mass of the body is believed
to be constant wherever it is taken and we consider it also the same whether the body is in
motion or at rest. However, Einstein considers the mass of a moving body not constant. Why
is this so can be understood by the law of acceleration. Recall that acceleration depends not
only on force but also on the mass of the object as well. Einstein believed that when work is
done on the object to increase its velocity, its mass increases as well. So, the force produces
less and less acceleration as velocity increases. The relationship between mass and velocity
is given in the following equation:
𝑚=
𝑀0
2
√1−𝑣 ⁄ 2
𝑐
= 𝛾𝑀0
Eqn. 2.7
where: m = the mass of the body when it is in motion, and
M0 = mass of the body when it is at rest
Example:
What is the mass of the electron traveling at half the speed of light? (Mass of an
electron at rest is 9.11 x 10-31 kg.)
Solution:
Given: M0 = 9.11 x 10-31 kg; v = 0.500c
Using Eqn. 2.7 we have
𝑚=
𝑀0
=
2
√1 − 𝑣 ⁄ 2
𝑐
9.11 𝑥 10−31 𝑘𝑔
√1 −
(0.500𝑐)2
= 1.05 𝑥 10−30 𝑘𝑔
⁄ 2
𝑐
2.3.1 Mass and Energy Relation
The famous Einstein equation that expresses the relationship between mass and
energy is E = mc2; where E stands for energy, m for mass, and c for the speed of light. The
equation further implies that mass and energy are not the same thing.
Mass can be changed into energy and energy can be changed into mass. For example,
when a nucleus of U-235 undergoes fission, the combined mass of the fission products is less
than the mass of the original uranium nucleus. Some mass has disappeared, and in its place
is an equivalent amount of energy in the form of kinetic energy of the moving fission products.
No protons or neutrons are destroyed during fission. This means that the total number of
protons and neutrons are the same before and after fission has occurred. However, there is a
rearrangement of protons and neutrons after the fission reaction and the arrangement have
different masses. It is the mass difference that appears as energy.
12
ASTROPHYSICS
In the early part of the nineteenth century, Robert Bunsen introduced the study of
materials by spectrum analysis, and Kichhoff investigated the meaning of the dark
lines in the solar spectrum. This investigation led to the study of the physical and
chemical constitution of heavenly bodies. This branch of astronomy is called
astrophysics. It is the astronomical study of the interactions of the matter-energy of the
universe with space-time.
How stars are formed?
People think that the space between stars is empty. Actually, space contains much gas
and dust. Large amounts of gas and dust collect and form stars. The huge cloud of gas and
dust is called a nebula. The clouds are believed to be at least a light year in diameter and as
massive as a thousand suns. The gravitational attraction draws the gas and dust together. As
the matter in the clustered mass concentrates, gravity between the particles increase. The
gas becomes more and more compressed as this process continues. The material in the
center of the mass becomes very hot. The temperature rises above 1,000,0000C. Atoms in
the gas split into nuclei and electrons. Some of the nuclei pass so close together that they
combine through nuclear fusion. This process causes a high pressure in the center of the star.
When pressure pushing out balances the gravity pulling the gas in, a star is born.
Black Hole
After a supernova (the stellar explosion), gravity causes stars to collapse completely. The
star’s mass is tightly packed to a small space as to allow it to have extremely strong gravity.
Astronomers think the pull of gravity from such an object would be strong enough to capture
anything, even light escaping from the dying star. Since no light could escape, we would not
see the object. It is called a black hole.
Quasars
Black holes can be considered the dimmest objects in the universe, but the brightest
objects could be the quasars. The energy output of these objects is tremendous – hundreds
of times than that of the entire solar system. Quasars were first believed to be just an ordinary
star in our galaxy, but in 1960 they were discovered to be emitting radio waves. Further
investigation revealed that these “radio stars” have patterns of spectral lines that could not be
interpreted. These objects become known as “Quasi-Stellar Sources.” In short, they are called
quasars.
13
Lesson
3
Relativistic Dynamics
What is it
3.1 Relativistic Velocity
In the special theory of relativity, it is important to know the velocity of an object relative
to an observer. It plays a very important role in attaining the effects of time dilation, length
contraction, relativistic momentum, and energy-mass transformation.
If we are to determine the velocity of an object relative to that of an observer, oftentimes
we have to add two or more velocities together in an equation. According to the special theory
of relativity, the velocities are related according to the velocity-addition equation:
𝑢=
𝑢′ +𝑣
𝑢′ 𝑣
Eqn. 3.1
1+ 2
𝑐
where u = the velocity of the object as measured by an observer on Earth,
u’ = the velocity of the object as measured by an observer in the moving frame
which itself is moving at a velocity, v, relative to Earth.
For motions along a straight line, the signs of the velocities can be positive or negative
provided they are directed along the positive or negative direction.
Example:
A car is approaching an observer on Earth with a velocity v = 0.85c. A person in the
car throws a ball towards the observer at a velocity of u’ = 0.60c relative to the car. At what
velocity does the observer on Earth see the ball approaching?
Solution:
Using Eqn. 3.1 we have,
𝑢=
𝑢′ + 𝑣
0.60𝑐 + 0.85𝑐
=
= 0.96𝑐
′
𝑢𝑣
(0.60𝑐)(0.85𝑐)
1+ 2
1+
𝑐
𝑐2
14
3.2 Relativistic Momentum
Special theory of relativity tells us that time, mass, and length are measured relative to
an observer. The theory also alters our understanding about momentum and energy.
According to the conservation of linear momentum principle, the total momentum of an isolated
system remains constant at all times. This principle is a law of physics; therefore, it is in
accordance with the postulate of relativity and is valid in all inertial frames of reference. The
total momentum of a system is conserved in an inertial frame of reference, as long as the
speeds of the objects do not approach the speed of light. However, when the speeds approach
the speed of light, the total linear momentum is not conserved in all inertial frames of reference
if one defines momentum as the product of mass and velocity. This is given in the following
equation:
𝑝′ =
𝑚𝑣
Eqn. 3.2
2
√1−𝑣 ⁄ 2
𝑐
Notice that the relativistic momentum differs from the non-relativistic momentum by a
2
factor of √1 − 𝑣 ⁄ 2 that is present in the time dilation and length contraction equations. Eqn.
𝑐
3.2 shows us that the relativistic momentum is always greater than the non-relativistic
momentum.
Example:
A particle accelerator is three kilometers long and accelerates electrons at the speed
of 0.999c, which is very nearly the speed of light. Calculate the relativistic momentum
emerging from the accelerator and compare its value with the nonrelativistic momentum (mass
of electron, me = 9.11 x 10-31 kg).
Solution:
Given: v = 0.999c
Unknown: relativistic momentum, p’
Using Eqn. 3.2 we have
′
𝑝 =
𝑚𝑣
2
√1 − 𝑣 ⁄ 2
𝑐
=
=
(9.11 𝑥 10−31 𝑘𝑔)(0.999𝑐 )
2
√1 − (0.999𝑐 ) ⁄ 2
𝑐
(9.02 𝑥 10−31 𝑘𝑔) 𝑐
= (4.53 𝑥 10−29 ) 𝑐 kg • m/s
0.0199
This value is 5,000 times greater than the nonrelativistic momentum, mv.
15
3.3 Relativistic Energy and Rest Energy
In section 2.3.1 we have discussed the mass and energy relation which says that a
gain or a loss of mass can be regarded as a gain or a loss of energy, and vice versa. According
to Einstein, an object of mass m traveling with a velocity v will have a total energy that is
related to its mass and speed. This is given by the following equation:
𝑚𝑐 2
𝐸=
2
√1 − 𝑣 ⁄ 2
𝑐
This is called the relativistic energy of the object. If the object is at rest, the equation
reduces to the famous 𝐸 = 𝑚𝑐 2 , which is called as the rest energy, E0. This rest energy
represents the energy equivalent of the mass of an object at rest.
What I Have Learned
1. A rocket ship has a mass of 1.50 x 10 5 kg. Its relativistic momentum is 3.00 x 1012 kg•m/s.
How fast is the rocket ship traveling?
2. How fast would a meterstick be moving so that its length will be observed to shrink to half
its original length?
3. How much work must be done to a proton to accelerate it from rest to 0.998c.
4. A person on Earth observes a space ship approaching from the right with a velocity of 0.75c
and a rocket ship approaching from the left at 0.50c. What is the relative velocity of the two
ships as measured by a passenger in each of them?
16
Summary
•
There are two postulates of the special theory of relativity:
a) The relativity postulate states that the laws of physics are the same in every inertial
frame of reference.
b) The speed of light postulate states that the speed of light in a vacuum, measured
in any inertial frame of reference, always has the same value of c no matter how fast
the source of light and the observers are moving relative to each other.
•
Two events are defined to be simultaneous if an observer measures them as occurring
at the same time. They are not necessarily simultaneous to all observers—simultaneity
is not absolute.
•
Time dilation is the phenomenon of time passing slower for an observer who is moving
relative to another observer.
•
Observers moving at a relative velocity v do not measure the same elapsed time for
an event. Proper time Δt0 is the time measured by an observer at rest relative to the
event being observed. Proper time is related to the time Δt measured by an Earthbound observer by the equation
∆𝑡 =
∆𝑡0
= 𝛾∆𝑡0
2
where
√1−𝑣2
𝑐
𝛾=
1
2
√1−𝑣2
𝑐
•
The equation relating proper time and time measured by an Earth-bound observer
implies that relative velocity cannot exceed the speed of light.
•
The twin paradox asks why a twin traveling at a relativistic speed away and then back
towards the Earth ages less than the Earth-bound twin. The premise to the paradox is
faulty because the traveling twin is accelerating. Special relativity does not apply to
accelerating frames of reference.
•
Time dilation is usually negligible at low relative velocities, but it does occur, and it has
been verified by experiment.
•
The proper length is the length between two points measured by an observer who is
at rest relative to the points. The relativistic length is given by the following equation:
𝐿 = 𝐿0 √1 −
•
The relativistic momentum p’ of an object of mass m and speed v is given by
𝑝′ =
•
𝑚𝑣
2
√1−𝑣 ⁄ 2
𝑐
The total energy of an object with mass m and speed v is given by
𝐸=
•
𝑣2
𝑐2
𝑚𝑐 2
2
√1−𝑣 ⁄ 2
𝑐
The speed of an object with a given mass cannot reach the speed of light c.
17
What I Know
A. Matching Type
1. K
2. J
3. G
4. E
5. F
6. C
7. A
8. H
9. B
10. I
B. Multiple Choice
1. C
2. D
3. C
4. A
5. A
Key to Answers
References:
Sears and Zemansky’s University Physics with Modern Physics Technology Update by Hugh
D. Young and Roger A. Freedman, 13th edition, pp. 1349-1388
The Basics of Physics by Arsenia V. Ferrer and Julieta dela Peña
Practical and Explorational Physics by Alicia L. Padua and Ricardo M. Crisostomo
https://courses.lumenlearning.com/physics/chapter/28-2-simultaneity-and-timedilation/
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