The document provides background information on Einstein's special theory of relativity. It discusses the two postulates of special relativity: 1) the principle of relativity, and 2) the constancy of the speed of light. It then summarizes some key consequences of special relativity, including time dilation, length contraction, relativistic Doppler effect, relativistic mass, mass-energy equivalence, and Lorentz transformations. Examples are provided to demonstrate calculations for these various consequences.
General Theory of Relativity is the geometric theory of gravitation published by Albert Einstein in 1915 and the current description of gravitation in modern physics.
General Theory of Relativity is the geometric theory of gravitation published by Albert Einstein in 1915 and the current description of gravitation in modern physics.
When we say something is moving, what we mean is that its position relative to something else is changing. So there is a frame of refernce. An inertial frame of reference is one which moves with a constant velocity. In such a reference frame, Newtons’s first law of motion holds true.
Einstein’s Theories of Relativity revolutionized how Today’s Scientific world thinks about Space, Time, Mass, Energy and Gravity. This is purely an imaginative Science that worked in the Laboratory of Einstein's Brain..
When we say something is moving, what we mean is that its position relative to something else is changing. So there is a frame of refernce. An inertial frame of reference is one which moves with a constant velocity. In such a reference frame, Newtons’s first law of motion holds true.
Einstein’s Theories of Relativity revolutionized how Today’s Scientific world thinks about Space, Time, Mass, Energy and Gravity. This is purely an imaginative Science that worked in the Laboratory of Einstein's Brain..
It should be helpful, special thanks to our teacher (whose name is in the power point and the one who made it) from whom I asked his permission to post it here.
Chapters
Reminders: light
speed of light in a vacuum
A brief historical reminder of the speed of light
Invariance of the speed of light in a vacuum
Influence of the propagation medium
Speed or celerity?
Speed, distance traveled, and duration
Relations including the speed of light
Faster than light?
Speed of light: did you know?
Reminders: light
Light is an electromagnetic wave, consisting of a magnetic field and an electric field oscillating perpendicular to each other in a plane perpendicular to the direction of propagation of the light wave. In a vacuum, light travels in a straight line at the speed of light noted c.
speed of light in a vacuum
Exact value
The exact value of the speed of light was fixed in 1983 by the Bureau of Weights and Measures at c = 299 792 458 m/s or c = 2.99792458 x 10 8 m/s, using the units of the international system. It can also be expressed in kilometers per hour by multiplying the value in m/s by 3.6: c = 1,079,252,848.8 km/h or c = 1.0792528488 x 10 9 km/h. This value, which represents a fundamental constant of physics, can be used for calculations requiring great precision. It is also used to define the meter in the international system of units: one meter corresponds to the length traveled in a vacuum by light for a duration of 1/299,792,458 seconds.
A brief historical reminder of the speed of light
The first conception concerning light suppose that it can be either present in a space, or absent: the light would therefore be instantaneous. The Arab scholar Alhazen (965-1039) was interested in optics and wrote reference treatises. He is the first to have the intuition that the appearance of light is not instantaneous, that it has a speed of propagation, but he cannot prove it.
Galileo (1564-1039) tries to measure the propagation time of light between two hills using two people a few kilometers apart and equipped with clocks. He fails to measure the speed of light (which, in the context of this experiment, takes 10 -5 seconds to travel the previously defined distance, not measurable for the time) and deduces from the failure of this experiment that the speed of propagation of light is very high.
Cassini (1625-1712) speculated that the irregularity in the movement of Io, a satellite of Jupiter, could come from a delay in the arrival of light from the satellite, "such that it takes 10 or 11 minutes for it travels a distance equal to the radius of the Earth's orbit". Römer (1644-1710) explains the discrepancy between the eclipses of Io (a satellite of Jupiter) and Cassini's predictions by assuming that light has a speed of propagation. It is the first to give an order of magnitude of the speed of light.
Bradley (1693-1762) confirms Römer's hypothesis and proposes a first estimate of the speed of light at approximately 10188 times that of the rotation of the Earth around the Sun, the latter being however poorly known. His discovery is linked to the aberration of light,
Telescope history
&facts,
Slide 1: Title Slide
Extrachromosomal Inheritance
Slide 2: Introduction to Extrachromosomal Inheritance
Definition: Extrachromosomal inheritance refers to the transmission of genetic material that is not found within the nucleus.
Key Components: Involves genes located in mitochondria, chloroplasts, and plasmids.
Slide 3: Mitochondrial Inheritance
Mitochondria: Organelles responsible for energy production.
Mitochondrial DNA (mtDNA): Circular DNA molecule found in mitochondria.
Inheritance Pattern: Maternally inherited, meaning it is passed from mothers to all their offspring.
Diseases: Examples include Leber’s hereditary optic neuropathy (LHON) and mitochondrial myopathy.
Slide 4: Chloroplast Inheritance
Chloroplasts: Organelles responsible for photosynthesis in plants.
Chloroplast DNA (cpDNA): Circular DNA molecule found in chloroplasts.
Inheritance Pattern: Often maternally inherited in most plants, but can vary in some species.
Examples: Variegation in plants, where leaf color patterns are determined by chloroplast DNA.
Slide 5: Plasmid Inheritance
Plasmids: Small, circular DNA molecules found in bacteria and some eukaryotes.
Features: Can carry antibiotic resistance genes and can be transferred between cells through processes like conjugation.
Significance: Important in biotechnology for gene cloning and genetic engineering.
Slide 6: Mechanisms of Extrachromosomal Inheritance
Non-Mendelian Patterns: Do not follow Mendel’s laws of inheritance.
Cytoplasmic Segregation: During cell division, organelles like mitochondria and chloroplasts are randomly distributed to daughter cells.
Heteroplasmy: Presence of more than one type of organellar genome within a cell, leading to variation in expression.
Slide 7: Examples of Extrachromosomal Inheritance
Four O’clock Plant (Mirabilis jalapa): Shows variegated leaves due to different cpDNA in leaf cells.
Petite Mutants in Yeast: Result from mutations in mitochondrial DNA affecting respiration.
Slide 8: Importance of Extrachromosomal Inheritance
Evolution: Provides insight into the evolution of eukaryotic cells.
Medicine: Understanding mitochondrial inheritance helps in diagnosing and treating mitochondrial diseases.
Agriculture: Chloroplast inheritance can be used in plant breeding and genetic modification.
Slide 9: Recent Research and Advances
Gene Editing: Techniques like CRISPR-Cas9 are being used to edit mitochondrial and chloroplast DNA.
Therapies: Development of mitochondrial replacement therapy (MRT) for preventing mitochondrial diseases.
Slide 10: Conclusion
Summary: Extrachromosomal inheritance involves the transmission of genetic material outside the nucleus and plays a crucial role in genetics, medicine, and biotechnology.
Future Directions: Continued research and technological advancements hold promise for new treatments and applications.
Slide 11: Questions and Discussion
Invite Audience: Open the floor for any questions or further discussion on the topic.
A brief information about the SCOP protein database used in bioinformatics.
The Structural Classification of Proteins (SCOP) database is a comprehensive and authoritative resource for the structural and evolutionary relationships of proteins. It provides a detailed and curated classification of protein structures, grouping them into families, superfamilies, and folds based on their structural and sequence similarities.
Professional air quality monitoring systems provide immediate, on-site data for analysis, compliance, and decision-making.
Monitor common gases, weather parameters, particulates.
This pdf is about the Schizophrenia.
For more details visit on YouTube; @SELF-EXPLANATORY;
https://www.youtube.com/channel/UCAiarMZDNhe1A3Rnpr_WkzA/videos
Thanks...!
Cancer cell metabolism: special Reference to Lactate PathwayAADYARAJPANDEY1
Normal Cell Metabolism:
Cellular respiration describes the series of steps that cells use to break down sugar and other chemicals to get the energy we need to function.
Energy is stored in the bonds of glucose and when glucose is broken down, much of that energy is released.
Cell utilize energy in the form of ATP.
The first step of respiration is called glycolysis. In a series of steps, glycolysis breaks glucose into two smaller molecules - a chemical called pyruvate. A small amount of ATP is formed during this process.
Most healthy cells continue the breakdown in a second process, called the Kreb's cycle. The Kreb's cycle allows cells to “burn” the pyruvates made in glycolysis to get more ATP.
The last step in the breakdown of glucose is called oxidative phosphorylation (Ox-Phos).
It takes place in specialized cell structures called mitochondria. This process produces a large amount of ATP. Importantly, cells need oxygen to complete oxidative phosphorylation.
If a cell completes only glycolysis, only 2 molecules of ATP are made per glucose. However, if the cell completes the entire respiration process (glycolysis - Kreb's - oxidative phosphorylation), about 36 molecules of ATP are created, giving it much more energy to use.
IN CANCER CELL:
Unlike healthy cells that "burn" the entire molecule of sugar to capture a large amount of energy as ATP, cancer cells are wasteful.
Cancer cells only partially break down sugar molecules. They overuse the first step of respiration, glycolysis. They frequently do not complete the second step, oxidative phosphorylation.
This results in only 2 molecules of ATP per each glucose molecule instead of the 36 or so ATPs healthy cells gain. As a result, cancer cells need to use a lot more sugar molecules to get enough energy to survive.
Unlike healthy cells that "burn" the entire molecule of sugar to capture a large amount of energy as ATP, cancer cells are wasteful.
Cancer cells only partially break down sugar molecules. They overuse the first step of respiration, glycolysis. They frequently do not complete the second step, oxidative phosphorylation.
This results in only 2 molecules of ATP per each glucose molecule instead of the 36 or so ATPs healthy cells gain. As a result, cancer cells need to use a lot more sugar molecules to get enough energy to survive.
introduction to WARBERG PHENOMENA:
WARBURG EFFECT Usually, cancer cells are highly glycolytic (glucose addiction) and take up more glucose than do normal cells from outside.
Otto Heinrich Warburg (; 8 October 1883 – 1 August 1970) In 1931 was awarded the Nobel Prize in Physiology for his "discovery of the nature and mode of action of the respiratory enzyme.
WARNBURG EFFECT : cancer cells under aerobic (well-oxygenated) conditions to metabolize glucose to lactate (aerobic glycolysis) is known as the Warburg effect. Warburg made the observation that tumor slices consume glucose and secrete lactate at a higher rate than normal tissues.
Earliest Galaxies in the JADES Origins Field: Luminosity Function and Cosmic ...Sérgio Sacani
We characterize the earliest galaxy population in the JADES Origins Field (JOF), the deepest
imaging field observed with JWST. We make use of the ancillary Hubble optical images (5 filters
spanning 0.4−0.9µm) and novel JWST images with 14 filters spanning 0.8−5µm, including 7 mediumband filters, and reaching total exposure times of up to 46 hours per filter. We combine all our data
at > 2.3µm to construct an ultradeep image, reaching as deep as ≈ 31.4 AB mag in the stack and
30.3-31.0 AB mag (5σ, r = 0.1” circular aperture) in individual filters. We measure photometric
redshifts and use robust selection criteria to identify a sample of eight galaxy candidates at redshifts
z = 11.5 − 15. These objects show compact half-light radii of R1/2 ∼ 50 − 200pc, stellar masses of
M⋆ ∼ 107−108M⊙, and star-formation rates of SFR ∼ 0.1−1 M⊙ yr−1
. Our search finds no candidates
at 15 < z < 20, placing upper limits at these redshifts. We develop a forward modeling approach to
infer the properties of the evolving luminosity function without binning in redshift or luminosity that
marginalizes over the photometric redshift uncertainty of our candidate galaxies and incorporates the
impact of non-detections. We find a z = 12 luminosity function in good agreement with prior results,
and that the luminosity function normalization and UV luminosity density decline by a factor of ∼ 2.5
from z = 12 to z = 14. We discuss the possible implications of our results in the context of theoretical
models for evolution of the dark matter halo mass function.
Deep Behavioral Phenotyping in Systems Neuroscience for Functional Atlasing a...Ana Luísa Pinho
Functional Magnetic Resonance Imaging (fMRI) provides means to characterize brain activations in response to behavior. However, cognitive neuroscience has been limited to group-level effects referring to the performance of specific tasks. To obtain the functional profile of elementary cognitive mechanisms, the combination of brain responses to many tasks is required. Yet, to date, both structural atlases and parcellation-based activations do not fully account for cognitive function and still present several limitations. Further, they do not adapt overall to individual characteristics. In this talk, I will give an account of deep-behavioral phenotyping strategies, namely data-driven methods in large task-fMRI datasets, to optimize functional brain-data collection and improve inference of effects-of-interest related to mental processes. Key to this approach is the employment of fast multi-functional paradigms rich on features that can be well parametrized and, consequently, facilitate the creation of psycho-physiological constructs to be modelled with imaging data. Particular emphasis will be given to music stimuli when studying high-order cognitive mechanisms, due to their ecological nature and quality to enable complex behavior compounded by discrete entities. I will also discuss how deep-behavioral phenotyping and individualized models applied to neuroimaging data can better account for the subject-specific organization of domain-general cognitive systems in the human brain. Finally, the accumulation of functional brain signatures brings the possibility to clarify relationships among tasks and create a univocal link between brain systems and mental functions through: (1) the development of ontologies proposing an organization of cognitive processes; and (2) brain-network taxonomies describing functional specialization. To this end, tools to improve commensurability in cognitive science are necessary, such as public repositories, ontology-based platforms and automated meta-analysis tools. I will thus discuss some brain-atlasing resources currently under development, and their applicability in cognitive as well as clinical neuroscience.
Observation of Io’s Resurfacing via Plume Deposition Using Ground-based Adapt...Sérgio Sacani
Since volcanic activity was first discovered on Io from Voyager images in 1979, changes
on Io’s surface have been monitored from both spacecraft and ground-based telescopes.
Here, we present the highest spatial resolution images of Io ever obtained from a groundbased telescope. These images, acquired by the SHARK-VIS instrument on the Large
Binocular Telescope, show evidence of a major resurfacing event on Io’s trailing hemisphere. When compared to the most recent spacecraft images, the SHARK-VIS images
show that a plume deposit from a powerful eruption at Pillan Patera has covered part
of the long-lived Pele plume deposit. Although this type of resurfacing event may be common on Io, few have been detected due to the rarity of spacecraft visits and the previously low spatial resolution available from Earth-based telescopes. The SHARK-VIS instrument ushers in a new era of high resolution imaging of Io’s surface using adaptive
optics at visible wavelengths.
Nutraceutical market, scope and growth: Herbal drug technologyLokesh Patil
As consumer awareness of health and wellness rises, the nutraceutical market—which includes goods like functional meals, drinks, and dietary supplements that provide health advantages beyond basic nutrition—is growing significantly. As healthcare expenses rise, the population ages, and people want natural and preventative health solutions more and more, this industry is increasing quickly. Further driving market expansion are product formulation innovations and the use of cutting-edge technology for customized nutrition. With its worldwide reach, the nutraceutical industry is expected to keep growing and provide significant chances for research and investment in a number of categories, including vitamins, minerals, probiotics, and herbal supplements.
Richard's entangled aventures in wonderlandRichard Gill
Since the loophole-free Bell experiments of 2020 and the Nobel prizes in physics of 2022, critics of Bell's work have retreated to the fortress of super-determinism. Now, super-determinism is a derogatory word - it just means "determinism". Palmer, Hance and Hossenfelder argue that quantum mechanics and determinism are not incompatible, using a sophisticated mathematical construction based on a subtle thinning of allowed states and measurements in quantum mechanics, such that what is left appears to make Bell's argument fail, without altering the empirical predictions of quantum mechanics. I think however that it is a smoke screen, and the slogan "lost in math" comes to my mind. I will discuss some other recent disproofs of Bell's theorem using the language of causality based on causal graphs. Causal thinking is also central to law and justice. I will mention surprising connections to my work on serial killer nurse cases, in particular the Dutch case of Lucia de Berk and the current UK case of Lucy Letby.
2. OBJECTIVES
To understand the different
consequences of special relativity
To familiarize different formulas and perform calculations
regarding the consequences of special relativity
To apply the lessons in real life
3. Did you know? If you could
travel at the speed of light
without breaking down to
your basic atoms, you’d be
able to go around the Earth
7 and a half times in a
single second
TRIVIA TRIVIATRIVIA
TRIVIATRIVIA
TRIVIA TRIVIATRIVIA
TRIVIA TRIVIATRIVIA TRIVIA TRIVIATRIVIA
4. BACKGROUND ON
SPECIAL RELATIVITY
Along with the nuclear physics, special
relativity is central to modern physics. It
describes the motion of particles with
speed ranging from zero to a value
close to the speed of light in vacuum.
In 1905, Albert Enstein published his special theory
of relativity. He based his theory on two postulates.
5. BACKGROUND ON
SPECIAL RELATIVITY
WHAT ARE THE TWO POSTULATES
OF SPECIAL RELATIVITY
Principle Of Relativity
Constancy Of The Speed Of Light
States that the laws of physics must be the same in all inertial frames
of reference. An inertial frame of reference is one where Newton’s
first law of motion is valid. It is a frame of reference where a particle is
observed to have no acceleration in the absence of an unbalanced
force, thus, an accelerating or rotating frame is not inertial.
States that the speed if light in a vacuum is constant in all inertial
frames of reference and is independent of the motion of the source.
6. BACKGROUND ON
SPECIAL RELATIVITY
LORENTZ
FACTOR
𝛾 =
1
1 −
𝑣
𝑐
2
This quantity is called the Lorentz factor
named after dutch physicist Hendrik
Lorentz. It is denoted by the symbol 𝛾
(read as gamma). The letter 𝑣 is the speed
of the object and 𝑐 is the speed of light
equal to 3𝑥108
𝑚/𝑠2
7. BACKGROUND ON
SPECIAL RELATIVITY
CONSEQUENCES FROM EINSTEIN’S
SPECIAL THEORY OF RELATIVITY
Relativistic Kinematics:
• Time dilation,
• Length contraction,
• Relativistic Doppler
effect, and
• Mass increase.
Relativistic Dynamics
• Relativistic mass and
momentum,
• mass-energy
equivalence, and
• Relativistic second law.
8. TIME DILATION
In 1971, physicist Joseph Hafele and
astronomer Richard Keating confirmed that
the time interval betweem two events
measured by a moving observer who views
these events as occuring in different places
is longer than the time interval measured by
an observer at rest, who views the events
as happening in the same place.
9. TIME DILATION
The first time interval is referred to as the dilated time
interval and the second time interval is the proper
time interval. The dilated time interval may be
computed using
𝛥𝑡 =
𝛥𝑡0
1 −
𝑣
𝑐
2
= 𝛾𝛥𝑡0
TIME
INTERVAL
𝛥𝑡
𝛥𝑡0
𝛾
𝑐
𝑣dilate time interval
proper time interval
speed of relative motion
speed of light in vacuum
Lorentz factor
10. TIME DILATION
TWIN
PARADOX
Imagine a pair of twins, one of whom is an
astronaut that travels to a distant star. The
stay-at-home twin will see his brother age
more slowly than him. You’d expect the space-
faring twin to see the same happen to his
Earth-bound counterpart since they’re moving
at the same speed relative to one another. But
if the twins are re-united, Einstein said that the
space-faring twin will have aged less than the
one on Earth, which is odd given that they’ve
both performed identical journeys relative to
each other.
11. TIME DILATION
TWIN
PARADOX
Sample Problem
Leo and Christian are twins. At the age of 30, Leo left for a round
trip to a distant star in a spaceship with a speed of 0.95𝑐 relative to
Earth. The whole trip took 20 years according to the shipboard
clock. Find their ages when Leo returns.
Given: 𝑣 = 0.95𝑐
Solution
The two events in this example are Leo’s departure from Earth and
his return to Earth. The proper time interval must be the time
recorded by the clock onboard the spaceship, that is 𝛥𝑡0=20 years
𝛥𝑡 =
𝛾𝛥𝑡0
1 −
𝑣
𝑐
2 𝛥𝑡 =
20𝑦𝑒𝑎𝑟𝑠
1 −
0.95𝑐
𝑐
2
𝛥𝑡 = 64.05𝑦𝑒𝑎𝑟𝑠 ≈ 64𝑦𝑒𝑎𝑟𝑠
12. LENGTH OF
CONTRACTION
Length of an object measured by an
observer in an inertial reference frame
that is moving with respect to the object
is less than its proper length
Happens only in dimensions along the
direction of motion
13. LENGTH OF
CONTRACTION
FORMULA FOR LENGTH
LENGTH OF CONTRACTION
𝐿′ = 𝐿 1 −
𝑣
𝑐
2
𝐿 =
𝐿
𝛾
where:
L = length if the object at rest
L’ = length of the object moving at speed 𝑣
𝛾 = the Lorentz Factor: 𝛾 =
1
1−
𝑣
𝑐
2
14. LENGTH OF
CONTRACTION
SAMPLE
PROBLEM
A spaceship traveling at 0.5c relative to earth is 45m long as
measured by its crew. How long is the spaceship as measured
by the mission control in Texas?
𝑣=0.5 𝑐
𝐿0=45m
Solution
𝐿′ = 𝐿 1 −
𝑣
𝑐
2
𝐿′ = 45𝑚 1 −
0.5𝑐
𝑐
2
𝐿′ = 38.97𝑚 ≈ 39𝑚
15. RELATIVISTIC
DOPPLER
EFFECT
Doppler effect for electromagnetic waves
Named after Austrian mathematician and
physicist Christian Johann Doppler
Doppler effect for electromagnetic waves
is given by:
- fobs is the observed frequency
- c is the speed of light
- v is the speed of source relative to the observer
- fs is frequency in the rest frame or source
- the upper sign is for observer moving toward the source
- the lower sign is for an observer moving away from the source
𝑓𝑜𝑏𝑠 = 𝑓𝑠
1 −
𝑣
𝑐
1 +
𝑣
𝑐
16. RELATIVISTIC
DOPPLER
EFFECT
DOPPLER EFFECT FOR LIGHT IS
GIVEN IN TERMS OF REDSHIFT OR
BLUESHIFT
RED SHIFT If the source of light is moving
away from the observer, the observed
wavelength is longer than the wavelength
emitted when the source is at rest
BLUE SHIFT if source
of light is moving
toward an observer,
the wavelength of light
is shorter than the
wavelength when the
source is at rest
17. RELATIVISTIC
DOPPLER
EFFECT
DOPPLER EFFECT FOR LIGHT IS
GIVEN IN TERMS OF REDSHIFT OR
BLUESHIFT
Doppler developed this theory in an
effort to explain the shift in frequency
light by moving atoms or astronomical
bodies
Edwin Hubble an American astronomer
used this theory to confirm that the
universe is expanding and that most
galaxies are receding from us.
Light emitted by
those galaxies
are redshifted
18. RELATIVISTIC
DOPPLER
EFFECT
Sample problem: suppose a space probe moves
away from earth at a speed of .350c. It sends a
radio-wave message back to earth at a frequency
of 1.50 GHz. At what frequency is message
received back on earth?
Solution
SAMPLE
PROBLEM
𝑓𝑜𝑏𝑠 = 𝑓𝑠
1 −
𝑣
𝑐
1 +
𝑣
𝑐
𝑓𝑜𝑏𝑠 = 150𝐺𝐻𝑧
1 −
0.35𝑐
𝑐
1 +
0.35𝑐
𝑐
𝑓𝑜𝑏𝑠 = 1.04𝐺𝐻𝑧
19. LORENTZ
TRANSFORMATIONS
Motion is relative and depends on the
frame of reference where motion is being
observed
Based from the figure: Galilean coordinate transformation
xB = xA – vt
yB = yA
zA = zB
20. LORENTZ
TRANSFORMATIONS
Based from the
figure: Galilean
coordinate
transformation
xB = xA – vt
yB = yA
zA = zB
differentiating the Galilean coordinate transformation with
respect to time
𝑑𝑥 𝐵
𝑑𝑡
=
𝑑𝑥 𝐴
𝑑𝑡
− 𝑣
𝑑𝑦 𝐵
𝑑𝑡
=
𝑑𝑦 𝐴
𝑑𝑡
𝑑𝑧 𝐵
𝑑𝑡
=
𝑑𝑧 𝐴
𝑑𝑡
21. LORENTZ
TRANSFORMATIONS
Let uA and uB be the velocity of the object as seen in frames A and
B, respectively
‣ Since frame B is just moving along the x-axis, the y- and z-
coordinates of the object viewed in the two frames are just the
same.
‣ Time tA in frame A is equal to time tb in frame B
If Speed of light measured in frame A is c, then, speed of light measured in frame B is:
uB = c – v
- this is contradictory in the speed of light (postulate 2) stating that the speed of light in
the vacuum is constant in all inertial frames of references and is independent of the
motion of the source
- postulate 2 requires that time dilation be considered in the coordinate and velocity
transformation
23. LORENTZ
TRANSFORMATIONS
LORENTZ VELOCITY
TRANSFORMATION
Velocity of (ux)B of the object along the x-axis as see in frame B:
𝑑𝑥 𝐵 =
𝑑𝑥 𝐴 − 𝑣𝑑𝑡 𝐴
1 −
𝑣2
𝑐2
𝑑𝑡 𝐵 =
𝑑𝑡 𝐴 −
𝑣𝑑𝑥 𝐴
𝑐2
1 −
𝑣2
𝑐2
Velocity of (ux)A of the object along the x-
axis as seen in frame A:
𝑢 𝑋 𝐴 =
𝑢 𝑥 𝐵 + 𝑣
1 +
𝑢 𝑥 𝐵 𝑣
𝑐2
24. LORENTZ
TRANSFORMATIONS
SAMPLE
PROBLEM
Sample problem: an electron is moving at 0l6c relative
to Lorentz. Einstein, who is inside a spaceship moving
at 0.8c relative to Lorentz, is also observing the
electron. What is the speed of the electron as
determined by Einstein?
Solution
𝑢 𝐵 =
𝑢 𝐴 − 𝑣
1 −
𝑢 𝐴 𝑣
𝑐2
Given: Lorentz is the fixed (stationary) frame and Einstein is in the moving frame.
uB = 0.6c – 0.8c / [1 – (0.6c)(0.8c)/ c2]
uB = -0.3846c ~-0.4c
- The negative sign means that relative to
Einstein, the electron is moving in the negative
direction.
v = 0.8c uA = 0.6c
solving for uB:
25. RELATIVISTIC
MASS
The mass of an object moving at a speed
v approaching that of light in vacuum (c) is
greater than its mass, called the rest
mass, when at rest relative to an observer.
The mass of a moving object, called the
relativistic mass is given by:
𝑚 =
𝑚0
1 −
𝑣
𝑐
2
= 𝛾𝑚0
Where m= relativistic mass and m0= rest mass. It follows that the relativistic
momentum p is given by:
𝑝 =
𝑚0 𝑣
1 −
𝑣
𝑐
2
= 𝛾𝑚0 𝑣
26. RELATIVISTIC
MASS
SAMPLE
PROBLEM
An electron is moving at 0.5c. What is the momentum
according to special relativity?
Given: v= 0.5c= 1.5 x 108 m/s, m0= 9.109 x 10-31kg
Solution
𝑝 =
𝑚0 𝑣
1 −
𝑣
𝑐
2
=
9.109x10−31
𝑘𝑔 (1.5x108 m
s
)
1 −
0.5𝑐
𝑐
2
= 1.5777 x 10-22 kg⋅m/s ≈ 1.6 x 10-22 kg⋅m/s
27. MASS-ENERGY
EQUIVALENCE
𝑬 = 𝒎0 𝒄 + 𝑲 = 𝒎𝒄2
Where E= Total energy, m0= rest
mass, c= speed of light in
vacuum, and K= relativistic
kinetic energy.
The equation shows that mass and energy are
equivalent. Therefore, a gain or loss in mass may
be considered a gain or loss in energy.
28. MASS-ENERGY
EQUIVALENCE
The relativistic kinetic energy is given as:
𝐾 =
𝑚0 𝑐2
1 −
𝑣
𝑐
2
− 𝑚0 𝑐2
=
1
1 −
𝑣
𝑐
2
− 1 𝑚0 𝑐2
= 𝛾 − 1 𝑚0 𝑐2
For the particle at rest, v= 0, and it
follows that K= 0. Therefore,
𝐸 = 𝐸0
𝑚0 𝑐2=
The energy 𝑬0is called rest energy
29. MASS-ENERGY
EQUIVALENCE
SAMPLE
PROBLEM
An object of mass 0.2 kg is moving at (2/3) c.
What is the kinetic energy in eV according to
special relativity?
Given: m= 0.2 kg, v= (2/3)c= 2 x 108 m/s
Solution
𝐾 = 𝛾 − 1 𝑚0 𝑐2
=
1
1 −
𝑣
𝑐
2
− 1 𝑚0 𝑐2
=
1
1 −
2
3
𝑐
𝑐
2
− 1 (0.2𝑘𝑔)(3x108
𝑚
𝑠
)
= 6.1495x1015
𝐽
1𝑒𝑉
1.602x10−19 𝐽
= 3.83866 x 1034 eV ≈ 3.8 x 1034 eV
30. RELATIVISTIC
SECOND LAW
In classical mechanics, force is
defined as the time rate of change
of momentum, that is
𝐹 =
𝑑𝑝
𝑑𝑡
This definition is still valid in relativistic
mechanics provided that the relativistic
momentum is used:
𝐹 =
𝑑
𝑑𝑡
𝑚𝑣
1 −
𝑣
𝑐
2
31. RELATIVISTIC
SECOND LAW
MORE
FORMULA
For force acting in the same direction,
𝐹 =
𝑚
1 −
𝑣
𝑐
2
3
2
𝑎
Solving for acceleration,
𝑎 =
𝐹
𝑚
1 −
𝑣
𝑐
2 3 2
This equation shows that is speed v is very small compared to the speed of light
c, the acceleration can be written as 𝑎 =
𝐹
𝑚
, which is Newton’s second law in
classical mechanics. As v approaches c, the acceleration approaches zero.
This implies that it is impossible to accelerate an object to a speed equal to or
greater than c (speed of light in vacuum).
32. RELATIVISTIC
SECOND LAW
SAMPLE
PROBLEM
A microsatellite has a mass of approximately 50 kg.
What is its acceleration when launched at a speed
of 0.6c by a force of 5000 N?
Given: m= 50kg, v= 0.6c, F= 5000N
Solution
𝑎 =
𝐹
𝑚
1 −
𝑣
𝑐
2 3 2
=
5000𝑁
50𝑘𝑔
1 −
0.6𝑐
𝑐
2 3 2
= 51.2 m/s2
Editor's Notes
Einstein’s prediction has been confirmed in experiments with atomic clocks, so what resolves this paradox? The answer lies in the fact that the twins don’t undertake identical journeys. To get back to Earth, the travelling twin experiences a force in order to slow down and reverse direction. The stay-at-home twin doesn’t, making their journeys fundamentally different. Not surprisingly, so are the relative travel times of the twins, thus one of them ages more.