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  • Describe proper length.
  • Calculate length contraction.
  • Explain why we don’t notice these effects at everyday scales.
A long isolated double-lane road banked by barren land on both sides.
People might describe distances differently, but at relativistic speeds, the distances really are different. (credit: Corey Leopold, Flickr)

Have you ever driven on a road that seems like it goes on forever? If you look ahead, you might say you have about 10 km left to go. Another traveler might say the road ahead looks like it’s about 15 km long. If you both measured the road, however, you would agree. Traveling at everyday speeds, the distance you both measure would be the same. You will read in this section, however, that this is not true at relativistic speeds. Close to the speed of light, distances measured are not the same when measured by different observers.

Proper length

One thing all observers agree upon is relative speed. Even though clocks measure different elapsed times for the same process, they still agree that relative speed, which is distance divided by elapsed time, is the same. This implies that distance, too, depends on the observer’s relative motion. If two observers see different times, then they must also see different distances for relative speed to be the same to each of them.

The muon discussed in [link] illustrates this concept. To an observer on the Earth, the muon travels at 0.950 c size 12{c} {} for 7.05 μ s size 12{c} {} from the time it is produced until it decays. Thus it travels a distance

L 0 = v Δ t = ( 0.950 ) ( 3.00 × 10 8 m/s ) ( 7.05 × 10 6 s ) = 2.01 km

relative to the Earth. In the muon’s frame of reference, its lifetime is only 2.20 μ s . It has enough time to travel only

L = v Δ t 0 = ( 0 . 950 ) ( 3 . 00 × 10 8 m/s ) ( 2 . 20 × 10 6 s ) = 0 .627 km .

The distance between the same two events (production and decay of a muon) depends on who measures it and how they are moving relative to it.

Proper length

Proper length L 0 size 12{L rSub { size 8{0} } } {} is the distance between two points measured by an observer who is at rest relative to both of the points.

The Earth-bound observer measures the proper length L 0 size 12{L rSub { size 8{0} } } {} , because the points at which the muon is produced and decays are stationary relative to the Earth. To the muon, the Earth, air, and clouds are moving, and so the distance L size 12{L} {} it sees is not the proper length.

In part a observer observes from ground frame of reference a muon above earth with speed v in the rightward direction. The distance between the muon and the place where it disintegrates is two point zero one. In part b the system is shown in motion having velocity v in the leftward direction. So, the cloud and ground are displaced zero point six two seven kilo meter in the opposite direction.
(a) The Earth-bound observer sees the muon travel 2.01 km between clouds. (b) The muon sees itself travel the same path, but only a distance of 0.627 km. The Earth, air, and clouds are moving relative to the muon in its frame, and all appear to have smaller lengths along the direction of travel.

Length contraction

To develop an equation relating distances measured by different observers, we note that the velocity relative to the Earth-bound observer in our muon example is given by

v = L 0 Δ t . size 12{v= { {L rSub { size 8{0} } } over {Δt} } } {}

The time relative to the Earth-bound observer is Δ t size 12{Δt} {} , since the object being timed is moving relative to this observer. The velocity relative to the moving observer is given by

v = L Δ t 0 . size 12{v= { {L rSub { size 8{0} } } over {Δt} } } {}

The moving observer travels with the muon and therefore observes the proper time Δ t 0 size 12{Δt rSub { size 8{0} } } {} . The two velocities are identical; thus,

L 0 Δ t = L Δ t 0 . size 12{ { {L rSub { size 8{0} } } over {Δt} } = { {L} over {Δt rSub { size 8{0} } } } } {}

We know that Δ t = γ Δ t 0 size 12{Δt=γΔt rSub { size 8{0} } } {} . Substituting this equation into the relationship above gives

Questions & Answers

what is heat and temperature
Kazeem Reply
how does sound affect temperature
Clement Reply
sound is directly proportional to the temperature.
juny
how to solve wave question
Wisdom Reply
I would like to know how I am not at all smart when it comes to math. please explain so I can understand. sincerly
Emma
Just know d relationship btw 1)wave length 2)frequency and velocity
Talhatu
First of all, you are smart and you will get it👍🏽... v = f × wavelength see my youtube channel: "mathwithmrv" if you want to know how to rearrange equations using the balance method
PhysicswithMrV
nice self promotion though xD
Beatrax
thanks dear
Chuks
hi pls help me with this question A ball is projected vertically upwards from the top of a tower 60m high with a velocity of 30ms1.what is the maximum height above the ground level?how long does it take to reach the ground level?
mahmoud
please guys help, what is the difference between concave lens and convex lens
Vincent Reply
convex lens brings rays of light to a focus while concave diverges rays of light
Christian
for mmHg to kPa yes
Matthew
it depends on the size
Matthew Reply
please what is concave lens
Vincent
a lens which diverge the ray of light
rinzuala
concave diverges light
Matthew
thank you guys
Vincent
A diverging lens
Yusuf
What is isotope
Yusuf
each of two or more forms of the same element that contain equal numbers of protons but different numbers of neutrons in their nuclei, and hence differ in relative atomic mass but not in chemical properties; in particular, a radioactive form of an element. "some elements have only one stable isotope
Karthi
what is wire wound resistors?
Naveedkhan Reply
What are the best colleges to go to for physics
Matthew Reply
I would like to know this too
Trevor
How do I calculate uncertainty in a frequency?
Rebecca Reply
Calculate . ..
Olufunsho
What is light wave
Sakeenah Reply
What is wave
Sakeenah
What is light
Sakeenah
okay
True
explain how neurons communicate feed and stimulate
Jeff
Great science students
Omo
A wave is a disturbance which travels through the medium transferring energy from one form to another without causing any permanent displacement of d medium itself
OGOR
Light is a form o wave
OGOR
Neurons communicate by sending message through nerves in coordination
OGOR
What are petrochemicals, give two examples
OGOR
light has dual nature, particle as well as wave. when we want to explain phenomena like Interference of light, then we consider light as wave.
Lalita
what is it as in the form of it or how to visualize it or what it contains
Matthew
particles of light are like small packets of energy called photons, and flow or motion of photons is wave like
Lalita
light is just the energy of which photons emit
Matthew
the wave is how they travel
Matthew
photons do not emitt energy, they are energy. They are massless particles.
Lalita
a wave is a disturbance through the medium. Have you ever thrown a stone in still water? the disturbance produced travels in form of wave, the wave produced by throwing stone in still water are circular in nature.
Lalita
a photon does contain mass when in motion. it doesnt contain mass when at rest
Matthew
when would it ever be at rest
Bob
a wave is a disturbance of which energy travels
Matthew
that's darkness. darkness has no mass because the photons within in aren't moving or producing energy
Matthew
Hi guys. Please I've been trying to understand the concept of SHM, but it's not been really easy, could someone please explain it to me or suggest a site I could visit? Thank you.
Odo
***google.com/url?sa=t&source=web&rct=j&url=***hyperphysics.phy-astr.gsu.edu/hbase/shm.html&ved=2ahUKEwiwu_PYqfzdAhXOmVkKHXi2CvkQFjAPegQIARAB&usg=AOvVaw0h5IZmrkQK4KtEMygT4ZGK
Matthew
effective mass of photons only comes into picture when we consider it accelerating in gravitational field, mass of photon has no meaning as it is always travelling with speed of light and is never at rest. with that high speed, Energy and momentum are equivalent. and darkness is absense of photons.
Lalita
darkness is absense of light. not the presence of 'resting photons'. photons are never at rest.
Lalita
photons are present in darkness but don't give off any light because they are stationary with no mass or energy. once a force makes them move again they will gain mass and give off light
Matthew
this theory is presented in Einsteins theory of special relativity
Matthew
A.The velocity Vo for the streamline flow of liquid in a small tube depends on the radius r of the tube,the density and the viscosity iter of the liquid .use the dimensional analysis to obtain an expression for the velocity . B.Given that Vo =r square ×p all over 4×iter ×l
True
A.The velocity Vo for the streamline flow of liquid in a small tube depends on the radius r of the tube,the density (rho)and the viscosity (iter)of the liquid. Use the method of dimensional analysis to obtain an expression for the velocity . B.Given that Vo =r square x p all over 4 x iter x l
True
Matthew, photons ARE light. there is no such thing as a photon that isn't moving. in fact the speed they move at is called C (for constant) in physics. through a vacuum they always travel at this speed no matter what. they can not slow down; except in another medium.
Brad
The reason why a photon can go at this speed is BECAUSE it had no mass. nothing can go this speed or faster because it needs to have no mass or negative mass. that's why it's called the constant.
Brad
when a photon hits something that is opaque, this is the only way to "stop"it. it isn't merely stopped but absorbed and turned into heat energy, then the remaining energy is reflected in different wavelengths. that reflection is what we call color. the darker something is, the less photons are ther
Brad
e. complete blackness is the absolute absence of photons altogether. I believe what you're referring to is not speed, but wavelength, which is indirectly proportional to the amount of energy a particular photon is made up of.
Brad
in order for a photon to have zero wavelength, it must (at least theoretically) have infinite energy.
Brad
about mass: you may have photons confused with electrons. elections have a mass so small that people say they are without mass, but they do. it is called electron mass or Me-.
Brad
you may also be getting electrons and photons confused because of the cherenkov effect. that is what happens when a particle travels faster than light IN THAT PARTICULAR MEDIUM. I emphasize that because no other particle besides photons can go the speed of c.
Brad
when a particle goes faster than light in a particular medium, a blue light is emitted, called cherenkov radiation. this is why nuclear reactors glow blue.
Brad
nuclear reactors release so much energy that when they emit electrons, those electrons are given enough energy to go faster than light in that medium (in this case water), releasing blue light. if you put the reactor in air or a vacuum, this effect wouldn't happen because the speed of light in air
Brad
is very close to c, which is the universal speed limit. I'd you did go faster than c, time would go backwards and you would have infinite theoretical mass and probably spagghettify, like with a black hole.
Brad
*if
Brad
*electrons
Brad
light waves can travel through a vacuum, and do not require a medium. In empty space, the wave does not dissipate (grow smaller) no matter how far it travels, because the wave is not interacting with anything else.
Salim
Please is there any instructional material for sounds Waves, Echo, light waves
Salami
how far there is hot topic that is boarding me now
Abraham
linear motion
Ahmed
kinematic
Abraham
tell us about it
Akinsanya
kinematic
Emma
kinematics disscuss the motion without cuases ...
ghulam
wow I like what am seeing here I need someone to brush me up in physics in fact I'll say I know nothing
Godslight
How does the Geiger tube works
Salma Reply
pls he do we find for tension
Belinda Reply
tension is equal to the weight of the object. so for example if something weighs 45 Newtons then the tension in the Rope holding it is 45 Newtons. and because it is in equilibrium if the object is 45N and there are three ropes holding it there would be 15 N of tension in each to equal the weight
Shii
does that work for you?
Shii
tnx
Belinda
very correct
Kudzy
A prankster applies 450 V to an 80.0 µF capacitor and then tosses it to an unsuspecting victim. The victim’s finger is burned by the discharge of the capacitor through 0.200 g of flesh. What is the temperature increase of the flesh? Is it reasonable to assume no phase change?
Khanh Reply
what is mass
Aliyu Reply
the quantity of matter that a body contains, as measured by its acceleration under a given force or by the force exerted on it by a gravitational field.
Aliyu
I agree wth aliyu shuwa
Nikita
correct
Jalil
or in easier terms the amount of stuff in an object (stuff meaning whatever element or material that makes the object heavy) the object composition
Shii
an object's resistance to change in motion?
Kudzy
your mass never changes but your weight changes based on the gravitational pull of a system or planet. your mass is just the amount of matter with a certain object
Matthew
please can someone help, why a bats can fly in the night without heating anything, how does he managed an see in darkness?
Vincent
...hit an obstacle and has bounced back.
Odo
Hi Vincent. From what I can remember and what I've learnt, bats do not have a good eye sight and so they make use of waves, when they send out waves and they do not return back to them, they realise that the site is free of obstacles but if it sends a wave and it returns back, it means it hit an
Odo
obstacle and bounced back
Odo
So bats make use of waves in place of eye sight. I think but confirm from a few sites .
Odo
they us echolocation to make a sort of mental map. many sea animals use the same method. this can also be compared to sonar which works in the same way
Matthew
y is atom d smallest particle
EDWIN Reply
yes
Lajpat
its a question Lajpat Rai
EDWIN
Google Quantum Physics or refer to the text
Shii
atoms are considered the smallest unit of matter. it is further divided into three subatomic particles: protons, neutrons, and electrons.
Arshiya
what's the dimensions of moment of inertia?
Rose Reply
Practice Key Terms 2

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Source:  OpenStax, College physics. OpenStax CNX. Jul 27, 2015 Download for free at http://legacy.cnx.org/content/col11406/1.9
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