# 28.3 Length contraction

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• Describe proper length.
• Calculate length contraction.
• Explain why we don’t notice these effects at everyday scales.

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 $0.950c$ for $7.05\phantom{\rule{0.25em}{0ex}}\mu s$ from the time it is produced until it decays. Thus it travels a distance

${L}_{0}=v\Delta t=\left(0.950\right)\left(3.00×{\text{10}}^{8}\phantom{\rule{0.25em}{0ex}}\text{m/s}\right)\left(7.05×{\text{10}}^{-6}\phantom{\rule{0.25em}{0ex}}\text{s}\right)=2.01\phantom{\rule{0.25em}{0ex}}\text{km}$

relative to the Earth. In the muon’s frame of reference, its lifetime is only $2.20\phantom{\rule{0.25em}{0ex}}\mu s$ . It has enough time to travel only

$L=v\Delta {t}_{0}=\left(0\text{.}\text{950}\right)\left(3\text{.}\text{00}×{\text{10}}^{8}\phantom{\rule{0.25em}{0ex}}\text{m/s}\right)\left(2\text{.}\text{20}×{\text{10}}^{-6}\phantom{\rule{0.25em}{0ex}}\text{s}\right)=0\text{.627 km}.$

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 ${L}_{0}$ 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 ${L}_{0}$ , 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 $L$ it sees is not the proper length.

## 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

$v=\frac{{L}_{0}}{\Delta t}.$

The time relative to the Earth-bound observer is $\Delta t$ , since the object being timed is moving relative to this observer. The velocity relative to the moving observer is given by

$v=\frac{L}{\Delta {t}_{0}}.$

The moving observer travels with the muon and therefore observes the proper time $\Delta {t}_{0}$ . The two velocities are identical; thus,

$\frac{{L}_{0}}{\Delta t}=\frac{L}{\Delta {t}_{0}}.$

We know that $\Delta t=\gamma \Delta {t}_{0}$ . Substituting this equation into the relationship above gives

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What are the two major system units in the world
What is physical quantities
Jizel
they are physical properties that can be measured or calculated
ZIFAC
Do you mean the SI system and Imperial system? SI is used by scientists the world over (with notable exception being the USA)
Who is the father of physics
Newton.Geliliyo and Einstein is called father of physics
Neha
Newtin
Jizel
Einstein
Jizel
Ancient - Archimedes Classical Physics - Newton Modern Physics - Einstein But each of these has built on the work of predecessors. No single personality defines physics.
Thanks
Hussain
Galileo Galilie must be. He was the creator of the scientific method
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Galileo was no doubt inspirational, but the scientific method has existed for millennia. What Galileo did was prove the superiority of experimentation to just thinking (Aristotle's way). Again all noteworthy scientists
Newton galileo and Einstein
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what is wave
a wave is a distirbance that transmits energy from one place ro another within or without a medium
Vincent
wave is the transfer of energy from one medium to another without the transfer of particles
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wave is a disturbance which transfer energy from one medium to another without causing any permanent displacement by itself
Joyfulsounds
wave is a disturbance or oscillation that travel through space and matter,accompanied by a transfer of energy
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A wave is any disturbances in an elastic medium which carries energy from one point to another through a medium
abdul
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a wave is a disturbance and there are trasfer energy from one medium to another and travel through any space without the trasfer of particles.
Neha
a wave is a disturbance that travels or carries energy from one point to another through a medium
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what is thermodynamics
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Charity
thermodynamics is a heat and energy significant physics
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It's the science of conversion between heat and other forms of energy
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Lalita
what is physics
physic is a pure science that deal with behavior of matter,energy & how it related to other physical properties
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Owk. But am are Art student.
Hussaini
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what are the differences between reflection and refraction ?
Matins
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what is physics
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sandhya
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lkpostpost2000@yahoo
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sandhya
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Phebilia
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Mr.
then due to lower pressure on upper part of wings helps producing an additional lift because air flows from areaof lower to the area of higher pressure
Mr.
The engines located under the wings generate thrust .. in relation thrust is a force ... which ovwrcomes or becomes greater than the weight of the plane.. remember weight is a force Weight = m x g-2 So therefore F(thrust) becomes greater than F(weight) Even if by 1Newton the plane starts lifting o
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what happens when a ship moves
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What is the sign of an acceleration that reduces the magnitude of a negative velocity? Of a positive velocity?
If it reduces the magnitude of the velocity, the acceleration sign is the opposite compared to the velocity.
Nicolas
yes
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what is accerelation
an objects tendency to speed up over time
RayRay
acceleration is the change in velocity over the change in time it would be written delta-v over delta-t.
Shii
the change in velocity V over a period of time T.
Matthew
Delta means "change in"...not period of
Shii
just kidding. it all works mathematically
Shii
except doesn't time really only change if the instantaneous speeds vary...?
Shii
and I assume we are all talking average acceleration
Shii
Hey shiii 😀
the rate of change of velocity is callaed acceleration
Amna
a=delta v/delta t
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the rate of change in velocity with respect to time is acceleration
Nana
nana you r right
Indrajit
good
oguji
what is meant by lost volt
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Theophilus
What is physics?
physics is brance science concerned with nature and properties of matter and energy
George
sure
Okpara
yah....
kashif
physics is study of the natural phenomenon on the basis of certain laws and principles. it's like watching a game of chess and trying to understand its rules how it's played.
Ajit
awesome
Okpara
physics is study of nature and it's law
AMRITA
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Lote
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Theophilus
what is a double-slit experiment?Explain.
when you pass a wave of any kind ie sound water light ect you get an interface pattern forming on a screen behind it, where the peaks and troughs add and cancel out due to the diffraction caused by a wave traveling through the slits
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navid
is formed and then the constructive interferences occur at 0 (which is the brightest band)... then a sequence of bright band (constructive interference) and dark band (destructive interference) happens and the further from the central band the lower the intensity of bright band(constructive interfe
navid