# 28.3 Length contraction

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## Learning objectives

By the end of this section, you will be able to:

• Describe proper length.
• Calculate length contraction.
• Explain why we do not 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,

#### Questions & Answers

Why does earth exert only a tiny downward pull?
Mya Reply
hello
Islam
Why is light bright?
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what is radioactive element
Attah Reply
an 8.0 capacitor is connected by to the terminals of 60Hz whoes rms voltage is 150v. a.find the capacity reactance and rms to the circuit
Aisha Reply
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Valdes Reply
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Nwafor Reply
is the study of matter in relation to energy
Kintu
a submersible pump is dropped a borehole and hits the level of water at the bottom of the borehole 5 seconds later.determine the level of water in the borehole
Obrian Reply
what is power?
aron Reply
power P = Work done per second W/ t. It means the more power, the stronger machine
Sphere
e.g. heart Uses 2 W per beat.
Rohit
A spherica, concave shaving mirror has a radius of curvature of 32 cm .what is the magnification of a persons face. when it is 12cm to the left of the vertex of the mirror
Alona Reply
did you solve?
Shii
1.75cm
Ridwan
my name is Abu m.konnek I am a student of a electrical engineer and I want you to help me
Abu
the magnification k = f/(f-d) with focus f = R/2 =16 cm; d =12 cm k = 16/4 =4
Sphere
what do we call velocity
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A weather vane is some sort of directional arrow parallel to the ground that may rotate freely in a horizontal plane. A typical weather vane has a large cross-sectional area perpendicular to the direction the arrow is pointing, like a “One Way” street sign. The purpose of the weather vane is to indicate the direction of the wind. As wind blows pa
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what about the wind vane
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If a prism is fully imersed in water then the ray of light will normally dispersed or their is any difference?
Anurag Reply
the same behavior thru the prism out or in water bud abbot
Ju
If this will experimented with a hollow(vaccum) prism in water then what will be result ?
Anurag
What was the previous far point of a patient who had laser correction that reduced the power of her eye by 7.00 D, producing a normal distant vision power of 50.0 D for her?
Jaydie Reply
What is the far point of a person whose eyes have a relaxed power of 50.5 D?
Jaydie
What is the far point of a person whose eyes have a relaxed power of 50.5 D?
Jaydie
A young woman with normal distant vision has a 10.0% ability to accommodate (that is, increase) the power of her eyes. What is the closest object she can see clearly?
Jaydie
29/20 ? maybes
Ju
In what ways does physics affect the society both positively or negatively
Princewill Reply
how can I read physics...am finding it difficult to understand...pls help
rerry Reply
try to read several books on phy don't just rely one. some authors explain better than other.
Ju
And don't forget to check out YouTube videos on the subject. Videos offer a different visual way to learn easier.
Ju
hope that helps
Ju
I have a exam on 12 february
David Reply
what is velocity
Jiti
the speed of something in a given direction.
Ju

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