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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

Describe an experiment to determine short half life
Tyson Reply
what is science
Kenedy Reply
A 20MH coil has a resistance of 50 ohms and us connected in series with a capacitor to a 520MV supply
Musa Reply
what is physics
Caya Reply
it is the science which we used in our daily life
Sujitha
Physics is the branch of science that deals with the study of matter and the interactions it undergoes with energy
Junior
it is branch of science which deals with study of happening in the human life
AMIT
it is when you get up of your arse and do some real work 😁
A 20MH coil has a resistance of 50 ohms and is connected in series with a capacitor to a 250MV supply if the circuit is to resonate at 100KHZ, Determine 1: the capacitance of the capacitor 2: the working voltage of the circuit, given that pie =3.142
Musa
Heat is transfered by thermal contact but if it is transfered by conduction or radiation, is it possible to reach in thermal equilibrium?
Eden Reply
Yes, It is possible by conduction if Surface is Adiabatic
Astronomy
what are the fundamentals qualities
Magret Reply
what is physic3
Kalilu
what is physic
Kalilu
Physics? Is a branch of science dealing with matter in relation to energy.
Moses
Physic... Is a purging medicine, which stimulates evacuation of the bowels.
Moses
are you asking for qualities or quantities?
Noman
give examples of three dimensional frame of reference
Ekwunazor Reply
your fat arse sitting all day is a good reference of three dimensional numbnut
Universe
Noman
Yes the Universe itself
Astronomy
Examine different types of shoes, including sports shoes and thongs. In terms of physics, why are the bottom surfaces designed as they are? What differences will dry and wet conditions make for these surfaces?
Lathan Reply
sports shoes are designed in such a way they are gripped well with your feet and their bases have and high friction surfaces, Thong shoes are for comfort, these are easily removed and light weight. these are usually low friction surfaces but in wet conditions they offer greater friction.
Noman
thong sleepers are usually used in restrooms.
Noman
what is wave
Ochigbo Reply
The phenomenon of transfer of energy
Noman
how does time flow in one dimension
Lord Reply
you mean in three dimensions......numbnut
yeah that was a mistake
Lord
if it flows in three dimensions does it mean if an object theoretically moves beyond the speed of light it won't experience time
Lord
time seems to flow in one direction...but I the past present and future happen every moment time flies regardless.
but if an object moves beyond the speed of light time stops right for it
Lord
yes but at light speed it ceases
Lord
yes it always flow from past to future.
Noman
if v=ktx Ly Mz find the value of x,y and z
Emmanuel Reply
x=v=ktx Ly Mz find the value of x,y and z
y=v=ktx Ly Mz find the value of x,y and z
z=v=ktx Ly Mz find the value of x,y and z
now get your lazy arse up and clean the kitchen 😁
I want to join the conversation
Subaba Reply
😂
hmm
Stephen
what conversation you talking about? .....numbnut
how do i calculate for period of the oscillation
Bridget Reply
T=2π√(m÷k).K is spring constance
Ambe
T=2π√m/k
Lord
does the force in a system result in the energy transfer?
Lebatam Reply
full meaning of GPS system
Anaele Reply
global positioning system
Noman
what's the use of the GPS
Matthew
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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