<< Chapter < Page Chapter >> Page >

Using the lorentz transformation for length

A surveyor measures a street to be L = 100 m long in Earth frame S. Use the Lorentz transformation to obtain an expression for its length measured from a spaceship S , moving by at speed 0.20 c , assuming the x coordinates of the two frames coincide at time t = 0 .

Solution

  1. Identify the known: L = 100 m ; v = 0.20 c ; Δ τ = 0 .
  2. Identify the unknown: L .
  3. Express the answer as an equation. The surveyor in frame S has measured the two ends of the stick simultaneously, and found them at rest at x 2 and x 1 a distance L = x 2 x 1 = 100 m apart. The spaceship crew measures the simultaneous location of the ends of the sticks in their frame. To relate the lengths recorded by observers in S and S, respectively, write the second of the four Lorentz transformation equations as:
    x 2 x 1 = x 2 v t 1 v 2 / c 2 x 1 v t 1 v 2 / c 2 = x 2 x 1 1 v 2 / c 2 = L 1 v 2 / c 2 .
  4. Do the calculation. Because x 2 x 1 = 100 m , the length of the moving stick is equal to:
    L = ( 100 m ) 1 v 2 / c 2 = ( 100 m ) 1 ( 0.20 ) 2 = 98.0 m.

    Note that the Lorentz transformation gave the length contraction equation for the street.
Got questions? Get instant answers now!

Lorentz transformation and simultaneity

The observer shown in [link] standing by the railroad tracks sees the two bulbs flash simultaneously at both ends of the 26 m long passenger car when the middle of the car passes him at a speed of c /2. Find the separation in time between when the bulbs flashed as seen by the train passenger seated in the middle of the car.

An observer on the ground watches a train car that is moving to the right with velocity v. Inside, at each end of the train car are lamps, each emitting a signal that is propagating toward the center of the car, and in the center of the car sits a passenger.
An person watching a train go by observes two bulbs flash simultaneously at opposite ends of a passenger car. There is another passenger inside of the car observing the same flashes but from a different perspective.

Solution

  1. Identify the known: Δ t = 0 .
    Note that the spatial separation of the two events is between the two lamps, not the distance of the lamp to the passenger.
  2. Identify the unknown: Δ t = t 2 t 1 .
    Again, note that the time interval is between the flashes of the lamps, not between arrival times for reaching the passenger.
  3. Express the answer as an equation:
    Δ t = Δ t + v Δ x / c 2 1 v 2 / c 2 .
  4. Do the calculation:
    0 = Δ t + c 2 ( 26 m ) / c 2 1 v 2 / c 2 Δ t = 26 m/s 2 c = 26 m/s 2 ( 3.00 × 10 8 m/s ) Δ t = −4.33 × 10 −8 s.

Significance

The sign indicates that the event with the larger x 2 , namely, the flash from the right, is seen to occur first in the S frame, as found earlier for this example, so that t 2 < t 1 .

Got questions? Get instant answers now!

Space-time

Relativistic phenomena can be analyzed in terms of events in a four-dimensional space-time . When phenomena such as the twin paradox, time dilation, length contraction, and the dependence of simultaneity on relative motion are viewed in this way, they are seen to be characteristic of the nature of space and time, rather than specific aspects of electromagnetism.

In three-dimensional space, positions are specified by three coordinates on a set of Cartesian axes, and the displacement of one point from another is given by:

( Δ x , Δ y , Δ z ) = ( x 2 x 1 , y 2 y 1 , z 2 z 1 ) .

The distance Δ r between the points is

Δ r 2 = ( Δ x ) 2 + ( Δ y ) 2 + ( Δ z ) 2 .

The distance Δ r is invariant under a rotation of axes. If a new set of Cartesian axes rotated around the origin relative to the original axes are used, each point in space will have new coordinates in terms of the new axes, but the distance Δ r given by

Questions & Answers

how does colour appear in thin films
Nwjwr Reply
in the wave equation y=Asin(kx-wt+¢) what does k and w stand for.
Kimani Reply
derivation of lateral shieft
James Reply
Hi
Amjad
Hi
Amjad
hi
ALFRED
how are you?
Amjad
hi
asif
hi
Imran
I'm fine
ALFRED
total binding energy of ionic crystal at equilibrium is
All Reply
How does, ray of light coming form focus, behaves in concave mirror after refraction?
Bishesh Reply
Refraction does not occur in concave mirror. If refraction occurs then I don't know about this.
Sushant
What is motion
Izevbogie Reply
Anything which changes itself with respect to time or surrounding
Sushant
good
Chemist
and what's time? is time everywhere same
Chemist
No
Sushant
how can u say that
Chemist
do u know about black hole
Chemist
Not so more
Sushant
Radioactive substance
DHEERAJ
These substance create harmful radiation like alpha particle radiation, beta particle radiation, gamma particle radiation
Sushant
But ask anything changes itself with respect to time or surrounding A Not any harmful radiation
DHEERAJ
explain cavendish experiment to determine the value of gravitational concept.
Celine Reply
For the question about the scuba instructor's head above the pool, how did you arrive at this answer? What is the process?
Evan Reply
as a free falling object increases speed what is happening to the acceleration
Success Reply
of course g is constant
Alwielland
acceleration also inc
Usman
which paper will be subjective and which one objective
jay
normal distributiin of errors report
Dennis
normal distribution of errors
Dennis
acceleration also increases
Jay
there are two correct answers depending on whether air resistance is considered. none of those answers have acceleration increasing.
Michael
Acceleration is the change in velocity over time, hence it's the derivative of the velocity with respect to time. So this case would depend on the velocity. More specifically the change in velocity in the system.
Big
photo electrons doesn't emmit when electrons are free to move on surface of metal why?
Rafi Reply
What would be the minimum work function of a metal have to be for visible light(400-700)nm to ejected photoelectrons?
Mohammed Reply
give any fix value to wave length
Rafi
40 cm into change mm
Arhaan Reply
40cm=40.0×10^-2m =400.0×10^-3m =400mm. that cap(^) I have used above is to the power.
Prema
i.e. 10to the power -2 in the first line and 10 to the power -3 in the the second line.
Prema
there is mistake in my first msg correction is 40cm=40.0×10^-2m =400.0×10^-3m =400mm. sorry for the mistake friends.
Prema
40cm=40.0×10^-2m =400.0×10^-3m =400mm.
Prema
this msg is out of mistake. sorry friends​.
Prema
what is physics?
sisay Reply
why we have physics
Anil Reply
because is the study of mater and natural world
John
because physics is nature. it explains the laws of nature. some laws already discovered. some laws yet to be discovered.
Yoblaze
physics is the study of non living things if we added it with biology it becomes biophysics and bio is the study of living things tell me please what is this?
tahreem
physics is the study of matter,energy and their interactions
Buvanes
all living things are matter
Buvanes
why rolling friction is less than sliding friction
tahreem
thanks buvanas
tahreem
is this a physics forum
Physics Reply
Practice Key Terms 4

Get the best University physics vol... course in your pocket!





Source:  OpenStax, University physics volume 3. OpenStax CNX. Nov 04, 2016 Download for free at http://cnx.org/content/col12067/1.4
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'University physics volume 3' conversation and receive update notifications?

Ask