<< Chapter < Page Chapter >> Page >

Velocity transformation equations for light

Suppose a spaceship heading directly toward Earth at half the speed of light sends a signal to us on a laser-produced beam of light ( [link] ). Given that the light leaves the ship at speed c as observed from the ship, calculate the speed at which it approaches Earth.

An illustration of a spaceship moving to the right with velocity v=0.500c and emitting a horizontal laser beam, which propagates to the right with velocity c.
How fast does a light signal approach Earth if sent from a spaceship traveling at 0.500 c ?

Strategy

Because the light and the spaceship are moving at relativistic speeds, we cannot use simple velocity addition. Instead, we determine the speed at which the light approaches Earth using relativistic velocity addition.

Solution

  1. Identify the knowns: v = 0.500 c ; u = c .
  2. Identify the unknown: u .
  3. Express the answer as an equation: u = v + u 1 + v u c 2 .
  4. Do the calculation:
    u = v + u 1 + v u c 2 = 0.500 c + c 1 + ( 0.500 c ) ( c ) c 2 = ( 0.500 + 1 ) c ( c 2 + 0.500 c 2 c 2 ) = c .

Significance

Relativistic velocity addition gives the correct result. Light leaves the ship at speed c and approaches Earth at speed c . The speed of light is independent of the relative motion of source and observer, whether the observer is on the ship or earthbound.

Got questions? Get instant answers now!

Velocities cannot add to greater than the speed of light, provided that v is less than c and u does not exceed c . The following example illustrates that relativistic velocity addition is not as symmetric as classical velocity addition.

Relativistic package delivery

Suppose the spaceship in the previous example approaches Earth at half the speed of light and shoots a canister at a speed of 0.750 c ( [link] ). (a) At what velocity does an earthbound observer see the canister if it is shot directly toward Earth? (b) If it is shot directly away from Earth?

The first illustration shows the spaceship moving to the right, toward the earth, with velocity v=0.500c, and a canister moving to the right with a velocity u prime = 0.750c. The second illustration shows the spaceship moving to the right, toward the earth, with velocity v=0.500c, and a canister moving to the left with a velocity u prime = -0.750c.
A canister is fired at 0.7500 c toward Earth or away from Earth.

Strategy

Because the canister and the spaceship are moving at relativistic speeds, we must determine the speed of the canister by an earthbound observer using relativistic velocity addition instead of simple velocity addition.

Solution for (a)

  1. Identify the knowns: v = 0.500 c ; u = 0.750 c .
  2. Identify the unknown: u .
  3. Express the answer as an equation: u = v + u 1 + v u c 2 .
  4. Do the calculation:
    u = v + u 1 + v u c 2 = 0.500 c + 0.750 c 1 + ( 0.500 c ) ( 0.750 c ) c 2 = 0.909 c .

Solution for (b)

  1. Identify the knowns: v = 0.500 c ; u = −0.750 c .
  2. Identify the unknown: u .
  3. Express the answer as an equation: u = v + u 1 + v u c 2 .
  4. Do the calculation:
    u = v + u 1 + v u c 2 = 0.500 c + ( −0.750 c ) 1 + ( 0.500 c ) ( −0.750 c ) c 2 = −0.400 c .

Significance

The minus sign indicates a velocity away from Earth (in the opposite direction from v ), which means the canister is heading toward Earth in part (a) and away in part (b), as expected. But relativistic velocities do not add as simply as they do classically. In part (a), the canister does approach Earth faster, but at less than the vector sum of the velocities, which would give 1.250 c . In part (b), the canister moves away from Earth at a velocity of −0.400 c , which is faster than the −0.250 c expected classically. The differences in velocities are not even symmetric: In part (a), an observer on Earth sees the canister and the ship moving apart at a speed of 0.409 c , and at a speed of 0.900 c in part (b).

Got questions? Get instant answers now!

Check Your Understanding Distances along a direction perpendicular to the relative motion of the two frames are the same in both frames. Why then are velocities perpendicular to the x -direction different in the two frames?

Although displacements perpendicular to the relative motion are the same in both frames of reference, the time interval between events differ, and differences in dt and d t lead to different velocities seen from the two frames.

Got questions? Get instant answers now!

Summary

  • With classical velocity addition, velocities add like regular numbers in one-dimensional motion: u = v + u , where v is the velocity between two observers, u is the velocity of an object relative to one observer, and u is the velocity relative to the other observer.
  • Velocities cannot add to be greater than the speed of light.
  • Relativistic velocity addition describes the velocities of an object moving at a relativistic velocity.

Problems

If two spaceships are heading directly toward each other at 0.800 c , at what speed must a canister be shot from the first ship to approach the other at 0.999 c as seen by the second ship?

Got questions? Get instant answers now!

Two planets are on a collision course, heading directly toward each other at 0.250 c . A spaceship sent from one planet approaches the second at 0.750 c as seen by the second planet. What is the velocity of the ship relative to the first planet?

0.615 c

Got questions? Get instant answers now!

When a missile is shot from one spaceship toward another, it leaves the first at 0.950 c and approaches the other at 0.750 c . What is the relative velocity of the two ships?

Got questions? Get instant answers now!

What is the relative velocity of two spaceships if one fires a missile at the other at 0.750 c and the other observes it to approach at 0.950 c ?

0.696 c

Got questions? Get instant answers now!

Prove that for any relative velocity v between two observers, a beam of light sent from one to the other will approach at speed c (provided that v is less than c , of course).

Got questions? Get instant answers now!

Show that for any relative velocity v between two observers, a beam of light projected by one directly away from the other will move away at the speed of light (provided that v is less than c , of course).

(Proof)

Got questions? Get instant answers now!

Questions & Answers

what is bohrs model for hydrogen atom
Swagatika Reply
hi
Tr
Hello
Youte
Hi
Nwangwu-ike
hi
Siddiquee
what is the value of speed of light
Propessor Reply
1.79×10_¹⁹ km per hour
Swagatika
what r dwarf planet
Sivalakshmi Reply
what is energy
Isiguzo Reply
কাজের একক কী
Jasim
কাজের একক কী
Jasim
friction ka direction Kaise pata karte hai
Rahul Reply
friction is always in the opposite of the direction of moving object
Punia
A twin paradox in the special theory of relativity arises due to.....? a) asymmetric of time only b) symmetric of time only c) only time
Varia Reply
b) symmetric of time only
Swagatika
fundamental note of a vibrating string
fasoyin Reply
every matter made up of particles and particles are also subdivided which are themselves subdivided and so on ,and the basic and smallest smallest smallest division is energy which vibrates to become particles and thats why particles have wave nature
Alvin
what are matter waves? Give some examples
mallam Reply
according to de Broglie any matter particles by attaining the higher velocity as compared to light'ill show the wave nature and equation of wave will applicable on it but in practical life people see it is impossible however it is practicaly true and possible while looking at the earth matter at far
Manikant
a centeral part of theory of quantum mechanics example:just like a beam of light or a water wave
Swagatika
Mathematical expression of principle of relativity
Nasir Reply
given that the velocity v of wave depends on the tension f in the spring, it's length 'I' and it's mass 'm'. derive using dimension the equation of the wave
obia Reply
What is the importance of de-broglie's wavelength?
Mukulika Reply
he related wave to matter
Zahid
at subatomic level wave and matter are associated. this refering to mass energy equivalence
Zahid
it is key of quantum
Manikant
how those weight effect a stable motion at equilibrium
Nonso Reply
how do I differentiate this equation- A sinwt with respect to t
Evans Reply
just use the chain rule : let u =wt , the dy/dt = dy/du × du/dt : wA × cos(wt)
Jerry
I see my message got garbled , anyway use the chain rule with u= wt , etc...
Jerry
de broglie wave equation
LoNE Reply
vy beautiful equation
chandrasekhar
what is electro statics
fitsum Reply
when you consider systems consisting of fixed charges
Sherly
hi ma new
adam
hello
Amal
Practice Key Terms 2

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