Relativistic Doppler Effect

In summary, the conversation discusses the problem of a shuttle launching from a space station and traveling towards a rocket at a different velocity. The given equations and formulas are used to determine the velocity of the shuttle relative to the Earth, which is found to be 3c/5. However, it is noted that the last step of transforming to the Earth's frame of reference may be incorrect due to the directions of the rocket and shuttle's velocities.
  • #1
mmh37
59
0
This is the problem

A shuttle is launched from a space station and travels away from it in a straight line. It rapidly accelerates and obtains a steady velocity of v = 4c/5 relative to the space station. The spaceship sends out radio signals of a frequency f. The spaceship is on a mission to dock with a rocket, which travels away from the Earth at velocity u. The rocket receives the shuttle's transmission at a frequency 3f/2. Show that u of the shuttle relative to the Earth is 3c/5.

GIVEN:

relativistic doppler effect; source moves towards the observer

[tex] f' = f \sqrt {\frac {c+v} {c-v}} [/tex]

velocity addition formula:

[tex] v' = \frac {v-u} {1 - v*u/c^2} [/tex]


My atempt

[tex] f' = f \sqrt {\frac {c+v} {c-v}} = 3f/2 [/tex]

hence: v = 5c/13

this is the relative speed between the shuttle and the rocket. Therefore in frame S' of the shuttle the rocket moves with a velociy u' = 5c/13

Now we have to transform to the Earth's frame of reference S, where v = 4c/5

using the velocity addition formula:

[tex] u = \frac {u' + v} {1 + v*u'/c^2} = 77/85 *c [/tex]

which is wrong.

Does anyone see why or can anyone give me a hint towards the right answer? That would be very helpful and much appreciated!
 
Last edited:
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  • #2
Only the last step is wrong, although it's hard to reason from the text since there's no picture showing the directions.

Anyway, the rocket and shuttle are approaching each other while the rocket is leaving Earth (so shuttle is approaching earth), so the velocity of the rocket wrt Earth and the velocity of the shuttle wrt the rocket have opposite signs.
 
  • #3
thanks very much for this. It does make sense (although I find it easy to miss) and one get's the correct answer by altering the signs of velocity u.
That was very helpful!:smile:
 

1. What is the Relativistic Doppler Effect?

The Relativistic Doppler Effect is a phenomenon in which the frequency and wavelength of a wave appear to change when observed by an observer moving relative to the source of the wave. It is a consequence of the theory of relativity, which states that the observed properties of an object depend on the relative motion between the observer and the object.

2. How does the Relativistic Doppler Effect differ from the Classical Doppler Effect?

The Classical Doppler Effect only takes into account the relative speeds between the source of the wave and the observer, while the Relativistic Doppler Effect also considers the effects of time dilation and length contraction due to the observer's relative motion. This makes the Relativistic Doppler Effect more accurate for high speeds and near the speed of light.

3. What is the formula for calculating the Relativistic Doppler Effect?

The formula for calculating the Relativistic Doppler Effect is: f' = f₀√[(1+v/c)/(1-v/c)]where f' is the observed frequency, f₀ is the emitted frequency, v is the relative velocity between the source and observer, and c is the speed of light.

4. How does the Relativistic Doppler Effect apply to light waves?

The Relativistic Doppler Effect applies to all types of waves, including light waves. It explains the observed changes in the frequency and wavelength of light emitted from a moving source or observed by a moving observer. This effect is also used in astronomy to determine the relative velocities and distances of celestial objects.

5. What are some real-life applications of the Relativistic Doppler Effect?

The Relativistic Doppler Effect has numerous real-life applications, including its use in radar and sonar technology, satellite communication, and medical imaging. It is also used in studying the expansion of the universe and in calculating the speeds of objects in space, among other things.

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