Worksheet: The Relativistic Doppler Effect

In this worksheet, we will practice calculating the change in the frequency and wavelength of electromagnetic radiation due to relative motion of the source and observer.

Q1:

A spaceship traveling from Earth to Alpha Centauri sends a transponder signal back to Earth every hour. The spaceship is moving at a speed of 0 . 6 𝑐 relative to Earth, and the frequency of the waves is 10 MHz in the rest frame of the spaceship. What is the frequency of the waves in the rest frame of Earth?

Q2:

In the spectrum of light from a star that is moving toward Earth, every wavelength of light has been shifted by a factor of 1 2 . At what speed must the star be moving toward Earth?

  • A 0 . 4 𝑐
  • B 0 . 5 𝑐
  • C 0 . 8 𝑐
  • D 0 . 7 𝑐
  • E 0 . 6 𝑐

Q3:

An emission line in the spectrum of the light from a star is at a frequency of 2 . 0 0 × 1 0 Hz in the rest frame of a laboratory on Earth. The star is moving away from Earth at a speed of 0 . 0 2 𝑐 . What was the frequency of the emission line in the rest frame of the star? Ignore the effect of cosmological redshift. Give your answer to 3 significant figures.

  • A 1 . 9 6 × 1 0 Hz
  • B 2 . 0 4 × 1 0 Hz
  • C 2 . 0 0 × 1 0 Hz
  • D 1 . 9 2 × 1 0 Hz
  • E 2 . 0 8 × 1 0 Hz

Q4:

An object that emits a specific wavelength of light is moving toward an observer at a speed of 0 . 9 5 𝑐 . The observer sees light of wavelength 200 nm coming from the object. What wavelength of light is the object emitting? Give your answer to 3 significant figures.

Q5:

In the spectrum of light coming from a star, the Hα emission line is found to be shifted toward the red end of the spectrum. In the rest frame of the source object, the H 𝛼 line has a frequency of 4 . 5 7 × 1 0 Hz. In the rest frame of Earth, the H 𝛼 line appears at 4 . 3 5 × 1 0 Hz. If all of this redshift is due to relativistic motion of the star relative to Earth, how fast is the star moving away from Earth? Give your answer to 3 significant figures as a multiple of the speed of light.

  • A 0 . 1 5 7 𝑐
  • B 0 . 0 2 4 7 𝑐
  • C 0 . 0 4 9 3 𝑐
  • D 0 . 2 2 2 𝑐
  • E 0 . 0 3 7 0 𝑐

Q6:

If you drive fast enough toward a red light, because of the relativistic Doppler effect, it will appear green. If a driver on a road drives fast enough toward a set of traffic lights, instead of seeing a red light, he will see a green light. How fast does the driver have to travel in order for the red light, with a wavelength of 650 nm, to appear green, with a wavelength of 520 nm? Give your answer, in kilometers per hour, to 3 significant figures.

  • A 2 . 3 7 × 1 0 km/h
  • B 1 . 2 0 × 1 0 km/h
  • C 5 . 0 6 × 1 0 km/h
  • D 3 . 6 0 × 1 0 km/h
  • E 1 . 4 1 × 1 0 km/h

Q7:

A spaceship is traveling from Earth to Epsilon Eridani. Every hour, the spaceship sends a transponder signal back to Earth. At what speed would the spaceship have to be traveling relative to Earth for the transponder signal to have a frequency in the rest frame of Earth that is 1 3 of the frequency of the signal in the rest frame of the spaceship?

  • A 0 . 8 𝑐
  • B 0 . 9 𝑐
  • C 0 . 6 𝑐
  • D 0 . 5 𝑐
  • E 0 . 7 𝑐

Q8:

The H- 𝛽 emission line has a wavelength of 486 nm, in the rest frame of the source object. In the spectrum of light coming from a star, the H- 𝛽 line is found to be at 489 nm.

Is the star moving toward or away from Earth?

  • AToward Earth
  • BAway from Earth

At what speed is the star moving relative to Earth? Give your answer as a multiple of the speed of light to 3 significant figures.

  • A 0 . 0 0 7 8 4 𝑐
  • B 0 . 0 0 3 0 8 𝑐
  • C 0 . 0 0 6 1 5 𝑐
  • D 0 . 0 0 5 4 6 𝑐
  • E 0 . 0 5 5 5 𝑐

Q9:

Consider two reference frames, 𝑆 and 𝑆 , where 𝑆 moves at a speed 𝑢 relative to 𝑆 . In 𝑆 , an object emits light at a frequency 𝑓 . In 𝑆 , this light has a frequency 𝑓 . The graph shows the ratio of 𝑓 to 𝑓 against 𝛽 , where 𝛽 = 𝑢 𝑐 , for 𝛽 = 1 to 𝛽 = 1 .

Which region of the graph shows redshift of the light emitted by the object?

  • A 1 < 𝛽
  • B 𝛽 > 1
  • C 0 < 𝛽 < 1
  • D 1 < 𝛽 < 1
  • E 1 < 𝛽 < 0

Which region of the graph shows blueshift of the light emitted by the object?

  • A 1 < 𝛽
  • B 1 < 𝛽 < 1
  • C 0 < 𝛽 < 1
  • D 𝛽 > 1
  • E 1 < 𝛽 < 0

At what value of 𝑢 would the light in frame 𝑆 have a frequency twice that of the light in frame 𝑆 ?

  • A 0 . 4 𝑐
  • B 0 . 5 𝑐
  • C 0 . 6 𝑐
  • D 0 . 3 𝑐
  • E 0 . 2 5 𝑐

At what value of 𝑢 would the light in frame 𝑆 have a frequency half that of the light in frame 𝑆 ?

  • A 0 . 6 𝑐
  • B 0 . 4 𝑐
  • C 0 . 3 𝑐
  • D 0 . 2 𝑐
  • E 0 . 5 𝑐

Q10:

Consider two reference frames, 𝑆 and 𝑆 , where 𝑆 moves at a speed 𝑢 relative to 𝑆 . In 𝑆 , an object emits light with a wavelength 𝜆 . In 𝑆 , this light has a wavelength 𝜆 . The graph shows the ratio of 𝜆 to 𝜆 against 𝛽 , where 𝛽 = 𝑢 𝑐 , for 𝛽 = 1 to 𝛽 = 1 .

Which region of the graph shows redshift of the light emitted by the object?

  • A 0 < 𝛽 < 1
  • B 𝛽 < 1
  • C 1 < 𝛽 < 1
  • D 0 < 𝛽
  • E 1 < 𝛽

Which region of the graph shows blueshift of the light emitted by the object?

  • A 0 < 𝛽 < 1
  • B 1 < 𝛽 < 1
  • C 1 < 𝛽
  • D 𝛽 < 1
  • E 1 < 𝛽 < 0

What is 𝜆 𝜆 when 𝛽 = 0 . 6 ?

If 𝑢 = 0 . 8 𝑐 and 𝜆 = 5 0 0 n m , what is 𝜆 ?

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