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Chapter 37: Problem 55

A spaceship, moving away from Earth at a speed of \(0.850 c\), reports back by transmitting at a frequency (measured in the spaceship frame) of \(100 \mathrm{MHz}\). To what frequency must Earth receivers be tuned to receive the report?

Short Answer

Expert verified
The Earth receivers must be tuned to approximately 28.47 MHz.

Step by step solution

01

Identify the Relevant Formula

To solve this problem, we will use the formula for the relativistic Doppler effect for frequency. The formula is \( f' = f \sqrt{\frac{1 - \beta}{1 + \beta}} \), where \( f' \) is the observed frequency, \( f \) is the emitted frequency (in the spaceship frame), and \( \beta = \frac{v}{c} \) is the ratio of the spaceship's speed \( v \) to the speed of light \( c \).
02

Calculate the Beta Value

Since the spaceship is moving away from Earth at a speed of \( 0.850 c \), we have \( \beta = 0.850 \).
03

Apply the Doppler Effect Formula

Substitute \( f = 100 \, \text{MHz} \) and \( \beta = 0.850 \) into the relativistic Doppler effect formula:\[ f' = 100 \, \text{MHz} \times \sqrt{\frac{1 - 0.850}{1 + 0.850}} \].
04

Calculate the Observed Frequency

Calculate the observed frequency \( f' \):\[ f' = 100 \, \text{MHz} \times \sqrt{\frac{0.150}{1.850}} \].First, calculate the fraction:\[ \frac{0.150}{1.850} \approx 0.0811 \].Next, find the square root:\[ \sqrt{0.0811} \approx 0.2847 \].Finally, calculate \( f' \):\[ f' = 100 \, \text{MHz} \times 0.2847 \approx 28.47 \, \text{MHz} \].

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Frequency Shift
Frequency shift occurs when the observed frequency of a wave differs from the frequency at which it was emitted. This can happen due to relative motion between the source and the observer.
In the context of the relativistic Doppler effect, the shift can be substantial when the velocities involved are at a significant fraction of the speed of light.

The formula for the relativistic Doppler effect for frequency is given by:
  • \(f' = f \sqrt{\frac{1 - \beta}{1 + \beta}}\)
Where \(f'\) is the observed frequency and \(f\) is the emitted frequency. The term \(\beta\) represents the ratio of the object's velocity to the speed of light, defined as \(\beta = \frac{v}{c}\).

Understanding this frequency shift is crucial for analyzing scenarios such as light from distant galaxies, where the redshift can inform us about their motion relative to Earth.
Special Relativity
Special relativity is a theory in physics that describes how objects in space and time are affected at high speeds.
Initiated by Albert Einstein, it introduces concepts different from classical mechanics, particularly noticeable as an object approaches the speed of light.

Key elements include:
  • The speed of light is constant for all observers, regardless of their motion.
  • Time and space are interwoven into a single continuum known as spacetime.
  • Moving objects will experience time dilation and length contraction.
In the case of the relativistic Doppler effect, special relativity dictates that not only the frequency but also time must be reconsidered when analyzing high-speed scenarios.
As the spaceship in the example moves away from Earth, the frequency of the radio transmission is shifted due to the relativistic effects predicted by this theory.
Speed of Light
The speed of light, denoted as \(c\), is a fundamental constant of nature with a value of approximately \(3 \times 10^8\) meters per second (or \(299,792,458\) m/s).
It's the maximum speed at which information or matter can travel through space.

Some key points to understand include:
  • Light travels at the same speed in a vacuum, irrespective of the observer's velocity.
  • This constancy is a cornerstone of Einstein's theory of special relativity, affecting how time and space are perceived at high speeds.
  • In practical terms, calculations involving the speed of light help us understand phenomena like Doppler shifts and time dilation.

When solving for the observed frequency in our example, the speed of light is essential in setting the baseline to compare the spaceship's velocity (\(0.850c\)) and calculate the frequency shift accurately using the relativistic Doppler formula.

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Most popular questions from this chapter

(a) The energy released in the explosion of \(1.00 \mathrm{~mol}\) of TNT is \(3.40 \mathrm{MJ}\). The molar mass of TNT is \(0.227 \mathrm{~kg} / \mathrm{mol}\). What weight of TNT is needed for an explosive release of \(2.50 \times 10^{14} \mathrm{~J} ?\) (b) Can you carry that weight in a backpack, or is a truck or train required? (c) Suppose that in an explosion of a fission bomb, \(0.080 \%\) of the fissionable mass is converted to released energy. What weight of fissionable material is needed for an explosive release of \(2.50 \times 10^{14} \mathrm{~J} ?\) (d) Can you carry that weight in a backpack, or is a truck or train required?

The mass of an electron is \(9.10938188 \times 10^{-31} \mathrm{~kg}\). To eight significant figures, find the following for the given electron kinetic energy: (a) \(\gamma\) and (b) \(\beta\) for \(K=2.0000000 \mathrm{keV}\), (c) \(\gamma\) and (d) \(\beta\) for \(K=2.0000000 \mathrm{MeV}\), and then (e) \(\gamma\) and (f) \(\beta\) for \(K=2.0000000 \mathrm{GeV}\).

How much work must be done to increase the speed of an electron from rest to (a) \(0.500 c\), (b) \(0.990 c\), and (c) \(0.99990 c\) ?

A sodium light source moves in a horizontal circle at a constant speed of \(0.200 c\) while emitting light at the proper wavelength of \(\lambda_{0}=589.00 \mathrm{~nm}\). Wavelength \(\lambda\) is measured for that light by a detector fixed at the center of the circle. What is the wavelength shift \(\lambda-\lambda_{0}\) ?

An unstable high-energy particle enters a detector and leaves a track of length \(0.856 \mathrm{~mm}\) before it decays. Its speed relative to the detector was \(0.992 c\). What is its proper lifetime? That is, how long would the particle have lasted before decay had it been at rest with respect to the detector?
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