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Chapter 38: Problem 45

A spaceship moving away from Earth at a speed \(0.900 c\) radios its reports back to Earth using a frequency of 100 MHz measured in the spaceship frame. To what frequency must Earth's receivers be tuned in order to receive the reports?

Short Answer

Expert verified
The receivers must be tuned to approximately 22.94 MHz.

Step by step solution

01

Understand the Doppler Effect

The problem involves the relativistic Doppler effect, which occurs when there's relative motion between a source and an observer. In this case, the spaceship (source) is moving away from Earth (observer) at a high speed.
02

Identify given values

Identify the given values: the frequency of the radio signal in the spaceship frame is 100 MHz, and the speed of the spaceship moving away from Earth is 0.900 c.
03

Use the Doppler effect formula

Use the formula for the relativistic Doppler effect for a source moving away: \[ f = f_0 \times \frac{\text{1}}{\text{纬}(1 + \frac{v}{c})} \] where \(f\) is the observed frequency, \(f_0\) is the emitted frequency (100 MHz), \(v\) is the velocity of the spaceship (0.900c), and \(c\) is the speed of light.
04

Calculate the Lorentz factor

Calculate the Lorentz factor \(纬\) using the formula: \[ \text{纬} = \frac{1}{\text{鈭殅(1 - (\frac{v}{c})^2)} \] Substituting \(v = 0.900c\): \[ \text{纬} = \frac{1}{\text{鈭殅(1 - 0.900^2)} = \frac{1}{\text{鈭殅(0.19)} 鈮� 2.294 \]
05

Substitute into the Doppler effect formula

Now substitute the known values into the Doppler effect formula: \[ f = 100 \text{ MHz} \times \frac{1}{2.294(1 + 0.900)} \] Simplify the expression: \[ f 鈮� 100 \text{ MHz} \times \frac{1}{2.294 \times 1.9} \] \[ f 鈮� 100 \text{ MHz} \times \frac{1}{4.3586} \] Finally, \[ f 鈮� 22.94 \text{ MHz} \]

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Key Concepts

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

Lorentz factor
The Lorentz factor (纬) is a crucial part of relativity. It measures the change in time and space when objects move close to the speed of light. This factor helps explain how moving objects experience time and space differently. The formula to calculate the Lorentz factor is: \[ 纬 = \frac{1}{\text{鈭殅(1 - (\frac{v}{c})^2)} \]Where:
  • 纬: Lorentz factor
  • v: Velocity of the moving object
  • c: Speed of light
For example, for a spaceship moving at 0.900c, the Lorentz factor becomes approximately 2.294. This means that time and distances for the spaceship are different than those for someone at rest. It's vital in calculating effects like time dilation and the Doppler effect in high-speed scenarios.
Radio frequency
Radio frequencies are used for communication over long distances. They're part of the electromagnetic spectrum and are measured in hertz (Hz). Common units include megahertz (MHz) and gigahertz (GHz). One megahertz is equal to one million hertz. In the exercise, the spaceship sends signals at a frequency of 100 MHz in its frame. However, due to the spaceship's high speed, the frequency changes when observed from Earth. This change is predicted by the relativistic Doppler effect. This shift in frequency is what engineers and scientists must account for when designing communication systems for high-speed travel, like satellites and space missions.
Relativistic calculations
When dealing with objects moving near the speed of light, standard calculations don't suffice. Instead, we use relativistic formulas to account for effects like time dilation and length contraction. Let鈥檚 dive into the relativistic Doppler effect, which shifts observed frequencies due to the relative motion between the source and the observer. The formula for a source moving away from the observer is:\[ f = f_0 \times \frac{\text{1}}{\text{纬}(1 + \frac{v}{c})} \]Where:
  • f: Observed frequency
  • f鈧�: Emitted frequency (100 MHz in the spaceship鈥檚 frame)
  • 纬: Lorentz factor (2.294 for 0.900c)
  • v: Velocity of spaceship (0.900c)
  • c: Speed of light
Using these values, we determine the frequency Earth must tune to is approximately 22.94 MHz. This example highlights the importance of understanding relativistic transformations in advanced physics and engineering.

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Most popular questions from this chapter

An unpowered rocket moves past you in the positive \(x\) direction at speed \(v^{\text {rel }}=0.9 c\). This rocket fires a bullet out the back that you measure to be moving at speed \(v_{\text {bullet }}=0.3 c\) in the positive \(x\) direction. With what speed relative to the rocket did the rocket observer fire the bullet out the back of her ship?

Astrophysicists describe the redshift of receding astronomical objects using the redshift factor \(z\), defined implicitly in the following equation: $$ \lambda_{\text {observed }} \equiv(1+z) \lambda_{\text {emitted }} $$ Here \(\lambda_{\text {observed }}\) is the wavelength of light observed from Earth, while \(\lambda_{\text {emitted }}\) is the wavelength of the light emitted from the source as measured in the rest frame of the source. The emitted wavelength is known if one knows the emitting atom, identified from the pattern of different wavelengths characteristic of that atom. Astrophysicists measuring the redshifts of light from extremely remote quasars calculate a \(z\) -factor in the neighborhood of \(z \approx 6 .\) Use the Dopplershift equations of special relativity to determine how fast such quasars are moving away from Earth. Note: Actually, for such distant objects the unmodified Doppler shift formula of special relativity does not apply. Instead, one thinks of the space between Earth and the source expanding as the universe expands; the wavelength of the light expands with this expansion of the universe as it travels from the source quasar to us.

(a) Find an equation for the unknown mass \(m\) of a particle if you know its momentum \(p\) and its kinetic energy \(K\). Show that this expression reduces to an expected result for nonrelativistic particle speeds. (b) Find the mass of a particle whose kinetic energy is \(K=55.0 \mathrm{MeV}\) and whose momentum is \(p=\) \(121 \mathrm{MeV} / \mathrm{c}\). Express your answer as a decimal fraction or multiple of the mass \(m_{\mathrm{e}}\) of the electron.

Review Problem 40 , in which we concluded that a limo of proper length \(30 \mathrm{~m}\) can fit into a garage of proper length \(6 \mathrm{~m}\) with room to spare. This result is possible because the speeding limo is observed by Garageman to be Lorentz -contracted. Carman protests that in the rest frame of the limo (in which the limo is its full proper length) it is the garage that is Lorentz-contracted. As a result, he claims, there is no possibility whatever that the limo can fit into the garage. What could be the possible basis for resolving this paradox? (Hint: Think about the space and time locations of two events: event A, front garage door closes and event \(\mathrm{B}\), rear garage door opens.)

An unstable high-energy particle is created in a collision inside a detector and leaves a track \(1.05 \mathrm{~mm}\) long before it decays while still in the detector. Its speed relative to the detector was \(0.992 c\). How long did the particle live as recorded in its rest frame?
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