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Chapter 35: Problem 38

A HeNe laser onboard a spaceship moving toward a remote space station emits a beam of red light toward the space station. The wavelength of the beam, as measured by a wavelength meter on board the spaceship, is \(632.8 \mathrm{nm}\). If the astronauts on the space station see the beam as a blue beam of light with a measured wavelength of \(514.5 \mathrm{nm},\) what is the relative speed of the spaceship with respect to the space station? What is the shift parameter \(z\) in this case?

Short Answer

Expert verified
Answer: The relative speed of the spaceship with respect to the space station is approximately \(1.881 * 10^8 ms^{-1}\), and the shift parameter \(z \approx -0.1869\).

Step by step solution

01

Convert wavelengths to frequencies.

Using the formula \(c = \lambda f\), where \(c\) is the speed of light (\(3 * 10^8 ms^{-1}\)), \(\lambda\) is the wavelength, and \(f\) is the frequency, we can convert the given wavelengths to their corresponding frequencies. Let's calculate the emitted and observed frequencies. Emitted wavelength (\(\lambda\)): \(632.8 nm\) = \(632.8 * 10^{-9} m\) Emitted frequency (\(f\)): \(f = \frac{c}{\lambda} = \frac{3 * 10^8 ms^{-1}}{632.8 * 10^{-9} m} = 4.738 * 10^{14} Hz\) Observed wavelength (\(\lambda'\)): \(514.5 nm\) = \(514.5 * 10^{-9} m\) Observed frequency (\(f'\)): \(f' = \frac{c}{\lambda'} = \frac{3 * 10^8 ms^{-1}}{514.5 * 10^{-9} m} = 5.834 * 10^{14} Hz\)
02

Use the Doppler effect formula to find the relative velocity.

Now that we have the emitted and observed frequencies, we can use the Doppler effect formula to find the relative velocity of the spaceship with respect to the space station: \(f' = \frac{f}{1+\frac{v}{c}}\) Rearranging for \(v\): \(v = c (\frac{f}{f'} - 1)\) Substitute the values of \(f, f', c\): \(v = 3 * 10^8 ms^{-1} (\frac{4.738 * 10^{14} Hz}{5.834 * 10^{14} Hz} - 1)\) \(v \approx -1.881 * 10^8 ms^{-1}\) The negative sign indicates that the spaceship is moving towards the space station. Therefore, the relative speed of the spaceship with respect to the space station is approximately \(1.881 * 10^8 ms^{-1}\).
03

Compute the shift parameter \(z\).

Finally, we will compute the shift parameter \(z = \frac{\Delta\lambda}{\lambda}\), where \(\Delta\lambda = \lambda'-\lambda\). \(\Delta\lambda = 514.5 nm - 632.8 nm = -118.3 nm\) \(z = \frac{-118.3 nm}{632.8 nm} \approx -0.1869\) The shift parameter \(z \approx -0.1869\).

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Most popular questions from this chapter

Use the relativistic velocity addition to reconfirm that the speed of light with respect to any inertial reference frame is \(c\). Assume one-dimensional motion along a common \(x\) -axis.

A square of area \(100 \mathrm{~m}^{2}\) that is at rest in the reference frame is moving with a speed \((\sqrt{3} / 2) c\). Which of the following statements is incorrect? a) \(\beta=\sqrt{3} / 2\) b) \(\gamma=2\) c) To an observer at rest, it looks like another square with an area less than \(100 \mathrm{~m}^{2}\) d) The length along the moving direction is contracted by a factor of \(\frac{1}{2}\)

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