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Chapter 22: Problem 56

Light of wavelength \(659.6 \mathrm{nm}\) is emitted by a star. The wavelength of this light as measured on Earth is \(661.1 \mathrm{nm} .\) How fast is the star moving with respect to Earth? Is it moving toward Earth or away from it?

Short Answer

Expert verified
Answer: The star is moving away from Earth with a relative speed of approximately 681 m/s.

Step by step solution

01

Calculate the change in observed wavelength

Using the given information, we can calculate the change in wavelength: \(\Delta \lambda = \lambda_{observed} - \lambda_{emitted} = 661.1 \,\text{nm} - 659.6 \,\text{nm} = 1.5 \,\text{nm}\) #Step 2: Use the Doppler effect formula for the change in wavelength#
02

Apply the Doppler effect formula

The Doppler effect formula for the change in wavelength due to relative motion is given by the following equation: \(\Delta \lambda = \frac{v}{c} \lambda_{emitted}\) where \(\Delta \lambda\) is the change in wavelength, \(v\) is the relative speed of the star with respect to Earth, \(c\) is the speed of light in vacuum, and \(\lambda_{emitted}\) is the wavelength of light emitted by the star. #Step 3: Solve for the speed of the star relative to Earth#
03

Calculate the relative speed

Rearrange the Doppler effect formula to solve for the speed \(v\): \(v = \frac{\Delta \lambda \cdot c}{\lambda_{emitted}}\) Now plug in the values for \(\Delta \lambda\), \(\lambda_{emitted}\), and \(c = 3.0 \times 10^8 \, \text{m/s}\) (converted to meters): \(v = \frac{(1.5 \times 10^{-9}\, \text{m}) \cdot (3.0 \times 10^8\, \text{m/s})}{(659.6 \times 10^{-9}\, \text{m})}\) #Step 4: Calculate the result#
04

Evaluate the expression

Calculate the relative speed of the star: \(v = \frac{(1.5 \times 10^{-9}\, \text{m}) \cdot (3.0 \times 10^8\, \text{m/s})}{(659.6 \times 10^{-9}\, \text{m})} \approx 681 \, \text{m/s}\) #Step 5: Determine the direction of motion#
05

Determine if the star is moving towards or away from Earth

Since \(\lambda_{observed} > \lambda_{emitted}\), the light has been redshifted, which means the star is moving away from Earth. #Solution# The star is moving away from Earth with a relative speed of approximately \(681 \,\text{m/s}\).

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