A source of electromagnetic radiation is moving in a radial direction relative to you. The frequency you measure is 1.25 times the frequency measured in the rest frame of the source. What is the speed of the source relative to you? Is the source moving toward you or away from you?

Verified step by step guidance

1. Identify the relevant concept: This problem involves the Doppler effect for electromagnetic waves, which describes how the observed frequency of a wave changes due to the relative motion between the source and the observer.
2. Write the formula for the relativistic Doppler effect: \( f_{obs} = f_{source} \sqrt{\frac{1 + v/c}{1 - v/c}} \), where \( f_{obs} \) is the observed frequency, \( f_{source} \) is the source frequency, \( v \) is the relative speed of the source, and \( c \) is the speed of light.
3. Substitute the given information: The observed frequency \( f_{obs} \) is 1.25 times the source frequency \( f_{source} \). This means \( \frac{f_{obs}}{f_{source}} = 1.25 \). Substitute this ratio into the Doppler effect formula: \( 1.25 = \sqrt{\frac{1 + v/c}{1 - v/c}} \).
4. Square both sides of the equation to eliminate the square root: \( (1.25)^2 = \frac{1 + v/c}{1 - v/c} \). Simplify this to \( 1.5625 = \frac{1 + v/c}{1 - v/c} \).
5. Solve for \( v/c \): Cross-multiply to get \( 1.5625(1 - v/c) = 1 + v/c \). Expand and rearrange terms to isolate \( v/c \). Once \( v/c \) is found, determine whether the source is moving toward or away from you based on the sign of \( v \). A positive \( v \) indicates the source is moving away, while a negative \( v \) indicates it is moving toward you.

Key Concepts

Doppler Effect

The Doppler Effect describes the change in frequency or wavelength of a wave in relation to an observer moving relative to the source of the wave. When a source of electromagnetic radiation moves toward an observer, the observed frequency increases (blue shift), while it decreases (red shift) when moving away. This effect is crucial for understanding how the motion of the source affects the frequency of the waves received.

Relativistic Doppler Shift

The relativistic Doppler shift accounts for the effects of special relativity when the source of electromagnetic radiation is moving at significant fractions of the speed of light. The formula for the observed frequency incorporates the speed of light and the relative velocity between the source and observer, allowing for accurate calculations of frequency changes due to high-speed motion.

Frequency and Wavelength Relationship

The frequency of a wave is inversely related to its wavelength, as described by the equation c = fλ, where c is the speed of light, f is the frequency, and λ is the wavelength. This relationship is essential for understanding how changes in frequency due to motion affect the wavelength of the emitted radiation, which can provide insights into the speed and direction of the source.