A police car's radar gun emits microwaves with a frequency...

A police car's radar gun emits microwaves with a frequency of $f_{1}=36.0$ GHz. The beam reflects from a speeding car, which is moving away at $43.0 \mathrm{m} / \mathrm{s}$ with respect to the police car. The frequency of the reflected microwave as observed by the police is $f_{2}$. (a) Which is larger, $f_{1}$ or $f_{2} ?$ (b) Find the frequency difference $f_{2}-f_{1}$ [Hint: There are two Doppler shifts. First think of the police as source and the speeder as observer. The speeding car "retransmits" a reflected wave at the same frequency at which it receives the incident wave. $.]$

Key Concepts

Electromagnetic Wave Propagation

Electromagnetic waves, such as microwaves, travel at the speed of light and exhibit wave properties including frequency and wavelength. Understanding these properties is essential for analyzing scenarios where the frequency of the electromagnetic waves might change due to motion, as seen in radar systems. This concept integrates with the Doppler effect principles to explain how variations in frequency can be used to measure speed.

Relative Motion and Velocity

The concept of relative motion is crucial in understanding Doppler shifts. The relative velocity between the source and the moving object determines the magnitude and direction of the frequency change. In such problems, it is the difference between the individual speeds of the source and the observer (or reflecting object) that directly influences the observed wave frequency.

Double Doppler Shift

In situations where a wave is both emitted and then reflected off a moving object, the Doppler effect occurs twice. The first shift happens when the moving object (acting as an observer) receives the wave, and the second shift occurs when it re-emits or reflects the wave back. This double shift must be considered when calculating the overall frequency change, as each Doppler shift contributes to the final observed frequency.

Doppler Effect

The Doppler Effect is the change in frequency or wavelength of a wave as observed by an observer moving relative to the source of the wave. This phenomenon explains why waves appear to be compressed (higher frequency) when the source and observer are moving towards each other, and stretched (lower frequency) when they are moving apart. It is a fundamental concept in understanding how the motion of a source or observer influences the perceived frequency of a signal.

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