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A state highway patrol car radar unit uses a frequency of \(8.00 \times 10^{9} \mathrm{Hz} .\) What frequency difference will the unit detect from a car receding at a speed of \(44.5 \mathrm{m} / \mathrm{s}\) from a stationary patrol car?

Short Answer

Expert verified
The frequency difference is approximately 1.19 Hz.

Step by step solution

01

Understand the Doppler Effect Formula

The Doppler effect formula relates the observed frequency, the source frequency, and the speed of the source and the observer. For electromagnetic waves, the formula for the observed frequency \( f' \) is given by: \[ f' = f \left( \frac{c}{c + v} \right) \] where \( f \) is the source frequency, \( c \) is the speed of light (approximately \( 3.00 \times 10^8 \) m/s), and \( v \) is the speed of the receding car.
02

Plug in Known Values

Substitute the given values into the Doppler effect formula. Here, the source frequency \( f \) is \( 8.00 \times 10^9 \) Hz, the speed of light \( c \) is \( 3.00 \times 10^8 \) m/s, and the speed of the receding car \( v \) is \( 44.5 \) m/s.
03

Calculate the Observed Frequency

Compute the observed frequency \( f' \) using the formula: \[ f' = 8.00 \times 10^9 \left( \frac{3.00 \times 10^8}{3.00 \times 10^8 + 44.5} \right) \] Simplify the terms to find \( f' \).
04

Find the Frequency Difference

The frequency difference \( \Delta f \) is given by the absolute difference between the source frequency and the observed frequency: \[ \Delta f = f - f' \] Substitute the values calculated in the previous steps to determine \( \Delta f \).

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Key Concepts

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

Frequency Calculation
The concept of frequency calculation plays a crucial role in understanding the Doppler Effect, especially when dealing with electromagnetic waves such as radar signals. In these scenarios, frequency helps to determine how vibrations or waves, either sound or electromagnetic, are perceived as they move past an observer. The cornerstone of calculating frequency in this situation is the Doppler Effect formula:
\[f' = f \left( \frac{c}{c + v} \right) \]This formula is specifically crucial when dealing with objects in motion, such as a car moving away from a radar unit. In this equation:
  • \(f\) represents the original frequency emitted, such as the initial radar frequency.
  • \(c\) symbolizes the constant speed of light, approximately \(3.00 \times 10^8\) meters per second, central to computations involving electromagnetic waves.
  • \(v\) is the speed of the moving object, like vehicles.
The observed frequency \(f'\) is calculated by substituting these values, showing how movement affects wave perception. The frequency difference \(\Delta f\) then reveals the change due to the Doppler Effect, crucial for accurate speed determination in technology like radar units.
Radar Technology
Radar technology utilizes the principles of the Doppler Effect to measure the velocity of objects by calculating changes in frequency. When a radar wave hits a moving object, such as a car, it reflects back with a frequency shift. This specific frequency change enables the determination of an object's speed. By employing radar technology:
  • Law enforcement can accurately measure vehicle speeds, helping monitor and manage traffic effectively.
  • It provides crucial data for weather monitoring, tracking rainstorms, or any objects moving within the atmosphere.
  • Radar is also employed in aviation for air traffic control, ensuring that aircraft remain well-monitored.
The critical component involving frequency calculation helps transform this technology into an essential tool across various fields. Understanding the relationship between frequency changes and movement allows radar to accurately gauge distances and speeds.
Electromagnetic Waves
Electromagnetic waves, like microwaves used in radar systems, travel at the speed of light and can bounce off objects to provide vital information back to the source. These waves span a broad spectrum, revealing their fascinating properties:
  • They consist of oscillating electric and magnetic fields which propagate perpendicularly to the field movements and travel through space.
  • This includes familiar parts of the spectrum such as visible light, radio waves, and x-rays.
  • Radar systems specifically use microwave frequencies due to their high penetration capabilities and minimal atmospheric interference.
By applying electromagnetic wave principles, radar technology can effectively track and analyze objects from vehicles to weather phenomena. Understanding these waves’ behavior and their interactions with objects helps to leverage their properties in practical applications, showcasing the importance of electromagnetic knowledge in radar technologies.

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