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Chapter 35: Problem 37

Radar-based speed detection works by sending an electromagnetic wave out from a source and examining the Doppler shift of the reflected wave. Suppose a wave of frequency \(10.6 \mathrm{GHz}\) is sent toward a car moving away at a speed of \(32.0 \mathrm{~km} / \mathrm{h}\). What is the difference between the frequency of the wave emitted by the source and the frequency of the wave an observer in the car would detect?

Short Answer

Expert verified
Answer: The difference between the emitted and detected frequencies is 37.27 kHz.

Step by step solution

01

Convert the given values to SI units

First, we need to convert the velocity of the car from kilometers per hour to meters per second. The conversion factor is: 1 km/h = (1000 m) / (3600 s) So the velocity of the car in meters per second is: \(v = 32.0\, km/h * \frac{1000\, m}{3600\, s} = 8.89\, m/s\)
02

Apply the Doppler effect formula

Now we can apply the Doppler effect formula to find the detected frequency. Since the car is moving away from the radar source, we use the minus sign in the denominator: \(f_{detected} = f_{emitted} \frac{c}{c - v} = (10.6 * 10^9\, Hz) \frac{3 * 10^8\, m/s}{3 * 10^8\, m/s - 8.89\, m/s}\)
03

Calculate the detected frequency

Now, we can calculate the detected frequency: \(f_{detected} = (10.6 * 10^9\, Hz) \frac{3 * 10^8\, m/s}{(3 * 10^8 - 8.89) m/s} = 10.60003727 * 10^9\, Hz\)
04

Determine the difference in frequencies

Finally, we need to subtract the emitted frequency from the detected frequency to find the difference: \(\Delta f = f_{detected} - f_{emitted} = (10.60003727 * 10^9\, Hz) - (10.6 * 10^9\, Hz) = 0.00003727 * 10^9\, Hz\)
05

Express the result in an appropriate unit

To express the result in a more readable unit, we can convert the difference from Hz to kHz: \(\Delta f = 0.00003727 * 10^9\, Hz * \frac{1\, kHz}{1 * 10^3\, Hz} = 37.27\, kHz\) Thus, the difference between the frequency of the wave emitted by the source and the frequency of the wave detected by the observer in the car is \(37.27\, kHz\).

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