Question

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

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

Step-by-step solution

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: \(v = 32.0\, km/h * \frac{1000\, m}{3600\, s} = 8.89\, m/s\)

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: \(f_{detected} = f_{emitted} \frac{c}{c - v} = (10.6 * 10^9\, Hz) \frac{3 * 10^8\, m/s}{3 * 10^8\, m/s - 8.89\, m/s}\)

Calculate the detected frequency

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

Determine the difference in frequencies

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

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 \(37.27\, kHz\).

Key concepts

Radar Speed Detection

Radar speed detection is a fascinating application of the Doppler effect. It is commonly used by law enforcement to measure a vehicle's speed. This technology operates by emitting electromagnetic waves from a radar device. When these waves hit a moving object, like a car, they bounce back to the radar device. The key to radar speed detection is the Doppler shift: a change in frequency caused by the object's motion. As a car moves, the frequency of the returning waves differs from the frequency of the emitted ones. If the car is moving away from the radar, the observed frequency is lower; conversely, if the car is approaching, the frequency is higher. This frequency change allows the radar to calculate the speed of the vehicle, showcasing the practical utility of the Doppler effect in everyday life.

Electromagnetic Waves

Electromagnetic waves are the backbone of radar technology. These waves travel at the speed of light and carry both electric and magnetic fields. In radar speed detection, electromagnetic waves, typically microwaves, are emitted towards moving objects to measure their speed. At a frequency of 10.6 GHz, these waves fall into the microwave category of the electromagnetic spectrum. Microwaves are ideal for radar because they can travel long distances and have the capability to reflect off objects effectively, which is a requirement for accurate speed measurements. The capability to emit these waves and analyze their reflections allows radar systems to measure speed based on changes in wave frequency, effectively using a property innate to electromagnetic waves—the Doppler effect.

Frequency Difference Calculation

Calculating the frequency difference between emitted and detected waves is at the heart of understanding the Doppler effect in radar. The Doppler formula provides a direct way to calculate this difference. The formula is: \[ f_{detected} = f_{emitted} \frac{c}{c - v} \] In this formula, \( f_{detected} \) is the frequency received by an observer moving away from the source, \( f_{emitted} \) is the emitted frequency, \( c \) represents the speed of light, and \( v \) is the velocity of the moving car.When calculating, as in the original exercise, first convert all units into standard units like meters per second for speed. Then, input these values into the equation to find the detected frequency. The next step involves calculating the difference, \( \Delta f \), which reveals how much the frequency changed due to the object’s motion. Finally, convert Hz to a more readable unit like kHz, if necessary. Through this calculation, the core concept of sound and wave behavior in motion is beautifully illustrated.