Chapter 16: Problem 52
While sitting in your car by the side of a country road, you are approached by your friend, who happens to be in an identical car. You blow your car's horn, which has a frequency of 260 Hz. Your friend blows his car's horn, which is identical to yours, and you hear a beat frequency of 6.0 Hz. How fast is your friend approaching you?
Short Answer
Step by step solution
Understand the Beat Frequency
Identify the Original Frequency
Use the Doppler Effect Formula for Frequency
Calculate the Observed Frequency
Solve for the Source Velocity
Perform the Calculation
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Beat Frequency
In our scenario, the car horns both originally have a frequency of 260 Hz. The beat frequency observed is 6 Hz. This indicates that there is a shift in the frequency due to the movement of your friend's car, altering what you hear to be either 254 Hz or 266 Hz. Since the car is approaching, the frequency increases to 266 Hz, leading to the beat frequency of 6 Hz.
Beat frequency is a practical application of sound wave interference and is useful in various fields, including music and engineering.
Sound Waves
The frequency of sound waves is measured in Hertz (Hz), and it determines the pitch of the sound. Higher frequencies result in higher pitches, while lower frequencies correspond to lower pitches. Our situation involves a car horn with a known frequency of 260 Hz.
Sound wave interactions, such as the Doppler Effect and beat frequency phenomena, depend heavily on these properties, applying physics principles to understand how sounds are perceived when in motion.
Velocity Calculation
- Start by recognizing the observed frequency, which was 266 Hz due to your friend's approach.
- The original frequency was 260 Hz, and since you're stationary, your velocity as an observer is 0 m/s.
Frequency Shift
The Doppler Effect formula describes how the frequency changes when there is relative motion between an observer and the source. If the source moves towards the observer, the frequency increases and shifts higher; if it moves away, the frequency decreases.
For two identical car horns, the shift is exactly what creates the beat frequency. Because the friend's car is moving towards you, the frequency rises from 260 Hz to 266 Hz, resulting in the beat frequency of 6 Hz. This frequency shift makes it possible to calculate the velocity at which your friend is approaching, using principles of wave dynamics.