The siren of a fire engine that is driving northward at 30.0 m/s emits a sound of frequency 2000 Hz. A truck in front of this fire engine is moving northward at 20.0 m/s. What wavelength would this driver measure for these reflected sound waves?

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First, understand that this problem involves the Doppler effect, which describes the change in frequency or wavelength of a wave in relation to an observer moving relative to the source of the wave.

Calculate the frequency of the sound waves as heard by the truck driver using the Doppler effect formula for a moving source and a moving observer. The formula is: \( f' = f \frac{v + v_o}{v + v_s} \), where \( f \) is the original frequency, \( v \) is the speed of sound in air (approximately 343 m/s), \( v_o \) is the speed of the observer (truck), and \( v_s \) is the speed of the source (fire engine).

Substitute the given values into the formula: \( f' = 2000 \text{ Hz} \times \frac{343 \text{ m/s} + 20 \text{ m/s}}{343 \text{ m/s} + 30 \text{ m/s}} \). This will give you the frequency of the sound as heard by the truck driver.

Next, consider that the sound waves are reflected off the truck and return to the fire engine. The truck now acts as the source of the sound waves, and the fire engine is the observer. Use the Doppler effect formula again to find the frequency of the reflected waves as heard by the fire engine: \( f'' = f' \frac{v + v_s}{v + v_o} \).

Finally, calculate the wavelength of the reflected sound waves using the formula \( \lambda = \frac{v}{f''} \), where \( \lambda \) is the wavelength, \( v \) is the speed of sound, and \( f'' \) is the frequency of the reflected waves. Substitute the values to find the wavelength.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Doppler Effect

The Doppler Effect refers to the change in frequency or wavelength of a wave in relation to an observer moving relative to the wave source. In this scenario, the fire engine and the truck are moving towards each other, affecting the frequency of the sound waves emitted by the siren as perceived by the truck driver.

Relative Velocity

Relative velocity is the velocity of an object as observed from a particular reference frame, and it is crucial in determining the effective speed of sound between the fire engine and the truck. Here, the relative velocity between the two vehicles affects the frequency and wavelength of the sound waves as they are reflected back to the fire engine.

Wave Equation

The wave equation relates the speed of a wave to its frequency and wavelength, expressed as v = fλ, where v is the wave speed, f is the frequency, and λ is the wavelength. This equation is essential for calculating the wavelength of the sound waves as perceived by the truck driver after considering the Doppler Effect and relative velocity.

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Textbook Question

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The siren of a fire engine that is driving northward at 30.0 m/s emits a sound of frequency 2000 Hz. A truck in front of this fire engine is moving northward at 20.0 m/s. (a) What is the frequency of the siren's sound that the fire engine's driver hears reflected from the back of the truck?

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