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Chapter 16: Problem 40

A train whistle emits a sound at a frequency \(f=3000 .\) Hz when stationary. You are standing near the tracks when the train goes by at a speed of \(v=30.0 \mathrm{~m} / \mathrm{s}\). What is the magnitude of the change in the frequency \((|\Delta f|)\) of the whistle as the train passes? (Assume that the speed of sound is \(v=343 \mathrm{~m} / \mathrm{s}\).)

Short Answer

Expert verified
Answer: To find the change in frequency (|Δf|), first determine the frequencies when the train is approaching and receding using the Doppler effect formula. Then, calculate the difference between these two frequencies.

Step by step solution

01

Determine frequency when the train is approaching

Using the Doppler effect formula for the frequency of the sound when the train is approaching: $$ f_{\text{approaching}} = f \times \frac{v_\text{sound}}{v_\text{sound} - v}. $$ Here, \(f=3000\) Hz, \(v_\text{sound}=343\) m/s, and \(v=30\) m/s. Plugging in these values, we can calculate the frequency when the train is approaching the observer.
02

Determine frequency when the train is receding

Using the Doppler effect formula for the frequency of the sound when the train is receding: $$ f_{\text{receding}} = f \times \frac{v_\text{sound}}{v_\text{sound} + v}. $$ Here, \(f=3000\) Hz, \(v_\text{sound}=343\) m/s, and \(v=30\) m/s. Plugging in these values, we can calculate the frequency when the train is moving away from the observer.
03

Calculate the change in frequency

We can find the magnitude of the change in frequency \((|\Delta f|)\) by taking the difference between the frequencies when the train is approaching and receding: $$ |\Delta f| = |f_{\text{approaching}} - f_{\text{receding}}|. $$ Using the values from Step 1 and Step 2, we can now find the magnitude of the change in frequency as the train passes by.

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Most popular questions from this chapter

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