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Chapter 9: Problem 302

A train is moving on a straight track with speed \(20 \mathrm{~ms}^{-1}\). It is blowing its whistle at the frequency of \(1000 \mathrm{~Hz}\). The percentage change in the frequency heard by a person standing near the track as the train passes him is (speed of sound \(=320 \mathrm{~ms}^{-1}\) ) close to (A) \(12 \%\) (B) \(18 \%\) (C) \(24 \%\) (D) \(6 \%\)

Short Answer

Expert verified
The percentage change in the frequency heard by a person standing near the track as the train passes him is approximately \(12\%\).

Step by step solution

01

Understand the Doppler Effect formula

The Doppler Effect formula for observed frequency (f') in terms of the source frequency (f), speed of sound (c), speed of the source (v), and the angle between the listener and the direction of the train (theta) is given by: \[f' = f \frac{c}{c \pm v\cos(\theta)}\] Here, the positive sign is used when the train is moving away from the person, and the negative sign is used when the train is approaching the person.
02

Calculate the observed frequencies when the train is approaching and moving away

When the train is approaching, we use a negative sign in the formula: \[f'_{approaching} = f \frac{c}{c - v\cos(\theta)}\] When the train is moving away, we use a positive sign in the formula: \[f'_{moving\_away} = f \frac{c}{c + v\cos(\theta)}\] Here, the train's speed on the track, v = 20ms^{-1}, f = 1000Hz, and the speed of sound, c = 320ms^{-1}. Also, assume the person is standing very close to the track, which means theta = 0 degrees, and cos(theta) = 1. Now, calculate the observed frequencies f'_{approaching} and f'_{moving\_away}: \[f'_{approaching} = 1000 \frac{320}{320 - 20} = \frac{1000 脳 320}{300} = 1066.67 Hz\] \[f'_{moving\_away} = 1000 \frac{320}{320 + 20} = \frac{1000 脳 320}{340} = 941.18 Hz\]
03

Determine the percentage change in the frequency

The percentage change in frequency can be represented as: \[\% \Delta f = \frac{f'_{approaching} - f'_{moving\_away}}{f} 脳 100\] Now, let's find out the percentage change in frequency: \[\%\Delta f = \frac{1066.67 - 941.18}{1000} 脳 100 = 12.55\%\] Since 12.55% is close to 12%, the answer is: (A) \(12\%\)

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