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Chapter 9: Problem 93

The driver of a car traveling with speed \(30 \mathrm{~m} / \mathrm{s}\) towards a hill sound a horn of frequency \(600 \mathrm{~Hz}\). If the velocity of sound in air is \(330 \mathrm{~m} / \mathrm{s}\), the frequency of reflected sound as heard by the driver is (A) \(720 \mathrm{~Hz}\) (B) \(555.5 \mathrm{~Hz}\) (C) \(550 \mathrm{~Hz}\) (D) \(500 \mathrm{~Hz}\)

Short Answer

Expert verified
The frequency of the reflected sound as heard by the driver is 720 Hz, which corresponds to option (A).

Step by step solution

01

1. Identifying variables and formula

We are given the speed of the car (v_car) is 30 m/s and the frequency of the horn (f_original) is 600 Hz. The velocity of sound in air (v_sound) is 330 m/s. We need to find the frequency of the reflected sound (f_reflected). We will use the Doppler effect formula: \(f'=f \frac{v \pm v_{listener}}{v \mp v_{source}}\)
02

2. Calculating the frequency reaching the hill

In this step, the car is the source, and the hill is the listener. Since the car is moving towards the hill, we will add the speed of the car at the denominator of the formula. So, we find the frequency reaching the hill (f_reaching_hill) as follows: \(f_{reaching\_hill} = f_{original} \frac{v_{sound}}{v_{sound} - v_{car}}\) \(f_{reaching\_hill} = 600 \frac{330}{330 - 30}\) \(f_{reaching\_hill} = 600 \frac{330}{300}\) \(f_{reaching\_hill} = 660 \, Hz\)
03

3. Calculating the frequency of reflected sound as heard by the driver

Now, the hill becomes the source, and the car is the listener. The frequency of the source is now the frequency reaching the hill. Since the car is still moving towards the hill when the sound reflection occurs, we will add the speed of the car at the numerator of the formula. So, we find the frequency of the reflected sound heard by the driver (f_reflected) as follows: \(f_{reflected} = f_{reaching\_hill} \frac{v_{sound} + v_{car}}{v_{sound}}\) \(f_{reflected} = 660 \frac{330 + 30}{330}\) \(f_{reflected} = 660 \frac{360}{330}\) \(f_{reflected} = 720 \, Hz\)
04

4. Identifying the correct answer

From the calculation in the previous step, we found that the frequency of the reflected sound as heard by the driver is 720 Hz. So, the correct answer is: (A) 720 Hz

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