Question

A person in a parked car sounds the horn. The frequency of the horn's sound is \(333 \mathrm{~Hz}\). A driver in an approaching car is moving at a speed of \(15.7 \mathrm{~m} / \mathrm{s}\). What is the frequency of the sound that she hears? (Use \(343 \mathrm{~m} / \mathrm{s}\) for the speed of sound.)

Short answer

Answer: The driver in the approaching car hears a frequency of approximately 347.56 Hz.

Step-by-step solution

Identify the given values

The given values in the problem are: - Source frequency, \(f_s = 333 \ Hz\) - Speed of the observer, \(v_o = 15.7 \ m/s\) - Speed of sound, \(v_s = 343 \ m/s\)

Doppler Effect Formula for Sound

We need to use the following Doppler effect formula for sound: \(f_o = \frac{v_s \pm v_o}{v_s \mp v_r} f_s\) Where: - \(f_o\) is the observed frequency - \(f_s\) is the source frequency - \(v_s\) is the speed of sound - \(v_o\) is the speed of the observer - \(v_r\) is the speed of the source - The plus sign in the numerator and the minus sign in the denominator are used for an approaching observer. The minus sign in the numerator and the plus sign in the denominator are used for a receding observer. Since the car is parked, the source is not moving, which means \(v_r = 0\).

Calculate the observed frequency

Now plug in the given values into the Doppler effect formula for sound: \(f_o = \frac{343 + 15.7}{343 - 0} (333)\) \(f_o = \frac{358.7}{343}(333)\) Calculate the observed frequency: \(f_o \approx 347.56 \ Hz\) The driver in the approaching car hears a frequency of approximately \(347.56 \ Hz\).