Question

Two vehicles carrying speakers that produce a tone of frequency \(1000.0 \mathrm{~Hz}\) are moving directly toward each other. Vehicle \(\mathrm{A}\) is moving at \(10.00 \mathrm{~m} / \mathrm{s}\) and vehicle \(\mathrm{B}\) is moving at \(20.00 \mathrm{~m} / \mathrm{s}\). Assume that the speed of sound in air is \(343.0 \mathrm{~m} / \mathrm{s},\) and find the frequencies that the driver of each vehicle hears.

Short answer

Answer: The driver of vehicle A will hear a frequency of approximately 1093.2 Hz, and the driver of vehicle B will hear a frequency of approximately 1090.1 Hz.

Step-by-step solution

Write down the Doppler effect formula

To solve this problem, we need to use the general Doppler effect formula for sound: $$ f' = f \frac{v \pm v_o}{v \pm v_s} $$ where \(f'\) is the observed frequency, \(f\) is the source frequency, \(v\) is the speed of sound in the medium, \(v_o\) is the speed of the observer, and \(v_s\) is the speed of the source. The plus/minus signs depend on whether the observer and source are moving toward or away from each other. In this problem, we have two cases to analyze: 1. The driver of vehicle A hears the sound produced by vehicle B. 2. The driver of vehicle B hears the sound produced by vehicle A.

Calculate the frequency heard by the driver of vehicle A

For the driver of vehicle A, the frequency produced by vehicle B is the source frequency which is \(f = 1000\,\text{Hz}\). Vehicle B is moving towards vehicle A, so its \(v_s = -20.0\,\text{m/s}\) (because it is moving in the opposite direction). Vehicle A is also moving towards Vehicle B, so its \(v_o = 10.0\,\text{m/s}\). The speed of sound in air is \(v = 343.0\,\text{m/s}\). Using the Doppler effect formula: $$ f'_{A} = f \frac{v + v_o}{v + v_s} = 1000 \frac{343.0 + 10.0}{343.0 - 20.0} $$ Now, we can calculate \(f'_{A}\): $$ f'_{A} = 1000 \frac{353.0}{323.0} = 1093.2\,\text{Hz} $$ So, the frequency heard by the driver of vehicle A is approximately \(1093.2\,\text{Hz}\).

Calculate the frequency heard by the driver of vehicle B

For the driver of vehicle B, the frequency produced by vehicle A is the source frequency, which is \(f = 1000\,\text{Hz}\). Vehicle A is moving towards vehicle B, so its \(v_s = -10.0\,\text{m/s}\). Vehicle B is also moving towards Vehicle A, so its \(v_o\) = \(20.0\,\text{m/s}\). The speed of sound in air is \(v = 343.0\,\text{m/s}\). Using the Doppler effect formula: $$ f'_{B} = f \frac{v + v_o}{v + v_s} = 1000 \frac{343.0 + 20.0}{343.0 - 10.0} $$ Now, we can calculate \(f'_{B}\): $$ f'_{B} = 1000 \frac{363.0}{333.0} = 1090.1\,\text{Hz} $$ So, the frequency heard by the driver of vehicle B is approximately \(1090.1\,\text{Hz}\). In conclusion, the driver of vehicle A will hear a frequency of approximately \(1093.2\,\text{Hz}\), and the driver of vehicle B will hear a frequency of approximately \(1090.1\,\text{Hz}\).