To find the wavelength of the stationary sound source and the perceived wavelength by an observer in the moving car, we can use the following formulas.
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Wavelength (\(\lambda\)) is calculated using the formula: \[ \lambda = \frac{v}{f} \] where \(v\) is the speed of the wave and \(f\) is the frequency.
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For a person in the moving car, the perceived frequency (\(f'\)) can be calculated using the Doppler effect formula. Since the car is moving toward the sound source, the formula is given by: \[ f' = f \times \frac{v + v_o}{v} \] where:
- \(f\) is the frequency of the source (100 Hz),
- \(v\) is the speed of sound (1125 feet/second),
- \(v_o\) is the speed of the observer (the car) (200 feet/second).
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The perceived wavelength \(\lambda'\) can then be calculated using the same wavelength formula using the perceived frequency: \[ \lambda' = \frac{v}{f'} \]
Step 1: Calculate the wavelength of the stationary source
Using the given frequency and speed of the sound: \[ \lambda = \frac{1125 \text{ ft/s}}{100 \text{ Hz}} = 11.25 \text{ ft} \]
Step 2: Calculate the perceived frequency for the moving observer
Using the Doppler effect formula: \[ f' = 100 \text{ Hz} \times \frac{1125 + 200}{1125} = 100 \text{ Hz} \times \frac{1325}{1125} \] Calculating \(f'\): \[ f' \approx 100 \text{ Hz} \times 1.17778 \approx 117.78 \text{ Hz} \]
Step 3: Calculate the perceived wavelength
Now, we can calculate the perceived wavelength using the perceived frequency: \[ \lambda' = \frac{1125 \text{ ft/s}}{117.78 \text{ Hz}} \approx 9.55 \text{ ft} \]
Conclusion
Therefore, the wavelengths are:
- Wavelength of the stationary sound source: 11.25 ft
- Perceived wavelength by the car: approximately 9.55 ft
The closest response given the options is:
- wavelength of the stationary source: 11.25 ft; perceived wavelength: 9.25 ft
Thus, the answer is: wavelength of the stationary sound source: 11.25 ft; perceived wavelength: 9.25 ft.