A stationary source produces a sound wave at a frequency of 100 Hz. The wave travels at 1125 feet per second. A car is moving toward the sound source at a speed of 100 feet per second.

Question

What is the wavelength of the stationary sound source and the wavelength that a person in the car perceives?

(1 point)
Responses

wavelength of the stationary source: 10.25 ft; perceived wavelength: 11.25 ft
wavelength of the stationary source: 10.25 ft; perceived wavelength: 11.25 ft

wavelength of the stationary source: 11.25 ft; perceived wavelength: 12.25 ft
wavelength of the stationary source: 11.25 ft; perceived wavelength: 12.25 ft

wavelength of the stationary source: 12.25 ft; perceived wavelength: 11.25 ft
wavelength of the stationary source: 12.25 ft; perceived wavelength: 11.25 ft

wavelength of the stationary source: 11.25 ft; perceived wavelength: 10.25 ft

Answer 1

To find the wavelength of the stationary sound source and the perceived wavelength by a person in the car, we can use the following formulas:

1. The wavelength \( \lambda \) of a wave is calculated using the formula: \[ \lambda = \frac{v}{f} \] where:

   • \( v \) is the speed of the wave,
   • \( f \) is the frequency of the wave.

2. For a person moving toward a stationary sound source, the perceived frequency (due to the Doppler effect) can be calculated using: \[ f' = f \frac{v + v_0}{v} \] where:

   • \( f' \) is the perceived frequency,
   • \( v_0 \) is the speed of the observer (the car),
   • \( v \) is the speed of sound.

Step 1: Calculate the Wavelength of the Stationary Sound Source

Given:

• Frequency \( f = 100 , \text{Hz} \)
• Speed of sound \( v = 1125 , \text{ft/s} \)

Using the wavelength formula: \[ \lambda = \frac{v}{f} = \frac{1125 , \text{ft/s}}{100 , \text{Hz}} = 11.25 , \text{ft} \]

Step 2: Calculate the Perceived Frequency

Given:

• Speed of observer \( v_0 = 100 , \text{ft/s} \)

Using the Doppler effect formula: \[ f' = f \frac{v + v_0}{v} = 100 , \text{Hz} \times \frac{1125 , \text{ft/s} + 100 , \text{ft/s}}{1125 , \text{ft/s}} = 100 , \text{Hz} \times \frac{1225}{1125} \]

Calculating it: \[ f' = 100 , \text{Hz} \times 1.08889 \approx 108.89 , \text{Hz} \]

Step 3: Calculate the Perceived Wavelength

Now we can calculate the perceived wavelength \( \lambda' \) using the formula: \[ \lambda' = \frac{v}{f'} \]

Substituting \( f' \): \[ \lambda' = \frac{1125 , \text{ft/s}}{108.89 , \text{Hz}} \approx 10.35 , \text{ft} \]

Summary of Results

• Wavelength of the stationary sound source: 11.25 ft
• Perceived wavelength by the person in the car: approximately 10.35 ft (although this should be calculated more precisely through the \( f' \)).

Based on the options given, it looks like the closest approach in the responses you provided is:

• Wavelength of the stationary source: 11.25 ft; perceived wavelength 10.25 ft (given options).

Therefore, the correct answer is: wavelength of the stationary source: 11.25 ft; perceived wavelength: 10.25 ft.