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 200 feet per second.
What is the wavelength of the stationary sound source and the wavelength that a person in the car perceives?
Which of the following is the best option?
a) wavelength of the stationary source: 11.25 ft; perceived wavelength: 13.25 ft
b) wavelength of the stationary source: 9.25 ft; perceived wavelength: 11.25 ft
c) wavelength of the stationary source: 11.25 ft; perceived wavelength: 9.25 ft
d) wavelength of the stationary source: 13.25 ft; perceived wavelength: 11.25 ft
The wavelength of a sound wave can be calculated using the formula:
wavelength = velocity / frequency
For the stationary source, the wavelength is:
wavelength = 1125 ft/s / 100 Hz = 11.25 ft
For the person in the car, we need to take into account the relative motion between the car and the sound source. The perceived frequency will be higher than the actual frequency, given by:
observed frequency = actual frequency x (velocity of sound + velocity of observer) / (velocity of sound)
Plugging in the values, we get:
observed frequency = 100 Hz x (1125 ft/s + 200 ft/s) / (1125 ft/s) = 106.67 Hz
Using the observed frequency, we can calculate the perceived wavelength:
perceived wavelength = velocity / observed frequency
perceived wavelength = 1125 ft/s / 106.67 Hz = 10.53 ft
Therefore, the correct option is:
b) wavelength of the stationary source: 11.25 ft; perceived wavelength: 10.53 ft
To find the wavelength of a sound wave, we can use the formula:
wavelength = wave speed / frequency
For the stationary sound source:
wavelength = 1125 ft/s / 100 Hz = 11.25 ft
For the sound wave perceived by a person in the car, we need to account for the fact that the car is moving towards the source. This is known as the Doppler effect. The perceived frequency can be calculated using the formula:
perceived frequency = (speed of sound + speed of observer) / (speed of sound + speed of source) * actual frequency
In this case, the speed of sound is given as 1125 ft/s, speed of observer (car) is 200 ft/s, and the actual frequency is 100 Hz. Plugging these values into the formula, we can solve for the perceived frequency:
perceived frequency = (1125 ft/s + 200 ft/s) / (1125 ft/s + 0 ft/s) * 100 Hz
perceived frequency = 1325 ft/s / 1125 ft/s * 100 Hz
perceived frequency = 118 Hz
Now, we can find the perceived wavelength using the same formula as before:
perceived wavelength = wave speed / perceived frequency
perceived wavelength = 1125 ft/s / 118 Hz
perceived wavelength ≈ 9.49 ft
Comparing the options given:
a) wavelength of the stationary source: 11.25 ft; perceived wavelength: 13.25 ft
b) wavelength of the stationary source: 9.25 ft; perceived wavelength: 11.25 ft
c) wavelength of the stationary source: 11.25 ft; perceived wavelength: 9.25 ft
d) wavelength of the stationary source: 13.25 ft; perceived wavelength: 11.25 ft
From the calculations above, it is clear that option c) wavelength of the stationary source: 11.25 ft; perceived wavelength: 9.25 ft is the correct answer.
To find the wavelength of the stationary sound source, we can use the formula:
wavelength = speed / frequency
Given that the frequency is 100 Hz and the speed is 1125 feet per second, we can calculate:
wavelength = 1125 ft/s / 100 Hz = 11.25 ft
So, the wavelength of the stationary sound source is 11.25 ft.
To find the wavelength that a person in the car perceives, we need to consider the Doppler effect. When a source of sound and an observer are moving towards each other, the perceived frequency and wavelength are affected.
The perceived frequency (fp) and perceived wavelength (wp) are related to the actual frequency (fa) and actual wavelength (wa) by the equation:
fp = fa * (v + va) / (v + vs)
wp = va / fp
where v is the speed of sound, va is the velocity of the observer (car), and vs is the velocity of the source (stationary).
Given that va is 200 ft/s and vs is 0 ft/s (since the source is stationary), we can calculate the perceived wavelength:
fp = 100 Hz * (1125 ft/s + 200 ft/s) / (1125 ft/s + 0 ft/s) = 300 Hz
wp = 200 ft/s / 300 Hz = 0.6667 ft
So, the perceived wavelength for a person in the car is approximately 0.6667 ft.
Comparing the options given:
a) wavelength of the stationary source: 11.25 ft; perceived wavelength: 13.25 ft - Incorrect
b) wavelength of the stationary source: 9.25 ft; perceived wavelength: 11.25 ft - Incorrect
c) wavelength of the stationary source: 11.25 ft; perceived wavelength: 9.25 ft - Incorrect
d) wavelength of the stationary source: 13.25 ft; perceived wavelength: 11.25 ft - Correct
Therefore, option (d) has the correct values for the wavelength of the stationary source (13.25 ft) and the perceived wavelength in the car (11.25 ft).