An airplane moves away from a stationary observer at 125 m/s and emits a frequency of 595 Hz on a day where the temperature is 25.0°C. What frequency does the observer hear?
http://hyperphysics.phy-astr.gsu.edu/hbase/sound/dopp.html
see the receeding source formula.
To calculate the frequency that the observer hears, we need to use the formula for Doppler effect. The formula is:
f' = f * (v + vo) / (v + vs)
Where:
f' is the observed frequency
f is the emitted frequency
v is the speed of sound in air
vo is the speed of the observer (stationary in this case)
vs is the speed of the source (the airplane in this case)
Step 1: Convert the temperature from Celsius to Kelvin
To convert Celsius to Kelvin, we add 273.15 to the given temperature.
Temperature in Kelvin = 25.0 + 273.15 = 298.15 K
Step 2: Calculate the speed of sound in air
The speed of sound in air can be calculated using the formula:
v = sqrt(γ * R * T)
Where:
v is the speed of sound
γ is the specific heat ratio of air (approximately 1.4)
R is the gas constant for air (approximately 287 J/(kg·K))
T is the temperature in Kelvin
Using the given values, we can calculate:
v = sqrt(1.4 * 287 * 298.15)
v ≈ 343.2 m/s (rounded to one decimal place)
Step 3: Substitute the given values into the Doppler effect formula
The given values are:
f = 595 Hz (emitted frequency)
vo = 0 m/s (speed of observer)
vs = 125 m/s (speed of source)
v = 343.2 m/s (speed of sound)
Substitute these values into the formula:
f' = 595 * (343.2 + 0) / (343.2 + 125)
f' ≈ 524.5 Hz (rounded to one decimal place)
Therefore, the observer hears a frequency of approximately 524.5 Hz.