At a factory, a noon whistle is sounding with a frequency of 484 Hz. As a car traveling at 77 km/h approaches the factory, the driver hears the whistle at frequency fi. After driving past the factory, the driver hears frequency ff. What is the change in frequency ff − fi heard by the driver? (Assume a temperature of 20° C.)
To find the change in frequency heard by the driver, we need to consider the Doppler effect. The Doppler effect describes the change in frequency of a wave when there is relative motion between the source of the wave and the observer.
First, let's define the variables:
- V₀ is the speed of sound in the medium (which is approximately 343 m/s at 20° C).
- v is the velocity of the car in m/s (which is 77 km/h converted to m/s).
- fs is the frequency of the source (noon whistle), which is given as 484 Hz.
To find the frequency fi heard by the driver as the car approaches the factory, we can use the following equation:
fi = fs((V₀ + v) / V₀) (Equation 1)
Next, to find the frequency ff heard by the driver after driving past the factory, we can use a similar equation:
ff = fs((V₀ - v) / V₀) (Equation 2)
Finally, to find the change in frequency ff − fi, we can subtract Equation 1 from Equation 2:
ff − fi = fs((V₀ - v) / V₀) - fs((V₀ + v) / V₀)
Now, let's plug in the given values and calculate the change in frequency:
V₀ ≈ 343 m/s
v = (77 km/h) * (1000 m/km) / (3600 s/h) ≈ 21.4 m/s
fs = 484 Hz
ff − fi = 484 * ((343 - 21.4) / 343) - 484 * ((343 + 21.4) / 343)
Simplifying the expression:
ff − fi ≈ 484 * (321.6 - 364.4) / 343
ff − fi ≈ -8.26 Hz
Therefore, the change in frequency ff − fi heard by the driver is approximately -8.26 Hz.