A siren, mounted on a tower, emits a sound whose frequency is 2140 Hz. A car isbeing driven away from the tower at a speed of 25.5 m s‒1, and, as the figureillustrates, the siren sound reaches the driver by two paths: sound coming directlyfrom the siren, and sound reaching the car after reflection from the building. Calculatethe frequency that the driver hears from (i) direct sound and (ii) reflected sound

To solve this problem, we need to use the Doppler effect formula for sound. The Doppler effect describes the change in frequency of a wave in relation to an observer moving relative to the source of the wave.

The formula is: f' = f * (v + vo) / (v + vs)

Where: f' is the observed frequency, f is the source frequency, v is the speed of sound in air (343 m/s), vo is the speed of the observer, vs is the speed of the source.

(i) For the direct sound, the car is moving away from the source, so the speed of the observer (vo) is -25.5 m/s (we take it as negative because the car is moving away from the source). The siren is not moving, so the speed of the source (vs) is 0.

Substituting these values into the formula, we get:

f' = 2140 Hz * (343 m/s + (-25.5 m/s)) / (343 m/s + 0) f' = 2140 Hz * (317.5 m/s) / 343 m/s f' ≈ 1980 Hz

So, the driver hears the direct sound at a frequency of approximately 1980 Hz.

(ii) For the reflected sound, the sound is first moving towards the building and then towards the car. So, we need to apply the Doppler effect formula twice.

First, when the sound is moving towards the building, the building is stationary (vo = 0) and the sound is moving (vs = -25.5 m/s, we take it as negative because the sound is moving towards the building). Applying the formula, we get:

f' = 2140 Hz * (343 m/s + 0) / (343 m/s + (-25.5 m/s)) f' = 2140 Hz * 343 m/s / 317.5 m/s f' ≈ 2300 Hz

Then, when the sound is moving towards the car, the car is moving away from the building (vo = -25.5 m/s) and the sound is not moving relative to the building (vs = 0). Applying the formula again, we get:

f'' = 2300 Hz * (343 m/s + (-25.5 m/s)) / (343 m/s + 0) f'' = 2300 Hz * 317.5 m/s / 343 m/s f'' ≈ 2140 Hz

So, the driver hears the reflected sound at a frequency of approximately 2140 Hz.