At rest, a car's horn sounds at a frequency of 365 Hz. The horn is sounded while the car is moving down the street. A bicyclist moving in the same direction with one-third the car's speed hears a frequency of 357 Hz. What is the speed of the car?

10.15m/s

Explanation:

The change in the frequency of sound (or any other wave) when the source of the sound and the receiver or observer of the sound move towards (or away from) each other is explained by the Doppler effect which is given by the following equation:

f₁ = [(v ± v₁) / (v ± v₂)] f            (i)

Where;

f₁ = frequency received by the observer or receiver

v = speed of sound in air

v₁ = velocity of the observer

v₂ = velocity of the source

f = original frequency of the sound

From the question, the observer is the bicyclist and the source is the car driver. Therefore;

f₁ = frequency received by the observer (bicyclist) = 357Hz

v = speed of sound in air = 330m/s

v₁ = velocity of the observer(bicyclist) = (1 / 3) v₂ = 0.33v₂

v₂ = velocity of the source (driver)

f = original frequency of the sound = 365Hz

Note: The speed of the observer is positive if he moves towards the source and negative if he moves away from the source. Also, the speed of the source is positive if it moves away from the listener and negative otherwise.

From the question, the cyclist and the driver are moving in the same direction. But then, we do not know which one is at the front. Therefore, two scenarios are possible.

i. The bicyclist is at the front. In this case, v₁ and v₂ are negative.

Substitute these values into equation (i) as follows;

357 = [(330 - 0.33v₂) / (330 - v₂)] * 365

(357 / 365) = [(330 - 0.33v₂) / (330 - v₂)]

0.98 =  [(330 - 0.33v₂) / (330 - v₂)]

0.98 (330 - v₂) =  (330 - 0.33v₂)

323.4 - 0.98v₂ = 330 - 0.33v₂

323.4 - 330 = (0.98 - 0.33)v₂

-6.6 = 0.65v₂

v₂ = -10.15

The value of v₂ is not supposed to be negative since we already plugged in the right value polarity into the equation.

iI. The bicyclist is behind. In this case, v₁ and v₂ are positive.

Substitute these values into equation (i) as follows;

357 = [(330 + 0.33v₂) / (330 + v₂)] * 365

(357 / 365) = [(330 + 0.33v₂) / (330 + v₂)]

0.98 =  [(330 + 0.33v₂) / (330 + v₂)]

0.98 (330 + v₂) =  (330 + 0.33v₂)

323.4 + 0.98v₂ = 330 + 0.33v₂

323.4 - 330 = (0.33 - 0.98)v₂

-6.6 = -0.65v₂

v₂ = 10.15

The value of v₂ is positive and that is a valid solution.

Therefore, the speed of the car is 10.15m/s

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