Question

Practice: The speed of sound at sea level is normally about 340 m/s. A car honks its horn as it drives toward an observer. The frequency of the horn is 800 Hz, but the observer hears an 860-Hz pitch.
What is the velocity of the car?



Show your work:

Answer 1

Answer:

25.5 m/s

Explanation:

The Doppler effect occurs when there is relative motion between a source of a wave and an observer. In such situation, there is a shift in the apparent frequency of the wave perceived by the observer.

The formula that gives the apparent frequency perceived by the observer is:

$f'=\frac{v\pm v_o}{v\pm v_s}f$

where

f is the real frequency of the wave

f' is the apparent frequency of the wave

v is the speed of the wave

$v_s$ is the velocity of the source (negative if the source is moving towards the observer, positive otherwise)

$v_o$ is the velocity of the observer (positive if the observer is moving towards the source, negative otherwise)

In this problem:

v = 340 m/s is the speed of sound

f = 800 Hz is the frequency of the horn

f' = 860 Hz is the apparent frequency

$v_o=0$ (the observer is at rest)

Re-arranging the equation for $v_s$, we can find the velocity of the horn and the driver:

$f'=\frac{v}{v-v_s}f\\(v-v_s)f'=vf\\vf'-v_sf'=vf\\v_s=v\frac{f'-f}{f'}=(340)\frac{860-800}{860}=25.5 m/s$

So, 25.5 m/s towards the observer.