You're standing by the roadside as a truck approaches, and you measure the dominant frequency in the truck noise at 1080 Hz. As the truck passes, the frequency drops to 895 Hz. What is the truck's speed in m/s?

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You're standing by the roadside as a truck approaches, and you measure the dominant frequency in the truck noise at 1080 Hz. As the truck passes, the frequency drops to 895 Hz. What is the truck's speed in m/s?

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## Answers (3)

When it's coming toward, you the frequency is:

fbig = f (c / c - v), where f is the true frequency, c is the speed of sound, and v is the speed

When it's going away, the frequency is

fsmall = f (c / c+ v)

Two equations, two unknowns. Let's solve the 2nd equation for the true frequency

f = fsmall (c+v) / c

Plug that into the first:

fbig = fsmall (c+v) / (c - v)

fbig/fsmall (c-v) = c+v

c (fbig/fsmall - 1) = v (1 + fbig/fsmall)

v = c (fbig/fsmall - 1) / (fbig/fsmall + 1)

They give you the frequencies. Look up the speed of sound. Plugnchug.

The approximation the others use:

true frequency = (fbig + fsmall) / 2

is probably close for this problem, but not so good if the doppler shift is large. Use the proper formula.

I'm not solving this for you, but here are the things you can use to calculate the answer:

Take the speed of sound, which is the velocity that a sound wave moves. If you know the frequency (e.g., 1080 Hz) , you can figure the spacing between the individual sound waves.

When the truck is approaching you, the frequency you hear is higher than the actual frequency because the waves caused by the truck noise are "packed" closer together. When the truck is receding, its sound waves are spread out and sound lower. If you find the halfway point frequency (1080 plus 895, divided by 2), you have the real frequency. By taking the difference between the highest frequency (or the lowest) from the real frequency, you can figure the truck's speed.

Years ago I did this when the driver of a car I was riding in made a mistake and crossed an intersection in the path of a car coming from the left. We barely made it past the path of the oncoming car and it sailed past us with the horn blaring. The pitch dropped and I made a point of remembering the musical interval that resulted. I later calculated that car's speed as 115 mph. No wonder my driver didn't see it coming.

Assuming the truck travels at a constant speed throughout the recording,

true frequency = (1080+895)/2 = 987.5

∆f = fv/c

92.5 = 987.5 x v(truck) / 330 m/s

v(truck) = 30.9 m/s

Reference:en.wikipedia.org/wiki/Doppler_effect