If a sound with frequency f s is produced by a source traveling along a line wit

Question

If a sound with frequency fs is produced by a source traveling along a line with speed vs and an observer is traveling with speed vo along the same line from the opposite direction toward the source, then the frequency of the sound heard by the observer is

f_0 =( (c+v_0)/(c-v_s))*f_s

where c is the speed of sound, about 332 m/s. (This is the Doppler effect.) Suppose that, at a particular moment, you are in a train traveling at 32 m/s and accelerating at 1.4 m/s2. A train is approaching you from the opposite direction on the other track at 45 m/s, accelerating at 1.5 m/s2, and sounds its whistle, which has a frequency of 454 Hz. At that instant, what is the perceived frequency that you hear? (Round your answer to one decimal place.)
Hz

How fast is it changing? (Round your answer to two decimal places.)
Hz/s

Explanation / Answer

f0 = ((c + v0) / (c - vs)) * fs

f0 = (332 + 32) / (332 - (45)) * 454

f0 = 575.8 Hz ---> ANSWER

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Lets derive expression for f0 with respect to t :

df0/dt :

u = (c + v0) , v = (c - vs)
du = dv0/dt = a0 , dv = -dvs/dt = -as

So,

((a0)(c - vs) - (-as)(c + v0)) / (c - vs)^2

Into this plug n the info :

(1.4(332 - 45) + 1.5(332 + 32)) / (332 - 45)^2

Now, this must be multiplied by fs....

0.0115067561825444*454

5.224

5.22 Hertz per second ---> ANSWER

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