Two train whistles have identical frequencies of 180Hz. When one train is at res

Question

Two train whistles have identical frequencies of 180Hz. When one train is at rest and the other is moving a person hears beats with a frequency of 2 beats per sec. when the whistles operate together. Using the speed of sound as 345 what are the two possible speeds and directions that the moving train can have? Two train whistles have identical frequencies of 180Hz. When one train is at rest and the other is moving a person hears beats with a frequency of 2 beats per sec. when the whistles operate together. Using the speed of sound as 345 what are the two possible speeds and directions that the moving train can have?

Explanation / Answer

The beat frequency equals the absolute value of the difference between the two source frequency, or :

f(b) = |f - f|

This allows for two different frequencies, one for approach and one for recession. The approach frequency is :

2 .00= |180Hz - f|
2.00 = f- 180Hz
f= 182Hz

The recession frequency is :

2.00 = |180Hz - f|
2.00 = 180Hz - f
f= 178Hz

Now, from the Doppler shift equation, the approach velocity is :

f = f(s)[v / v + v(s)]
v(s) = v[f - f(s)] / f
= 345m/s[182Hz – 180Hz] / 182Hz
= 3.79m/s

The velocity of recession is (from the same equation) :

v(s) = 345m/s[178Hz - 180Hz] / 178Hz
= -3.87m/s

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