A train sounds its whistle as it approaches an observer standing at a point near the track. The observer measures a frequency of 216 Hz as the train approaches and a frequency of 184 Hz as the train leaves. What is the frequency of its whistle?

Question

A train sounds its whistle as it approaches an observer standing at a point near the track. The observer measures a frequency of 216 Hz as the train approaches and a frequency of 184 Hz as the train leaves. What is the frequency of its whistle? (A) 210 Hz (B) 190 Hz (C) 205 Hz (D) 202 Hz

Tardigrade Admin · Accepted Answer

Case I When train is moving towards a stationary observer.
According to Doppler's effect of sound,
apparent frequency heard,

n^{'} = \frac{n v}{v - v _{s}}

where,

n =

frequency of source,

v =

speed of sound and

v_{s} =

speed of source.

\Rightarrow v_{s} = v [1 - \frac{n}{n ^{'}}]

...(i)
Case II When train is leaving the standing observer.
Again, according to the Doppler's effect of sound,
Apparent frequency heard,

n^{''} = \frac{n v}{v + v _{s}} \Rightarrow v_{s} = v [\frac{n}{n ^{'}} - 1]

...(ii)
From Eqs. (i) and (ii), we get

\frac{n}{n ^{'}} + \frac{n}{n ^{''}} = 2

Given,

n^{'} = 216 Hz

and

n^{''} = 184 Hz

Substituting these values in the above equation, we get

n = \frac{2 \times 216 \times 184}{( 216 + 184 )} \approx 200 Hz

Q. A train sounds its whistle as it approaches an observer standing at a point near the track. The observer measures a frequency of 216Hz as the train approaches and a frequency of 184Hz as the train leaves. What is the frequency of its whistle?

210 Hz

190 Hz

205 Hz

202 Hz

200 Hz

Q. A train sounds its whistle as it approaches an observer standing at a point near the track. The observer measures a frequency of $216 Hz$ as the train approaches and a frequency of $184 Hz$ as the train leaves. What is the frequency of its whistle?