A stationary source emits sound waves of frequency 500 Hz. Two observers moving along a line passing through the source detect sound to be of frequencies 480 Hz and 530 Hz. Their respective speeds are, in ms^-1,

(Given speed of sound = 300 m/s)

Concept:

Apparent frequency:

Apparent frequency is the frequency perceived by an outside observer. It may or may not match the actual frequency

Apparent frequency according to the Doppler Effect is given by the formula,

\(f = \frac{{{f_0}\left( {v - {v_{obs}}} \right)}}{v}\)

f₀ is the Frequency of sound source.

Calculation:

Given,

Frequency of sound source (f₀) = 500 Hz

When observer moves away from the source, the frequency of the sound detected will decrease. Thus, f₁ = 480 Hz.

When observer moves towards the source, the frequency of the sound detected will increase. Thus, f₂ = 530 Hz.

Speed of the sound, v = 300 m/s

Apparent frequency according to the Doppler Effect is given by the formula,

\(f = \frac{{{f_0}\left( {v - {v_{obs}}} \right)}}{v}\)

When observer is moving away from the source

\({{\rm{f}}_1} = {{\rm{f}}_0}\left( {\frac{{v - v_0^{'}}}{v}} \right)\)

\(480 = {{\rm{f}}_0}\left( {\frac{{v - v_0^{'}}}{v}} \right)\)

\(480 = 500\left( {\frac{{300 - v_0^{'}}}{{300}}} \right)\)

\(480 \times \frac{{300}}{{500}} = 300 - {v_0}'\)

96 × 3 = 300 – v₀'

v₀' = 300 – 288

v₀' = 12 m/s

And when observer is moving towards the source

\({{\rm{f}}_2} = {{\rm{f}}_0}\left( {\frac{{{\rm{v}} - {\rm{v}}_0^{''}}}{{\rm{v}}}} \right)\)

\(530 = {{\rm{f}}_0}\left( {\frac{{{\rm{v}} - {\rm{v}}_0^{''}}}{{\rm{v}}}} \right)\)

\(530 = 500\left( {\frac{{300 - {\rm{v}}_0^{''}}}{{300}}} \right)\)

\(530 \times \frac{{300}}{{500}} = 300 - {\rm{v}}_0^{''}\)

\(530 \times \frac{3}{5} = 300 - {\rm{v}}_0^{''}\)

106 × 3 = 300 – v₀''

v₀'' = 300 – 318 = -18

∴ v₀'' = 18 m/s