Question

A source of sound and an observer are moving with velocities \( \overrightarrow{v_{s}}=(2 \hat{i}+4 \hat{j}) m / s \) and \( \overrightarrow{v_{0}}=(6 \hat{i}-3 \hat{j}) m / s \). If frequency of source is \( 400 Hz \), then frequency observed by the observer will be 

(1) \( 360 Hz \) 

(2) \( 380 Hz \) 

(3) \( 400 Hz \) 

(4) \( 440 Hz \)

Answer 1

Correct option is (3) 400 Hz

\(\vec{v_s}\) = 2\(\hat i\) + 4\(\hat j\) m/s

\(\vec{v_0}\) = 6\(\hat i\) - 3 m/s 

\(|\vec{v_s}|\) = \(\sqrt{4+16}\)

\(|\vec{v_s}|\) = \(\sqrt{20}\) m/s

\(|\vec{v_0}|\) = \(\sqrt{45}\) m/s

f' = f \(\left(\cfrac{v-v_0}{v-v_s}\right)\)

= 400 x \(\left(\cfrac{331-\sqrt{45}}{331-\sqrt{20}}\right)\)

= 400 x \(\cfrac{324.5}{326.5}\)

f' = 397.5 Hz