Question

A whistle S of sound frequency f is oscillating with angular frequency 𝜔 along the x-axis. Its instantaneous position and the velocity are given by 𝑥(𝑡) = 𝑎 sin(𝜔𝑡) and 𝑣(𝑡)=𝑣0 cos (𝜔𝑡), respectively. An observer P is located on the y-axis at a distance L from the origin (see figure). Let 𝑣𝑃𝑆(𝑡) be the component of 𝑣(𝑡) along the line joining the source and the observer. Choose the correct option(s):
(Here 𝑎 and 𝑣0 are constants)

• A. 𝑣𝑃𝑆(𝑡)=$\frac{1}{2}\frac{av_o}{\sqrt{a^2-sin^2wt+L^2}}sin(2wt)$
• B. The observed frequency will be f when the source is at 𝑥 = 0 and 𝑥 = ±a
• C. The observed frequency will be f when the source is at position 𝑥 = ± $\frac{a}{2}$
• D. 𝑣𝑃𝑆(𝑡)=$\frac{1}{2}\frac{av_o}{\sqrt{a^2+L^2}}sin(2wt)$

Answer 1

Answer: (A), (B)

𝑣𝑃𝑆(𝑡)=$\frac{1}{2}\frac{av_o}{\sqrt{a^2-sin^2wt+L^2}}sin(2wt)$

Answer 2

The correct options are: A and B