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 𝑣_𝑃𝑆(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. 𝑣_𝑃𝑆(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. 𝑣_𝑃𝑆(𝑡)=\(\frac{1}{2}\frac{av_o}{\sqrt{a^2+L^2}}sin(2wt)\)

Answer 1

The Correct Option is A, B

Velocity Component Along the Line of Sight 

The velocity component v_PS(t) along the line joining the source and the observer is given by:

v_PS(t) = v(t) cosϕ

where ϕ is the angle between the velocity vector and the line joining S and P.

Distance Between the Source and Observer

From the geometry of the problem, the distance between the source and the observer is:

r(t) = √(x(t)² + L²) = √(a² sin²(ωt) + L²)

The angle ϕ satisfies:

cosϕ = L / r(t)

Velocity Component Substitution

Substituting v(t) = v₀ cos(ωt) and cosϕ = L / r(t), we have:

v_PS(t) = v₀ cos(ωt) × (L / √(a² sin²(ωt) + L²))

Using Trigonometric Identities

The component along the line of sight becomes:

v_PS(t) = (1/2) × (a v₀ / √(a² sin²(ωt) + L²)) sin(2ωt)

Thus, statement (A) is correct.

Doppler Effect and Observed Frequency

The observed frequency is given by the Doppler effect, which depends on the relative velocity component v_PS(t).

• At x = 0 (midpoint of oscillation) and x = ±a (extremes of oscillation), the relative velocity component becomes zero, and the observed frequency equals the original frequency f.

Hence, statement (B) is correct.

Analyzing Other Statements

• At x = ±a/2, the velocity component v_PS(t) is nonzero, so the observed frequency differs from f.
• Statement (D) is incorrect because v_PS(t) depends on the varying distance r(t), not just √(a² + L²).

Conclusion

The correct statements are (A) and (B).