Doppler shift occurs due to relative motion; which expression gives the Doppler frequency shift?

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Multiple Choice

Doppler shift occurs due to relative motion; which expression gives the Doppler frequency shift?

Explanation:
Doppler shift comes from the part of the relative motion that is along the line of sight. The wave has speed v and wavelength λ in the medium, so its base frequency is f = v/λ. Only the component of motion toward or away from the observer matters, which is v cos θ. Thus the amount by which the frequency shifts is f_d = (v/λ) cos θ. This shows why the shift is largest when the motion is directly along the line of sight (θ = 0) and zero when the motion is perpendicular (θ = 90°). The other forms misplace v and λ or omit the angular projection, leading to incorrect units or missing the directional factor.

Doppler shift comes from the part of the relative motion that is along the line of sight. The wave has speed v and wavelength λ in the medium, so its base frequency is f = v/λ. Only the component of motion toward or away from the observer matters, which is v cos θ. Thus the amount by which the frequency shifts is f_d = (v/λ) cos θ.

This shows why the shift is largest when the motion is directly along the line of sight (θ = 0) and zero when the motion is perpendicular (θ = 90°). The other forms misplace v and λ or omit the angular projection, leading to incorrect units or missing the directional factor.

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