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Question #42867

Difficulty: 8/10 | 🟢 Free

The Question

An observer is stationary at the origin $(0,0)$ of a Cartesian coordinate system. At time $t = 0$ , two identical sound sources, $S_1$ and $S_2$ , are located at a height $H = 110 \text{ m}$ on the y-axis (initial position $(0, 110 \text{ m})$ ). Source $S_1$ is projected horizontally with a velocity $\vec{u}_1 = 10 \hat{i} \text{ m/s}$ . Source $S_2$ is simultaneously dropped from rest. Both sources continuously emit sound of proper frequency $f_0 = 1215 \text{ Hz}$ . The speed of sound in air is $v_s = 340 \text{ m/s}$ , and the acceleration due to gravity is $g = 10 \text{ m/s}^2$ (acting in the $-\hat{j}$ direction). Calculate the beat frequency detected by the observer corresponding to the sound waves emitted by the sources at the exact instant $t = 4 \text{ s}$ .

Options

A.
102 \text{ Hz}
B.
0 \text{ Hz}
C.
36 \text{ Hz}
D.
54 \text{ Hz}

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Correct Answer

Step-by-Step Solution

Explanation

How the Examiner Thinks

Multi-Concept_Bridge

Master problems that require linking physics and math concepts to solve complex competitive questions.

Concepts Behind This Question

Beats and Frequency Difference|Relative Velocity in 2D|Projectile Path Equation

Why This Question Is Tricky

Common Mistakes

This question tests the student's ability to correctly apply the Doppler effect in two-dimensional kinematics.

The primary trap lies in calculating the effective source velocity.

Students frequently misapply the Doppler formula by using either the absolute speed of the projectile ( $|\vec{v}|$ ) or merely its vertical velocity component ( $v_y$ ), ignoring the geometric relationship between the source and the observer.

The correct physical principle requires projecting the instantaneous velocity vector onto the line-of-sight unit vector connecting the source to the observer ( $\vec{v} \cdot \hat{n}$ ).

By setting up a parabolic trajectory, the problem forces a rigorous vector dot product to find the true radial velocity ( $v_{\parallel} = 16 \text{ m/s}$ ), distinguishing it from the trivial 1D free-fall case of the second source.

Quick Facts About This Question

Question Information

Subject	Physics
Chapter	Waves - Class 11 Part 2
Concept	Beats and Frequency Difference
Difficulty	8
Cognitive Load	7
Target Solve Time	160 sec
Access	Free