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A man is standing between a stationary s...

A man is standing between a stationary source and cliff. When he starts moving along line joining him and source, he hears 10 beats per second. The velocity of man is [frequency of source=600Hz, velocity of sound=330`ms^(-1)`]

A

5.5 m/s

B

11m/s

C

16/5m/s

D

2.75m/s

Text Solution

AI Generated Solution

The correct Answer is:
To solve the problem, we will use the concept of the Doppler effect and the information given about beats. Here’s a step-by-step breakdown of the solution: ### Step 1: Understand the Problem We have a stationary sound source emitting a frequency \( f_s = 600 \, \text{Hz} \) and a man moving towards the source. The speed of sound is \( v = 330 \, \text{m/s} \). The man hears 10 beats per second, which indicates that there are two different frequencies being heard: one from the source and one reflected off the cliff. ### Step 2: Identify the Frequencies 1. **Frequency heard directly from the source** (\( f_1 \)): \[ f_1 = f_s \frac{v - v_m}{v} \] where \( v_m \) is the velocity of the man. 2. **Frequency heard after reflecting off the cliff** (\( f_2 \)): \[ f_2 = f_s \frac{v + v_m}{v} \] ### Step 3: Set Up the Beat Frequency Equation The beat frequency \( f_b \) is given by the difference between the two frequencies: \[ f_b = |f_2 - f_1| = 10 \, \text{Hz} \] Substituting the expressions for \( f_1 \) and \( f_2 \): \[ 10 = \left( f_s \frac{v + v_m}{v} - f_s \frac{v - v_m}{v} \right) \] ### Step 4: Simplify the Equation Factoring out \( f_s \): \[ 10 = f_s \left( \frac{(v + v_m) - (v - v_m)}{v} \right) \] This simplifies to: \[ 10 = f_s \left( \frac{2v_m}{v} \right) \] Substituting \( f_s = 600 \, \text{Hz} \): \[ 10 = 600 \left( \frac{2v_m}{330} \right) \] ### Step 5: Solve for the Velocity of the Man Rearranging the equation gives: \[ 10 = \frac{1200 v_m}{330} \] Multiplying both sides by 330: \[ 3300 = 1200 v_m \] Now, solving for \( v_m \): \[ v_m = \frac{3300}{1200} = \frac{11}{4} \, \text{m/s} \] Calculating the numerical value: \[ v_m = 2.75 \, \text{m/s} \] ### Final Answer The velocity of the man is \( \boxed{2.75 \, \text{m/s}} \). ---
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