Question

A submarine A, traveling at \(17 { m/s}\), is being chased along the line of its velocity by another submarine B, traveling at \(34 { m/s}\). Submarine B sends a sonar signal of \(600 { Hz}\) to detect A and receives a reflected sound of frequency \( v \). The value of \( v \) is: \[ {[Speed of sound in water} = 1500 { m/s}] \]

• A. \(613.7 { Hz}\)
• B. \(6137 { Hz}\)
• C. \(62 { Hz}\)
• D. \(539 { Hz}\)

Hint:

In two-step Doppler problems: - Apply the Doppler formula twice: once for the moving observer and once for the reflected sound. - Carefully assign the observer and source velocities in each step.

Answer 1

The Correct Option is A

Given: Velocity of submarine A, \( v_A = 17 \) m/s 
Velocity of submarine B, \( v_B = 34 \) m/s 
Signal sent by submarine B is detected by submarine A can be shown as: 

 Frequency of the signal, \( f_0 = 600 \) Hz 
Speed of sound in water \( v_s = 1500 \) m/s 
Step 1: Calculate the frequency received by submarine A 
\[ f_1 = \left( \frac{v_s - v_A}{v_s - v_B} \right) f_0 \] Substituting values: \[ f_1 = \left( \frac{1500 - 17}{1500 - 34} \right) \times 600 \] \[ f_1 = \frac{1483}{1466} \times 600 \] \[ f_1 \approx 600 \quad {(i)} \] 
Step 2: Calculate the frequency received by submarine B 
\[ f_2 = \left( \frac{v_s + v_B}{v_s + v_A} \right) f_1 \] Substituting values and using \( f_1 \) from Eq. (i), we get: \[ f_2 = \left( \frac{1500 + 34}{1500 + 17} \right) \times \left( \frac{1483}{1466} \times 600 \right) \] \[ f_2 = 1.0112 \times 1.0115 \times 600 \] \[ f_2 = 613.7 { Hz} \]