Question

Calculate the percentage increase in apparent frequency for an observer moving towards a stationary sound source with \( \frac{1}{5} \)th the velocity of sound.

• A. \(10%\)
• B. \(20%\)
• C. \(25%\)
• D. \(30%\)

Hint:

For a moving observer and stationary source, the Doppler formula simplifies to \[ f' = f\left(\frac{v + v_o}{v}\right). \] The fractional increase in frequency equals \( \frac{v_o}{v} \).

Answer 1

The Correct Option is B

Concept: According to the Doppler Effect, when an observer moves towards a stationary source, the apparent frequency is given by \[ f' = f\left(\frac{v + v_o}{v}\right) \] where

• \(f'\) = apparent frequency
• \(f\) = actual frequency
• \(v\) = velocity of sound
• \(v_o\) = velocity of the observer

The percentage increase in frequency is calculated from the ratio \( \frac{f' - f}{f} \times 100 \).
Step 1: {Substitute the given velocity.} The observer moves with velocity \[ v_o = \frac{v}{5} \] Thus \[ f' = f\left(\frac{v + \frac{v}{5}}{v}\right) \]
Step 2: {Simplify the expression.} \[ f' = f\left(\frac{6v}{5v}\right) \] \[ f' = \frac{6}{5}f \]
Step 3: {Find the increase in frequency.} \[ f' - f = \frac{6}{5}f - f \] \[ = \frac{1}{5}f \]
Step 4: {Calculate percentage increase.} \[ \frac{f' - f}{f} \times 100 \] \[ = \frac{1}{5} \times 100 \] \[ = 20% \]