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

Step 1: Understanding the Question:
This problem deals with the Doppler Effect in sound.
The observer is moving toward the source, which causes the perceived frequency to increase because more wave crests are encountered per unit time.
Step 2: Key Formula or Approach:
1. Doppler formula for moving observer and stationary source: \( f' = f \left( \frac{v + v_o}{v} \right) \).
2. Percentage increase: \( \frac{f' - f}{f} \times 100% \).
Where \(v\) is speed of sound and \(v_o\) is speed of observer.
Step 3: Detailed Explanation:
Given:
Velocity of observer \( v_o = \frac{v}{5} \).
The apparent frequency \(f'\) is:
\[ f' = f \left( \frac{v + v/5}{v} \right) \]
\[ f' = f \left( \frac{6v/5}{v} \right) = \frac{6}{5} f \]
\[ f' = 1.2 f \]
The increase in frequency is:
\[ \Delta f = f' - f = 1.2f - f = 0.2f \]
Percentage increase:
\[ % \text{ Increase} = \frac{0.2f}{f} \times 100 = 20% \]
Step 4: Final Answer:
The percentage increase in apparent frequency is \(20%\).