Question

A source of sound of frequency \(500\ \text{Hz}\) is moving towards an observer with velocity \(30\ \text{m/s}\). The speed of sound is \(330\ \text{m/s}\). The frequency heard by the observer will be

• A. \(450\ \text{Hz}\)
• B. \(550\ \text{Hz}\)
• C. \(600\ \text{Hz}\)
• D. \(500\ \text{Hz}\)

Hint:

When source moves towards observer, apparent frequency increases; use denominator \(v-v_s\).

Answer 1

The Correct Option is B

Step 1: Understanding the Question:
The Doppler Effect describes the change in observed frequency when there is relative motion between the source and the observer.
When a source moves towards a stationary observer, the observed frequency increases because the sound waves are compressed.
Step 2: Key Formula or Approach:
Apparent frequency \(f' = f \cdot \left( \frac{v \pm v_o}{v \mp v_s} \right)\).
Here:
\(f = 500 \text{ Hz}\) (actual frequency).
\(v = 330 \text{ m/s}\) (speed of sound).
\(v_o = 0\) (stationary observer).
\(v_s = 30 \text{ m/s}\) (source moving towards observer).
Step 3: Detailed Explanation:

For a source moving towards a stationary observer, the formula simplifies to:
\[ f' = f \left( \frac{v}{v - v_s} \right) \]

Plug in the given values:
\[ f' = 500 \left( \frac{330}{330 - 30} \right) \]

Simplify the denominator:
\[ f' = 500 \left( \frac{330}{300} \right) \]

Reduce the fraction:
\[ \frac{330}{300} = \frac{33}{30} = \frac{11}{10} = 1.1 \]

Calculate the final frequency:
\[ f' = 500 \times 1.1 = 550 \text{ Hz} \]

Step 4: Final Answer:
Due to the Doppler effect, the frequency perceived by the stationary observer increases to 550 Hz as the source approaches.