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Question

A train is moving at $30\,m{s^{ - 1}}$ in still air the frequency of the locomotive whistle is 500Hz and the speed of sound is $345\,m{s^{ - 1}}.$ The apparent wavelengths of sound in front of and behind the locomotive are respectively.

NAGA SAI · Accepted Answer

## CORE INFORMATION
✓ **Answer**: B
## CONCEPT FRAMEWORK
	**Primary Concept**: Doppler Effect for moving source.
	**Related Topics**: Wave speed, Wavelength, Frequency.
	**Prerequisites**: Understanding of wave properties and Doppler effect.
## SOLUTION PATHWAY
**Given Information**:
	* Speed of train (source), $v_s = 30\,m/s$
	* Frequency of whistle (source), $f_s = 500\,Hz$
	* Speed of sound, $v = 345\,m/s$
	* Observer is stationary
**Method & Steps**:
	1. **Calculate the apparent frequency in front of the train (approaching):**
		Reasoning: When the source is moving towards the observer, the apparent frequency is higher.
		Formula/Rule: $f_{front} = f_s \frac{v}{v-v_s}$
		Working: $f_{front} = 500 \frac{345}{345-30} = 500 \frac{345}{315} \approx 547.62\,Hz$
	2. **Calculate the apparent wavelength in front of the train:**
		Reasoning: Using the relationship between wave speed, frequency, and wavelength. $\lambda = v/f$.
		Working: $\lambda_{front} = \frac{345}{547.62} \approx 0.63\,m$
	3. **Calculate the apparent frequency behind the train (receding):**
		Reasoning: When the source is moving away from the observer, the apparent frequency is lower.
		Formula/Rule: $f_{behind} = f_s \frac{v}{v+v_s}$
		Working: $f_{behind} = 500 \frac{345}{345+30} = 500 \frac{345}{375} = 460\,Hz$
	4. **Calculate the apparent wavelength behind the train:**
		Reasoning: Using the relationship between wave speed, frequency, and wavelength. $\lambda = v/f$.
		Working: $\lambda_{behind} = \frac{345}{460} = 0.75\,m$
	5. **Final Result**: The apparent wavelengths in front and behind the train are approximately $0.63\,m$ and $0.75\,m$ respectively.
 ## OPTION ANALYSIS
[A] :x: Incorrect wavelength behind the train.
[B] ✓: Correct wavelengths for both in front and behind the train.
[C] :x: Incorrect wavelengths for both.
[D] :x: Incorrect wavelengths for both.
 ## LEARNING AIDS
	**Common Mistakes**:
		1. **Sign Convention**: Forgetting the correct sign in the Doppler effect equations. Use '+' for receding and '-' for approaching.
		2. **Incorrect Calculation**: Mistakes in arithmetic calculations.
	**Quick Tips**:
		1. **Approaching Higher**: When the source approaches, the frequency is higher and the wavelength is shorter.
		2. **Receding Lower**: When the source recedes, the frequency is lower and the wavelength is longer.

NAGA SAI · Answer

0.63 m, 0.80 m

NAGA SAI · Answer

The correct Option is "0.63 m, 0.75 m"

NAGA SAI · Answer

0.60 m, 0.85 m

NAGA SAI · Answer

0.60 m, 0.75 m

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