Question

A train is moving with a speed of \(10 m/s\) towards a platform and blows a horn with frequency \(400 Hz\). Find the frequency heard by a passenger standing on the platform. Take speed of sound = \(310 m/s\).

• A. 405 Hz
• B. 425 Hz
• C. 380 Hz
• D. 413 Hz

Answer 1

The Correct Option is D

To find the frequency heard by a passenger standing on the platform while the train is approaching, we need to use the Doppler Effect formula for sound. The Doppler Effect describes the change in frequency or wavelength of a wave in relation to an observer moving relative to the source of the waves.

Step-by-step Solution:

1. The formula for the frequency heard by an observer when the source is moving towards the observer is given by: \(f' = \frac{f(v + v_o)}{v - v_s}\)
   • Here, \(f'\) is the frequency heard by the observer.
   • \(f\) is the frequency of the source (400 Hz).
   • \(v\) is the speed of sound in air (310 m/s).
   • \(v_o\) is the speed of the observer. Since the observer is stationary, \(v_o = 0\).
   • \(v_s\) is the speed of the source (train), which is 10 m/s towards the observer.
2. Substitute the values into the formula: \(f' = \frac{400 \times (310 + 0)}{310 - 10}\)
3. Calculate the denominator and the fraction:
   • Denominator: \(310 - 10 = 300\)
   • Frequency heard by passenger: \(f' = \frac{400 \times 310}{300}\)
   • Simplify the equation: \(f' = \frac{124000}{300} = 413.33 \, \text{Hz}\)
4. Rounding to the nearest whole number gives the approximate frequency heard as 413 Hz.
5. Thus, the correct answer is 413 Hz.

This calculation is consistent with the Doppler Effect, where the frequency increases as the source approaches the observer. Hence, option "413 Hz" is correct.