Question

A motor-cyclist moving towards a huge cliff with a speed of 18 kmh^-1, blows a horn of source frequency 325 Hz. If the speed of the sound in air is 330 ms^-1, the number of beats heard by him is

• A. 5
• B. 4
• C. 10
• D. 7

Answer 1

The Correct Option is A

This is a case involving the Doppler effect, where the motor-cyclist hears a different frequency than the source frequency due to his motion towards the cliff.

Given:

• Source frequency (\( f_s \)) = 325 Hz
• Speed of the motor-cyclist (\( v_s \)) = 18 km/h = 5 m/s
• Speed of sound in air (\( v \)) = 330 m/s

Step 1: Calculate the apparent frequency (\( f' \)) of the sound waves when they are reflected back towards the motor-cyclist.

The formula for the Doppler effect when the source is moving towards the observer is:

\[ f' = f_s \times \frac{v + v_s}{v} \]

Substituting the given values:

\[ f' = 325 \times \frac{330 + 5}{330} = 325 \times \frac{335}{330} \approx 329.92 \text{ Hz} \]

Step 2: Calculate the number of beats heard by the motor-cyclist.

The number of beats is the difference between the two frequencies:

\[ \text{Number of beats} = |f' - f_s| = |329.92 - 325| \approx 5 \text{ beats per second} \]

The number of beats heard by the motor-cyclist is (A) 5.

Answer 2

Using the Doppler effect formula: \[ \Delta f = f \left( \frac{V}{V - V_s} - 1 \right) \] where: f = source frequency = 325 Hz V = speed of sound in air = 330 m/s V_s = speed of the motor-cyclist = 18 km/h = 5 m/s Substitute the values: \[ \Delta f = 325 \left( \frac{330}{330 - 5} - 1 \right) \] \[ \Delta f = 325 \left( \frac{330}{325} - 1 \right) = 325 \times \left( 1.0154 - 1 \right) = 325 \times 0.0154 \approx 5 \, \text{beats} \]