Question

A source (S) of sound has frequency\( 240 Hz\). When the observer (O) and the source move towards each other at a speed \(v\) with respect to the ground (as shown in Case 1 in the figure), the observer measures the frequency of the sound to be \(288 Hz\). However, when the observer and the source move away from each other at the same speed v with respect to the ground (as shown in Case 2 in the figure), the observer measures the frequency of sound to be \(n\) Hz. The value of \(n\) is _____.

Answer 1

Correct Answer: 200

Case 1: Observer and Source Moving Towards Each Other 

The apparent frequency is given by:

\[ f_{\text{app}} = f_0 \cdot \frac{v + u}{v - u} \]

Substituting the given values:

\[ 288 = 240 \cdot \frac{v + u}{v - u} \]

Case 2: Observer and Source Moving Away From Each Other

The apparent frequency is given by:

\[ h = f_0 \cdot \frac{v - u}{v + u} \]

Substituting the given values:

\[ h = 240 \cdot \frac{v - u}{v + u} \]

Solving for \( v \) and \( u \):

From equation (i):

\[ \frac{v + u}{v - u} = \frac{288}{240} = \frac{6}{5} \]

Simplifying:

\[ v + u = 6k, \quad v - u = 5k \]

Adding and subtracting these equations:

\[ 2v = 11k \Rightarrow v = 5.5k, \quad 2u = k \Rightarrow u = 0.5k \]

Finding \( h \):

Substituting \( v + u = 6k \) and \( v - u = 5k \) into equation (ii):

\[ h = 240 \cdot \frac{5}{6} \]

Simplifying:

\[ h = 200 \text{ Hz} \]

Final Answer:

\( h = 200 \) Hz