A source of sound is moving with velocity u/2 and two observers A and B are moving with velocity u (In same direction). Find the ratio of wavelenght received by A and B. (Velocity of sound is 10u.)

Dear Student,

There can be four cases in this problem:
Case 1: When A and B are approaching the source from one side and source is also approaching them
Case 2: When A and B are moving in the same direction from one side as the source
Case 3: When A is approaching to the source and B is moving away from it
Case 4: When B is approaching to the source and A is moving away from it

Let $\lambda \mathrm{and} \nu$ be the wavelength and the frequency of the wave sound wave emitted by the source. 

Case 1: In this case both observers will receive the same wavelength say $\lambda \text{'}$. 

Thus, $\frac{\lambda {\text{'}}_A}{\lambda {\text{'}}_B}=1$

Case 2: In this case both observers will receive the same wavelength say $\lambda \text{'}$. 

Thus, $\frac{\lambda {\text{'}}_A}{\lambda {\text{'}}_B}=1$

Case 3: For A

$$\begin{gathered} \frac{1}{{\lambda }_A\text{'}}=\left[\frac{v-v_L}{v-v_s}\right]\frac{1}{\lambda }=\left[\frac{10u-u}{10u-0.5u}\right]\frac{1}{\lambda } \\ \Rightarrow \frac{1}{{\lambda }_A\text{'}}=\frac{90}{95}\frac{1}{\lambda } ....\left(i\right) \end{gathered}$$

For B
$$\begin{gathered} \frac{1}{{\lambda }_B\text{'}}=\left[\frac{v-v_L}{v-(-v_s)}\right]\frac{1}{\lambda }=\left[\frac{10u-u}{10u+0.5u}\right]\frac{1}{\lambda } \\ \Rightarrow \frac{1}{{\lambda }_B\text{'}}=\frac{90}{105}\frac{1}{\lambda } ....\left(ii\right) \end{gathered}$$
From (i) and (ii), we get
$\frac{\lambda {\text{'}}_A}{\lambda {\text{'}}_B}=\frac{19}{21}$

Case 4: Similar to case 3
$\frac{\lambda {\text{'}}_A}{\lambda {\text{'}}_B}=\frac{21}{19}$

Regards,