Speed of Whistle Sound Observed by Train Passengers

To find the speed of sound of the whistle observed by the passengers in trains $A$ and $B$, we consider the relative velocity of sound with respect to the observers in each train.

1. Context and Given Parameters:

• Train $A$ and train $B$ are moving in the same direction (let this be along the positive $x$-axis, $+\hat{i}$) on parallel tracks with speeds:
  $v_A = 20\text{ m/s}, \quad v_B = 30\text{ m/s}$
• Train $A$ is behind train $B$.
• The speed of sound in still air is $v = 340\text{ m/s}$.

2. Relative Speed of Sound for Passengers in Train $A$:

• The sound waves reaching the passengers in train $A$ from the whistle travel in the backward direction (opposite to the direction of motion of the trains), so the velocity of sound in the air frame is:
  $\vec{v}_{\text{sound}} = -340\,\hat{i}\text{ m/s}$
• The velocity of passengers in train $A$ is:
  $\vec{v}_A = 20\,\hat{i}\text{ m/s}$
• The relative velocity of the sound wave with respect to the passengers in train $A$ is:
  $\vec{v}_{\text{sound}, A} = \vec{v}_{\text{sound}} - \vec{v}_A = -340\,\hat{i} - 20\,\hat{i} = -360\,\hat{i}\text{ m/s}$
• Thus, the observed speed of sound for passengers in train $A$ is:
  $|\vec{v}_{\text{sound}, A}| = 360\text{ m/s}$

3. Relative Speed of Sound for Passengers in Train $B$:

• For passengers in train $B$, the whistle is on train $B$ and the sound traveling forward along the train moves in the direction of the train's motion, so:
  $\vec{v}_{\text{sound}} = 340\,\hat{i}\text{ m/s}$
• The velocity of passengers in train $B$ is:
  $\vec{v}_B = 30\,\hat{i}\text{ m/s}$
• The relative velocity of the sound wave with respect to the passengers in train $B$ is:
  $\vec{v}_{\text{sound}, B} = \vec{v}_{\text{sound}} - \vec{v}_B = 340\,\hat{i} - 30\,\hat{i} = 310\,\hat{i}\text{ m/s}$
• Thus, the observed speed of sound for passengers in train $B$ is:
  $|\vec{v}_{\text{sound}, B}| = 310\text{ m/s}$

Therefore, the speed of sound of the whistle is $360\text{ m/s}$ for passengers in train $A$ and $310\text{ m/s}$ for passengers in train $B$.

Correct Option: (B)