Two trains are moving towards each other at speed of 144 km/hr and 54 km/hr relative to the ground. The second sounds a whistle of frequency 710 Hz, the frequency of this whistle as heard by a passenger in the first train after the trains have crossed each other is `x xx 10^2Hz` , what is value of x. (v = 340 m/s)

Answer

To solve the problem, we need to find the frequency of the whistle as heard by a passenger in the first train after the trains have crossed each other. We will use the Doppler effect formula for sound.

### Step 1: Convert the speeds of the trains from km/hr to m/s
- The speed of the first train (observer) is 144 km/hr.
- The speed of the second train (source) is 54 km/hr.

**Conversion formula**:
\[ \text{Speed in m/s} = \text{Speed in km/hr} \times \frac{5}{18} \]

Calculating the speeds:
- For the first train:
\[ V_o = 144 \times \frac{5}{18} = 40 \text{ m/s} \]
- For the second train:
\[ V_s = 54 \times \frac{5}{18} = 15 \text{ m/s} \]

### Step 2: Identify the velocities in the Doppler effect formula
After the trains cross each other, the source and observer are moving away from each other. Thus:
- Velocity of sound, \( V = 340 \text{ m/s} \)
- Velocity of observer, \( V_o = 40 \text{ m/s} \) (moving away)
- Velocity of source, \( V_s = 15 \text{ m/s} \) (also moving away)

### Step 3: Use the Doppler effect formula
The apparent frequency \( f' \) can be calculated using the formula:
\[ f' = f \times \frac{V - V_o}{V + V_s} \]
Where:
- \( f = 710 \text{ Hz} \) (original frequency)

Substituting the values:
\[ f' = 710 \times \frac{340 - 40}{340 + 15} \]

### Step 4: Calculate the values in the formula
Calculating the numerator and denominator:
- Numerator:
\[ 340 - 40 = 300 \]
- Denominator:
\[ 340 + 15 = 355 \]

Now substituting these into the formula:
\[ f' = 710 \times \frac{300}{355} \]

### Step 5: Perform the multiplication and division
Calculating:
\[ f' = 710 \times 0.84507 \approx 599.99 \text{ Hz} \]

### Step 6: Express the frequency in the required format
We need to express \( f' \) in the form \( x \times 10^2 \text{ Hz} \):
\[ f' \approx 600 \text{ Hz} = 6.00 \times 10^2 \text{ Hz} \]

Thus, the value of \( x \) is:
\[ x = 6 \]

### Final Answer:
The value of \( x \) is **6**.

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