Two trains are moving towards each other at speeds of 144 km/hr and 54 km/hr relative to the ground. The first train sounds a whistle of frequency 600 Hz. Find the frequency of the whistle as heard by a passenger in the second train before the trains meet. (y =340m/s)

Options

A. 610 Hz
B. 510 Hz
C. 710 Hz
D. 170 Hz

Answer

To find the frequency of the whistle as heard by a passenger in the second train before the trains meet, we can use the Doppler Effect formula for sound. Here’s the step-by-step solution:

### Step 1: Convert the speeds of the trains from km/hr to m/s
- The speed of the first train (V1) is given as 144 km/hr. To convert this to m/s:
\[ V1 = 144 \times \frac{5}{18} = 40 \, \text{m/s} \]

- The speed of the second train (V2) is given as 54 km/hr. To convert this to m/s:
\[ V2 = 54 \times \frac{5}{18} = 15 \, \text{m/s} \]

### Step 2: Identify the frequency of the whistle
- The frequency of the whistle emitted by the first train (f0) is given as:
\[ f0 = 600 \, \text{Hz} \]

### Step 3: Apply the Doppler Effect formula
- Since the trains are moving towards each other, we can use the following Doppler Effect formula to find the observed frequency (f'):
\[ f' = f0 \times \frac{Vs + V2}{Vs - V1} \]
where:
- Vs = speed of sound = 340 m/s
- V1 = speed of the source (first train) = 40 m/s
- V2 = speed of the observer (second train) = 15 m/s

### Step 4: Substitute the values into the formula
- Now substituting the values into the formula:
\[ f' = 600 \times \frac{340 + 15}{340 - 40} \]
\[ f' = 600 \times \frac{355}{300} \]

### Step 5: Calculate the observed frequency
- Simplifying the fraction:
\[ f' = 600 \times \frac{355}{300} = 600 \times 1.1833 \approx 710 \, \text{Hz} \]

### Conclusion
- The frequency of the whistle as heard by a passenger in the second train is approximately **710 Hz**.