Two trains are moving towards each other on parallel tracks at speeds of `144kmph` and `54kmph`. The first train sounds a whistle of frequency `600Hz`. Frequency of the whistle as heard by a passenger in the second train is (`V=340m//s`)

Options

A. `510Hz`
B. `610Hz`
C. `710Hz`
D. `810Hz`

Explanation

To solve the problem of finding the frequency of the whistle as heard by a passenger in the second train, we can use the Doppler effect formula for sound. Here's a step-by-step solution:

### Step 1: Convert Speeds from km/h to m/s
First, we need to convert the speeds of both trains from kilometers per hour (km/h) to meters per second (m/s).

- For the first train (Train 1):
\[ V_1 = 144 \text{ km/h} = 144 \times \frac{5}{18} = 40 \text{ m/s} \]

- For the second train (Train 2):
\[ V_2 = 54 \text{ km/h} = 54 \times \frac{5}{18} = 15 \text{ m/s} \]

### Step 2: Identify the Variables
We have the following variables:
- Frequency of the whistle (source frequency), \( f_s = 600 \text{ Hz} \)
- Speed of sound, \( V = 340 \text{ m/s} \)
- Speed of the observer (Train 2), \( V_o = 15 \text{ m/s} \)
- Speed of the source (Train 1), \( V_s = 40 \text{ m/s} \)

### Step 3: Apply the Doppler Effect Formula
Since both the source and observer are moving towards each other, we use the following formula for the apparent frequency \( f' \):
\[ f' = f_s \left( \frac{V + V_o}{V - V_s} \right) \]

### Step 4: Substitute the Values
Now we substitute the known values into the formula:
\[ f' = 600 \left( \frac{340 + 15}{340 - 40} \right) \]

### Step 5: Calculate the Numerator and Denominator
Calculate the numerator and denominator:
- Numerator: \( 340 + 15 = 355 \)
- Denominator: \( 340 - 40 = 300 \)

### Step 6: Final Calculation
Now plug these values back into the equation:
\[ f' = 600 \left( \frac{355}{300} \right) \]

Calculating this gives:
\[ f' = 600 \times 1.1833 \approx 710 \text{ Hz} \]

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

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