Two trains are travelling towards each other both at a speed of `90 km h^-1`. If one of the trains sounds a whistle at 500 Hz, what will be the apparent frequency heard in the other train ? Speed of sound in air `= 350 m s^-1`.

Answer

To solve the problem of finding the apparent frequency heard in one train when the other train sounds a whistle, we can follow these steps:

### Step 1: Understand the Given Data
- Speed of both trains (source and observer) = 90 km/h
- Frequency of the whistle (source frequency) = 500 Hz
- Speed of sound in air = 350 m/s

### Step 2: Convert the Speed of the Trains from km/h to m/s
To convert the speed from kilometers per hour to meters per second, we use the conversion factor:
\[ 1 \text{ km/h} = \frac{1}{3.6} \text{ m/s} \]
Thus,
\[ 90 \text{ km/h} = 90 \times \frac{1}{3.6} \text{ m/s} = 25 \text{ m/s} \]

### Step 3: Identify the Variables for the Apparent Frequency Formula
The formula for the apparent frequency when both the source and observer are moving towards each other is:
\[ f' = \frac{V + V_o}{V - V_s} \times f \]
Where:
- \( f' \) = apparent frequency
- \( V \) = speed of sound in air = 350 m/s
- \( V_o \) = speed of the observer = 25 m/s
- \( V_s \) = speed of the source = 25 m/s
- \( f \) = frequency of the source = 500 Hz

### Step 4: Substitute the Values into the Formula
Substituting the known values into the formula:
\[ f' = \frac{350 + 25}{350 - 25} \times 500 \]
Calculating the numerator and denominator:
\[ f' = \frac{375}{325} \times 500 \]

### Step 5: Calculate the Apparent Frequency
Now, calculate the fraction:
\[ \frac{375}{325} \approx 1.1538 \]
Now multiply by the frequency of the source:
\[ f' \approx 1.1538 \times 500 \approx 576.9 \text{ Hz} \]
Rounding to the nearest whole number, we find:
\[ f' \approx 577 \text{ Hz} \]

### Final Answer
The apparent frequency heard in the other train is approximately **577 Hz**.

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