On a certain day sound is found to travel 1.4 km in 4 second in air. Calculate the average temperature of air. `V_(0)="331 m/s"^(-1)`.

Answer

To solve the problem, we need to calculate the average temperature of air based on the speed of sound. Here’s a step-by-step solution:

### Step 1: Calculate the speed of sound
We are given the distance traveled by sound (d) and the time taken (t).

- Distance, \( d = 1.4 \, \text{km} = 1.4 \times 1000 \, \text{m} = 1400 \, \text{m} \)
- Time, \( t = 4 \, \text{s} \)

The speed of sound \( v \) can be calculated using the formula:
\[ v = \frac{d}{t} \]
Substituting the values:
\[ v = \frac{1400 \, \text{m}}{4 \, \text{s}} = 350 \, \text{m/s} \]

### Step 2: Relate speed of sound to temperature
The speed of sound in air can be expressed in terms of temperature using the formula:
\[ v = V_0 \sqrt{\frac{T}{T_0}} \]
Where:
- \( V_0 = 331 \, \text{m/s} \) (speed of sound at 0°C)
- \( T_0 = 273 \, \text{K} \) (temperature in Kelvin at 0°C)
- \( T \) is the temperature in Kelvin we want to find.

### Step 3: Set up the equation
From the formula, we can rearrange it to find \( T \):
\[ v = V_0 \sqrt{\frac{T}{T_0}} \implies 350 = 331 \sqrt{\frac{T}{273}} \]

### Step 4: Square both sides
To eliminate the square root, we square both sides:
\[ 350^2 = 331^2 \cdot \frac{T}{273} \]
Calculating \( 350^2 \) and \( 331^2 \):
\[ 122500 = 109561 \cdot \frac{T}{273} \]

### Step 5: Solve for T
Now, we can solve for \( T \):
\[ T = \frac{122500 \cdot 273}{109561} \]
Calculating the right-hand side:
\[ T \approx \frac{33472500}{109561} \approx 305.3 \, \text{K} \]

### Step 6: Convert to Celsius
To convert Kelvin to Celsius:
\[ T_{Celsius} = T - 273 \approx 305.3 - 273 \approx 32.3 \, \text{°C} \]

### Final Answer
The average temperature of air is approximately \( 305.3 \, \text{K} \) or \( 32.3 \, \text{°C} \).

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