The velocity of sound in a container of air at `-73^@C` is `300m//s` It temp. of container were raised to `127^@C` what would be the velocity of sound?

Options

A. `300m//s`
B. `300sqrt2 m//s`
C. `300//sqrt2 m//s`
D. `600 m//s`

Explanation

To solve the problem of finding the velocity of sound in air at a new temperature, we can follow these steps:

### Step 1: Understand the relationship between temperature and the velocity of sound
The velocity of sound in a gas is proportional to the square root of the absolute temperature (in Kelvin). This can be expressed as:
\[ v \propto \sqrt{T} \]
where \( v \) is the velocity of sound and \( T \) is the absolute temperature in Kelvin.

### Step 2: Convert the initial temperature from Celsius to Kelvin
The initial temperature is given as \(-73^\circ C\). To convert this to Kelvin:
\[ T_1 = 273 + (-73) = 200 \, K \]

### Step 3: Convert the final temperature from Celsius to Kelvin
The final temperature is given as \(127^\circ C\). To convert this to Kelvin:
\[ T_2 = 273 + 127 = 400 \, K \]

### Step 4: Use the proportionality of the velocities and temperatures
Using the relationship established in Step 1, we can write:
\[ \frac{v_1}{v_2} = \sqrt{\frac{T_1}{T_2}} \]
where \( v_1 = 300 \, m/s \) (the initial velocity) and \( v_2 \) is the velocity we want to find.

### Step 5: Substitute the values into the equation
Now we can substitute the known values into the equation:
\[ \frac{300}{v_2} = \sqrt{\frac{200}{400}} \]

### Step 6: Simplify the equation
The right side simplifies as follows:
\[ \sqrt{\frac{200}{400}} = \sqrt{\frac{1}{2}} = \frac{1}{\sqrt{2}} \]
Thus, the equation becomes:
\[ \frac{300}{v_2} = \frac{1}{\sqrt{2}} \]

### Step 7: Solve for \( v_2 \)
Cross-multiplying gives:
\[ 300 \cdot \sqrt{2} = v_2 \]
Now calculating \( v_2 \):
\[ v_2 = 300 \cdot \sqrt{2} \approx 300 \cdot 1.414 \approx 424.26 \, m/s \]

### Final Answer
The velocity of sound in the container at \(127^\circ C\) is approximately \(424.26 \, m/s\).

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