If `v_(a),v_(h) and v_(m)` and are the speeds of sound in air, hydrogen and a metal at the same temperature, then

Options

A. A) `v_(h) gt v_(a) gt v_(m)`
B. B) `v_(m) gt v _(h) gt v_(a)`
C. C) `v_(h) gt v_(m) gt v_(a)`
D. D) `v_(a) gt v_(h) gt v_(m)`

Answer

To solve the problem regarding the speeds of sound in air (v_a), hydrogen (v_h), and a metal (v_m) at the same temperature, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Speed of Sound**:
The speed of sound in a medium is influenced by several factors, primarily the medium's density and temperature. The formula for the speed of sound in a medium can be expressed as:
\[ v = \sqrt{\frac{E}{\rho}} \]
where \(E\) is the modulus of elasticity (or bulk modulus) of the medium and \(\rho\) is the density of the medium.

2. **Comparing Densities**:
At the same temperature, we need to consider the densities of the three media:
- Density of air (\(\rho_a\))
- Density of hydrogen (\(\rho_h\))
- Density of metal (\(\rho_m\))

Generally, we know that:
\[ \rho_m > \rho_a > \rho_h \]
This means that the density of metal is greater than that of air, which in turn is greater than that of hydrogen.

3. **Analyzing the Speed of Sound**:
Since the speed of sound is inversely proportional to the square root of the density (assuming similar elastic properties), we can conclude:
- The speed of sound in hydrogen will be the highest because it has the lowest density.
- The speed of sound in air will be in the middle.
- The speed of sound in metal will be the lowest due to its higher density.

Therefore, we can summarize:
\[ v_h > v_a > v_m \]

4. **Conclusion**:
Based on our analysis, we can conclude that the correct order of speeds of sound in the three media is:
\[ v_h > v_a > v_m \]

### Final Answer:
The correct option is that the speed of sound in hydrogen (v_h) is greater than the speed of sound in air (v_a), which is greater than the speed of sound in metal (v_m).