By using (i) Newton's formula and (ii) Laplace's formula, calculate the speed of sound in air at standard pressure and temperature. The density of air is `1.293` `kg//m^(3)`. `gamma` = 1.4 for air.

Answer

To calculate the speed of sound in air at standard pressure and temperature using both Newton's formula and Laplace's formula, we can follow these steps:

### Step 1: Using Newton's Formula
1. **Identify the formula**: According to Newton's formula, the speed of sound \( v \) is given by:
\[ v = \sqrt{\frac{P}{\rho}} \]
where \( P \) is the pressure and \( \rho \) is the density of the medium.

2. **Substitute the values**:
- Given:
- Pressure \( P = 1.013 \times 10^5 \, \text{N/m}^2 \) (1 atmospheric pressure)
- Density \( \rho = 1.293 \, \text{kg/m}^3 \)

- Plugging in the values:
\[ v = \sqrt{\frac{1.013 \times 10^5}{1.293}} \]

3. **Calculate**:
- First, calculate the fraction:
\[ \frac{1.013 \times 10^5}{1.293} \approx 78316.5 \]
- Now take the square root:
\[ v \approx \sqrt{78316.5} \approx 280 \, \text{m/s} \]

### Step 2: Using Laplace's Formula
1. **Identify the formula**: According to Laplace's formula, the speed of sound \( v \) is given by:
\[ v = \sqrt{\frac{\gamma P}{\rho}} \]
where \( \gamma \) is the adiabatic index (ratio of specific heats).

2. **Substitute the values**:
- Given:
- \( \gamma = 1.4 \)
- Pressure \( P = 1.013 \times 10^5 \, \text{N/m}^2 \)
- Density \( \rho = 1.293 \, \text{kg/m}^3 \)

- Plugging in the values:
\[ v = \sqrt{\frac{1.4 \times 1.013 \times 10^5}{1.293}} \]

3. **Calculate**:
- First, calculate the numerator:
\[ 1.4 \times 1.013 \times 10^5 \approx 1.4192 \times 10^5 \]
- Now calculate the fraction:
\[ \frac{1.4192 \times 10^5}{1.293} \approx 109749.7 \]
- Now take the square root:
\[ v \approx \sqrt{109749.7} \approx 331 \, \text{m/s} \]

### Final Answers
- Speed of sound using Newton's formula: \( v \approx 280 \, \text{m/s} \)
- Speed of sound using Laplace's formula: \( v \approx 331 \, \text{m/s} \)