The escape velocity is the minimum speed needed for an object to break free from the gravitational pull of a massive body without any further propulsion. It can be calculated using the following formula:
\[ v_e = \sqrt{\frac{2GM}{r}} \]
where:
- \( v_e \) is the escape velocity,
- \( G \) is the gravitational constant (\(6.67430 \times 10^{-11} \, \text{m}^3 \text{kg}^{-1} \text{s}^{-2}\)),
- \( M \) is the mass of the massive body,
- \( r \) is the distance from the center of the massive body to the point of escape.
### Steps to Calculate Escape Velocity:
1. **Identify the mass of the massive body (\( M \))**:
- For Earth, \( M \approx 5.972 \times 10^{24} \, \text{kg} \).
- For the Sun, \( M \approx 1.989 \times 10^{30} \, \text{kg} \).
2. **Determine the distance (\( r \))**:
- This is the distance from the center of the massive body to the point of escape. For a point on the surface, \( r \) would be the radius of the body.
- For Earth, the average radius is \( r \approx 6.371 \times 10^6 \, \text{m} \).
- For the Sun, the average radius is \( r \approx 6.957 \times 10^8 \, \text{m} \).
3. **Plug the values into the formula**:
- For Earth:
\[ v_e = \sqrt{\frac{2 \times 6.67430 \times 10^{-11} \, \text{m}^3 \text{kg}^{-1} \text{s}^{-2} \times 5.972 \times 10^{24} \, \text{kg}}{6.371 \times 10^6 \, \text{m}}} \]
\[ v_e \approx \sqrt{\frac{7.973 \times 10^{14} \, \text{m}^2 \text{s}^{-2}}{6.371 \times 10^6 \, \text{m}}} \]
\[ v_e \approx \sqrt{1.252 \times 10^8 \, \text{m}^2 \text{s}^{-2}} \]
\[ v_e \approx 11.2 \times 10^3 \, \text{m/s} \]
\[ v_e \approx 11.2 \, \text{km/s} \]
- For the Sun:
\[ v_e = \sqrt{\frac{2 \times 6.67430 \times 10^{-11} \, \text{m}^3 \text{kg}^{-1} \text{s}^{-2} \times 1.989 \times 10^{30} \, \text{kg}}{6.957 \times 10^8 \, \text{m}}} \]
\[ v_e \approx \sqrt{\frac{2.652 \times 10^{20} \, \text{m}^2 \text{s}^{-2}}{6.957 \times 10^8 \, \text{m}}} \]
\[ v_e \approx \sqrt{3.812 \times 10^{11} \, \text{m}^2 \text{s}^{-2}} \]
\[ v_e \approx 617.6 \times 10^3 \, \text{m/s} \]
\[ v_e \approx 617.6 \, \text{km/s} \]
### Conclusion
The escape velocity depends on the mass and radius of the massive body. For Earth, the escape velocity is approximately 11.2 km/s, while for the Sun, it is significantly higher at about 617.6 km/s. This formula and method can be applied to any massive body to calculate its escape velocity.
