Escape velocity is the minimum speed that an object needs to achieve to break free from a celestial body's gravitational influence without further propulsion. This concept is crucial in space exploration and astrophysics, determining the requirements for launching spacecraft and understanding planetary formation.

The escape velocity is given by the following equation:

\[ v_e = \sqrt{\frac{2GM}{r}} \]

Where:

- \( v_e \) is the escape velocity (m/s)
- \( G \) is the gravitational constant (\( 6.67430 \times 10^{-11} \text{ N} \cdot \text{m}^2/\text{kg}^2 \))
- \( M \) is the mass of the celestial body (kg)
- \( r \) is the distance from the center of the celestial body (m)

Let's calculate the escape velocity from Earth's surface:

- Identify the known values:
- \( G = 6.67430 \times 10^{-11} \text{ N} \cdot \text{m}^2/\text{kg}^2 \)
- \( M = 5.97 \times 10^{24} \text{ kg} \) (mass of Earth)
- \( r = 6.37 \times 10^6 \text{ m} \) (radius of Earth)

- Apply the escape velocity formula: \[ v_e = \sqrt{\frac{2GM}{r}} \]
- Substitute the known values: \[ v_e = \sqrt{\frac{2(6.67430 \times 10^{-11})(5.97 \times 10^{24})}{6.37 \times 10^6}} \]
- Perform the calculation: \[ v_e \approx 11,186 \text{ m/s} \text{ or } 11.2 \text{ km/s} \]

Let's visualize the escape velocity concept for Earth:

This diagram illustrates:

- The Earth represented by the blue circle
- The radius (r) from the center of Earth to its surface
- The escape velocity (v_e) trajectory, showing the minimum speed needed to escape Earth's gravity

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