Escape Velocity
Escape velocity is the minimum velocity needed for an object to break free from the gravitational attraction of a planet or other celestial body without further propulsion. It is a crucial concept in astrophysics and space exploration.

Formula for Escape Velocity
The formula to calculate escape velocity is given by:
\[ v = \sqrt{\frac{2GM}{R}} \]
where
- \( v \) is the escape velocity,
- \( G \) is the gravitational constant,
- \( M \) is the mass of the celestial body,
- \( R \) is the distance from the center of the body to the point where the object is located.

Derivation of Escape Velocity Formula
To derive the formula for escape velocity, we consider the conservation of mechanical energy. At the surface of the celestial body, the object has kinetic energy \(\frac{1}{2}mv^2\) and potential energy \(-\frac{GMm}{R}\), where \(m\) is the mass of the object. At an infinite distance from the body, the object has zero kinetic and potential energy.
By equating the initial and final energies, we get:
\[ \frac{1}{2}mv^2 - \frac{GMm}{R} = 0 \]
Solving for \(v\), we get the formula for escape velocity.

Significance of Escape Velocity
Escape velocity is crucial for space missions as it determines the speed required for spacecraft to leave Earth's or any other celestial body's gravitational pull. It helps in planning trajectories for space missions and ensuring the success of space exploration endeavors.