Escape Velocity:
The escape velocity is the minimum velocity needed for an object to break free from the gravitational attraction of a planet or other body without any additional propulsion. It is an important concept in physics and astronomy.

Expression for Escape Velocity:
The escape velocity can be calculated using the formula:
\[ v = \sqrt{\frac{2GM}{r}} \]
Where:
- \( v \) is the escape velocity
- \( G \) is the universal gravitational constant (\( 6.67430 × 10^{-11} m^3 kg^{-1} s^{-2} \))
- \( M \) is the mass of the planet or body
- \( r \) is the distance from the center of the planet or body to the object

Explanation:
The expression for escape velocity is derived from the conservation of mechanical energy. When an object is at a distance \( r \) from the center of a planet with mass \( M \), its total mechanical energy is the sum of its kinetic energy and its gravitational potential energy:
\[ E = \frac{1}{2}mv^2 - \frac{GMm}{r} \]
Where:
- \( m \) is the mass of the object
For the object to escape the gravitational pull of the planet, its total mechanical energy must be zero (since it will have zero kinetic and zero potential energy at infinity). Setting \( E = 0 \) and rearranging the terms, we get:
\[ \frac{1}{2}mv^2 = \frac{GMm}{r} \]
Solving for \( v \), we arrive at the expression for escape velocity as mentioned above.
In conclusion, the escape velocity is a crucial concept in understanding the dynamics of objects in gravitational fields, and its expression is derived from the conservation of mechanical energy.