Escape velocity is the minimum velocity required for an object to escape the gravitational pull of a planet or any other celestial body. It is an important concept in physics and is calculated using various formulas, such as √(2GM/R) and √2gR.

I. Escape Velocity Formula: √(2GM/R)
The escape velocity formula √(2GM/R) is derived using the gravitational potential energy and kinetic energy of the object. Here's a breakdown of the variables:

- G: the gravitational constant
- M: the mass of the celestial body
- R: the distance from the center of the celestial body to the object

The escape velocity is inversely proportional to the square root of the distance from the center of the celestial body (R). This means that as the distance from the center increases, the escape velocity decreases. In other words, the farther an object is from the center of a celestial body, the less velocity it needs to escape its gravitational pull.

II. Alternative Escape Velocity Formula: √2gR
Another formula for escape velocity is √2gR, where g represents the acceleration due to gravity. This formula is derived by considering the gravitational force acting on the object and equating it to the centripetal force required for circular motion.

- g: acceleration due to gravity at the surface of the celestial body
- R: radius of the celestial body

This formula shows that the escape velocity is directly proportional to the square root of the radius (R) of the celestial body. As the radius increases, the escape velocity also increases. This implies that larger celestial bodies require more velocity to escape their gravitational pull.

III. Choosing the Appropriate Formula
When solving problems involving escape velocity, it is essential to consider the specific context and information provided. Here are a few guidelines to consider:

1. Use √(2GM/R) formula when:
- Mass (M) of the celestial body is given.
- Distance (R) from the center of the celestial body to the object is given.
- The problem involves calculating the escape velocity from a specific distance.

2. Use √2gR formula when:
- Acceleration due to gravity (g) at the surface of the celestial body is given.
- Radius (R) of the celestial body is given.
- The problem involves calculating the escape velocity at the surface of the celestial body.

By carefully analyzing the problem and the information provided, you can determine which formula to use and solve for the escape velocity accordingly.

Root 2 gR vala