Derivation of escape velocity? Plz sir help me in this.?

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Derivation of Escape Velocity:

Escape velocity is the minimum velocity required for an object to escape the gravitational pull of a massive body, such as a planet or a star. It is the speed at which the object needs to be launched so that its kinetic energy is equal to the gravitational potential energy at that point.

Formula:
The formula for escape velocity can be derived using the principle of conservation of energy. The total mechanical energy (E) of an object in a gravitational field is given by the sum of its kinetic energy (K) and gravitational potential energy (U):

E = K + U

When an object is at the surface of a planet or a star, its gravitational potential energy is given by:

U = -GMm/r

Where G is the gravitational constant, M is the mass of the planet or star, m is the mass of the object, and r is the distance between the object and the center of the planet or star.

At a distance r from the center of the planet or star, the object has a velocity v. The kinetic energy of the object is given by:

K = (1/2)mv²

Derivation:
To derive the escape velocity, we set the total mechanical energy of the object equal to zero, since at escape velocity, the object just barely escapes the gravitational field and its kinetic energy becomes zero. Therefore, we have:

0 = K + U

0 = (1/2)mv² - GMm/r

Simplifying the equation, we get:

(1/2)mv² = GMm/r

v² = 2GM/r

Taking the square root of both sides, we obtain:

v = √(2GM/r)

This is the formula for the escape velocity, where G is the gravitational constant, M is the mass of the planet or star, and r is the distance between the object and the center of the planet or star.

Summary:
Escape velocity is derived using the principle of conservation of energy. By setting the total mechanical energy of the object equal to zero and solving for the velocity, we obtain the formula v = √(2GM/r), where G is the gravitational constant, M is the mass of the planet or star, and r is the distance between the object and the center of the planet or star. This formula gives the minimum velocity required for an object to escape the gravitational pull of a massive body.

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