Define escape velocity. Obtain an expression for the escape velocity of a body from the surface of the earth.

Escape velocity is the minimum speed required for a body to escape the Earth’s gravitational field. When a body is thrown upwards, it rises to a certain height and falls back. However, if it is thrown with sufficient speed, it can escape the Earth’s gravitational pull.

To derive the expression for escape velocity, consider the Earth as a sphere of mass \( M \) and radius \( R \). The gravitational force at a point a distance \( x \) from the Earth’s center is proportional to the inverse square of the distance. The small work done in moving a body against this gravitational force is calculated, and the total work needed to move the body from the Earth’s surface to infinity is derived.

The work done is equal to the kinetic energy required for the body to escape, and by equating this work to the kinetic energy, the expression for escape velocity is obtained. This velocity is found to depend on the Earth’s gravitational constant and radius, but not on the mass of the body.

The escape velocity can be expressed in different forms, based on the gravitational constant, Earth’s radius, and mean density. The key point is that escape velocity is independent of the mass of the body being projected.