To derive the expression for the gravitational potential energy of a body with mass m located at a distance r from the center of the Earth, we begin by understanding that gravitational potential energy is the energy associated with the position of an object in a gravitational field.

When an object is at a distance r from the center of the Earth, it feels a gravitational force due to the mass of the Earth. To determine the gravitational potential energy we consider the work done against this gravitational force in moving an object from a reference point, which is often considered to be infinity, at which the potential energy is zero, to the distance r.

The gravitational force acting on the mass is given by Newton’s law of gravitation as is proportional to the product of the masses and inversely proportional to the square of the distance between them. In order to obtain the expression for potential energy, it is necessary to integrate that force over the distance from infinity to. This results in an expression for gravitational potential energy that tells the amount of energy stored for being at a certain location within the gravitational field.