How are gravitational force and potential energy affected when a rocket reaches one Earth radius above the Earth's surface?

When analyzing gravitational force and potential energy as a rocket ascends to an altitude that is one Earth radius above the Earth's surface, it's essential to recall the relationship between gravitational force and distance from the center of the Earth.

Gravitational force can be calculated using Newton's law of gravitation:

[ F = \frac{G \cdot M \cdot m}{r^2} ]

where ( F ) is the gravitational force, ( G ) is the universal gravitational constant, ( M ) is the mass of the Earth, ( m ) is the mass of the object, and ( r ) is the distance from the center of the Earth to the object.

At the Earth's surface, the distance ( r ) is equal to the Earth's radius (let's denote it as ( R )). When the rocket ascends to one Earth radius above the surface, the new distance from the center of the Earth becomes ( 2R ) (the original radius plus one additional radius).

Thus, the gravitational force at this new distance can be calculated as:

[ F_{\text{new}} = \frac{G \cdot M \cdot m}{(2R)^2} = \frac