Energy and Work

All the intriguing questions actually stem from the peculiar framework of physics where both energy ad work are measured in the same unit, the joules. While work is defined as the force acting through a displacement and is directly linked to change in a system's momentum, energy is only a state function that has shown to only quantify the fraction of work exerted in the past or exerted in the future and does not by itself convey any details of the trajectory.

Let's recall how we have been taught about energy. Potential energy depends solely on an object's final position in a gravitational field, independent of the trajectory taken to reach that position. However, when an object moves to that height, the work performed by or against gravity depends on the actual path taken, because work involves all the forces acting along that path. Thus, while potential energy only quantifies the difference between the initial and final states, work reveals all the trace through the path the object takes to reach that point.