Final answer:
Work done in a gravitational field is independent of the path because gravity is a conservative force where work done depends only on the starting and ending points, not on the path taken between them.
Step-by-step explanation:
The work done in moving a body in a gravitational field is independent of the path followed because gravity is a conservative force. By definition, a conservative force means that the work done by or against it depends only on the starting and ending points and not on the actual path taken. This is why the change in gravitational potential energy (APEg) between two points, which is the work done by gravity, can be represented as mgh (where m is mass, g is the acceleration due to gravity, and h is the height change) regardless of the path.
When we analyze the work done in a gravitational field, we can see that it can be represented as the change in gravitational potential energy between two points. Since this potential energy is a function of only the position (defined by the height), the total work done depends solely on the difference in the potential energy (U) at these two positions, which is path independent. From the work-energy theorem, this is further supported as work done is equal to the change in mechanical energy (kinetic plus potential energy), and for gravity, this is independent of the path.