 Let's look at a specific example of the work done on and by a gravitational field. In this example, we're sending a probe to do a flyby of the Sun to gather data. The probe starts its flyby at an initial distance from the Sun Ri and with some initial velocity, and it swings through the field around the Sun with a minimum distance of Rf, and then it continues back around on its way. How much work is done by the field from where the probe starts to its closest point? We could try to figure this out by calculating the force on the probe at every point and looking for the components in the direction of motion. This would be really hard because we'd have to cut up the path into a bunch of small segments that are roughly straight and get the angle between the force and the motion and the strength of the force for the entire path. Alternatively, we can just look at the change in potential energy between the starting point and the endpoint. This change in potential energy represents the work done by the field. As the probe gets closer to the Sun, the potential energy decreases. The gravitational force between the objects is causing the objects, mostly the much smaller probe, to accelerate as the distance decreases in its approach to the Sun, and this is transforming the potential energy to kinetic energy. As the probe comes into line with the Sun, the component of the gravitational force accelerating the probe is reducing. Once the probe has passed the Sun, it decelerates. With the velocity and thus kinetic energy decreasing, we know that the gravitational potential energy must be increasing. So given that it starts at a distance ri, and the turning point is at a distance rf, the change in potential energy will be the general potential energy equation we saw before. Due to conservation of energy, we know that this loss of potential energy is the work done on the probe by the field. So, without needing to worry about the initial velocity, or the strength of the field at all these points, or the direction of the field, we were able to look at the change in potential energy and figure out how much work the field did on the object, moving it from the initial position to the turning point. Finally, let's consider where the potential energy actually went. Well, we know the potential energy was used to do work on the probe. If you do work on an object, you add to its speed, which means you increase its kinetic energy. Thus, the field does work on the probe converting the potential energy into kinetic energy. If we were to look at the reverse situation with the probe coming back around the sun and travelling away, we'd find that the probe is doing work on the field by converting its own kinetic energy back into potential energy, and thus conserving total energy of the system.