|There are two distinct ways that gravitation is modeled for the two different contexts that are the scenarios for most problem solving. In basic mechanics, the gravitational force on an object is often assumed to be constant, and called simply 'the weight'. With astronomical distances, though, gravitational force is seen as something which changes according to the position of the interacting masses.|
The kinematic model of idealized projectile motion occurs within the practically uniform force environment of the Earth's surface, and so the force is often taken as constant in this context. However, the Law of Universal Gravitation predicts that an object an top of a mountain will weigh slightly less than in a valley because the distance is greater from the center of the Earth on top of the mountain.
The gravitational field of the Earth is slightly decreased the further the object is from the center of the Earth, and the object weighs slightly less on top of the mountain. The difference in weight between these two positions is less, however, than the range of error for any normal method of measurement so it would make little sense to try to account for how much less the weight of a normal projectile is at the peak of trajectory.
Imagine the Earth's gravitational field lines permeating the space of the room, vectors pointing from the ceiling to the floor. To all accounts, the field lines will appear to be strictly parallel, but in fact, the field lines are further apart from each other at the ceiling than the floor. To our eyes, the distance is not significant relative to the divergence of the field, so for many problems, the field may be taken as a constant. For 99.9% of problems, the weight of an object on the Earth can be assumed to be the same for all positions.