The Newtonian approximation, also known as classical mechanics or Newtonian physics, is a valid approximation under certain circumstances. It is applicable when:
The speeds involved are much smaller than the speed of light (i.e., non-relativistic speeds). In situations where objects are not approaching the speed of light, such as everyday motions and most macroscopic objects, the Newtonian approximation is valid.
The sizes of objects involved are much larger than atomic or subatomic scales. Classical mechanics works well for objects that are significantly larger than atoms and molecules. At the atomic and subatomic level, quantum mechanics becomes necessary.
The gravitational field is not extremely strong. For everyday situations on Earth, the gravitational field is not intense enough to require the use of general relativity. Newtonian physics is sufficient for most gravitational interactions, such as objects falling under gravity.
The system being considered does not involve high precision or extreme accuracy. Newtonian physics provides accurate results for many practical applications and engineering purposes. However, in scenarios where very high precision is required or when dealing with highly accurate measurements, relativistic or quantum mechanical effects may need to be taken into account.
It's important to note that while the Newtonian approximation is widely applicable and extremely useful in many situations, there are cases where it fails to accurately describe the behavior of objects and phenomena. In those cases, more advanced theories like relativity and quantum mechanics must be employed.