Assumptions and Conditions

All physics formulas, equations, laws, or any other mathematical forms are the ways to express physical phenomena in a very compact and consistent manner. However, they are often misunderstood as merely the relations of variables more like in mathematics rather than in physics.

The facts that there are stories, conditions, assumptions underlying the mathematical appearance are very crucial to determine the extent, limitations, and the environment where the formulas may be applied. Let us take an example to grasp the main ideas of this section: the trajectory of a projectile.

In high-school physics, the trajectory is described as a parabolic path starting from a point on the ground up to a maximum in the air that serves as an axis symmetrically halves the whole trajectory. The symmetry is not only in space, but also in time, as well as the energy (kinetic and gravitational potential energy). So far it does not seem there is anything wrong. The question is whether it is true in the real world that the symmetry does happen?

Of course not. The parabolic trajectory is actually an oversimplified solution to the projectile problem. Let us list a few assumptions embedded in it:

1. no air resistance or wind
2. no effect from earth rotation
3. the gravitational field is constant along the trajectory
4. the earth is spherical

	The assumptions provide some restrictions to simplify the problem. Now watch out if an assumption is removed. Say, air resistance is added, then the path is asymmetric, shorter in the destination part due to some amount of energy is given up to the air friction (the broken-line curve).
	How if the earth rotation is taken into account? There will be an easterly accelaration known as Coriolis effect that will deviate the trajectory of falling bodies to the east. If the trajectory from the north down south, this Coriolis accelaration causes the trajectory out of plane (the broken-line curve) that would have been otherwise for a parabolic path.
	We know from Newton's gravitational law that the gravitational field (accelaration) changes with the distance from the center of the earth. It means that it changes along the trajectory, which in fact gives us the curve shape of a section of an ellipse rather than a parabola for the trajectory.
	The earth is definitely not spherical, but an oblate spheroid, so that the gravitational field around it is slightly different from the central force approximation proposed by Newton.

We can see from this example, that the situation becomes more and more complicated when we relax the assumption. Fortunately, though the parabolic trajectory problem is very limited in application, that is, only when the consequences of the assumptions listed above are all negligible, it is a pretty accurate approximation in our daily life where those assumptions are acceptable. However, more accurate approach must be applied for launching missiles, for example.