Most physical situations are quite complicated, involving a number of forces or interactions even in the simplest of cases. Much of the praxis of physics is breaking a problem into parts, treating some parts exactly and ignoring other parts. Once we have solved the idealized problem, we can use the solution to evaluate the appropriateness of the idealizing assumptions.
A simple example of this is the analysis of a tennis ball falling from a height at the surface of the Earth. The ball consists of many atoms, connected by chemical forces. In addition to the chemical forces, each atom is pulled by all the pieces of the Earth. The first idealization we make is that we can treat the ball as a point mass located at its center and the Earth as a point mass located at its center. Second, we ignore the gradient of the gravitational field, so that allows us to approximate the force of gravitation on the ball as Fgrav = mg, where g ≈ 9.8 m/s2 is a constant.
The third effect we generally ignore is air resistance. If we ignore air resistance, we can calculate the idealized maximum velocity of the falling ball and then calculate the force of air drag. This is given by
Fdrag ≈ CρAv2 (1)
where C (= 0.2) is a constant, ρ(= 1.3 kg/m3) is the density of air, A is the cross-sectional area of the ball, and v is its velocity relative to the air. If the air resistance is small, then we were justified in ignoring it.
If air resistance is important, it is possible that we can still do the problem. If the ball falls far enough for there to be a force balance Fnet = 0, then we can use equation (1) to solve the problem. (Actually we can only require that Fnet be small compared to the other forces in the problem.)
For the following problems, consider a ball of radius 0.03 m and mass 0.05 kg which is tossed upward at initial velocity 3 m/s.
If we are going to ignore air resistance, the drag force must be small compared to