Can we ignore air resistance?

A pendulum is a practical example when it is easy to test the accuracy of ignoring air resistance.

If you measure the period of oscillation $T = 2 pi sqrt {frac{L}{g}}$ with a chronometer, the measurement error will be probable bigger than the combined effect of air resistance and the approximation of small angle used for the formula.

Not surprisingly they were used so long to measure time.

The Reynolds number characterizes how important viscous forces are relative to inertial forces.

At higher Reynolds numbers, the viscous losses are small. At lower Reynolds numbers, the viscous losses affect the motion more.

It's defined as

$$Re = {rm{inertial forces} over {rm{viscous forces}}} = {{rho u L}over{mu}} = {{u L}over nu} $$

where $mu$ is the dynamic viscosity and $nu$ is the kinematic viscosity, $u$ is the flow speed and $L$ is a typical dimension.

So the answer to the question is: When the speed and size are large compared to the viscosity, you can ignore the air resistance.

For air, the kinematic viscosity is roughly $15 times 10^{-6} m^2/s$ so if the product of size and speed is very large compared to that, the air resistance is small.

No, you cannot ignore it if you want to get exactly right. In fact, you need precise measurements of every single atom in the system in order to be exactly right. And even then, quantum mechanics says you will only be statistically right.

We do not attempt to get exactly right. We don't even try. What we try to do is get right enough that the predictions we make are meaningful.

In many situations, the predictions you make ignoring air resistance are highly meaningful. They are useful to such a degree that we don't even teach real air resistance until several years into someone's physics career.