When I create a projectile motion problem for students (in a course where air drag is neglected) I usually solve the problem numerically including drag. Then I plot the trajectory and compare it with that one obtained by neglecting the air drag.
However is there a more simple method to calculate if air drag matters or not?
If there is some air drag, you have a nonconservative system that includes a loss of energy. This loss of energy is given by
$<P> \Delta t$
with $<P>$ the average power of the air drag and the time interval $\Delta t$ of the process. It holds further for the power at some instant of time
$P = Fv$
with drag force $F$ and velocity $v$. Typically, the drag force increases with velocity. Example: If $F = cv^2$, then loss of energy is
$\Delta E = c <v^3>\Delta t$.
If $\Delta E$ is small in comparison with the sum of total kinetic and potential energy, you can neglect the air drag. That may be the case for small constants $c$, for sufficiently small velocities (where $v^2$ kinetic energy term is greater than $v^3$ energy loss term) or very small time intervals.
