The drag formula has to be built up out of individual arrow parts. The shaft Reynolds number is of the order of millions and it is therefore treated as a turbulent flow problem. While the fletching seems to be doing surprisingly little. For our model this does not matter very much because the two coupled differential equations won't be analytically solvable anyway. We'll have to resort to numerical methods.
Arrow reaches max.
For those who want to calculate some trajectories without having to program it themselves, I included the crude pascal program I used. As said before a real arrow isn't rigid at all. At the launch, it is forced to accelerate so fast that it buckles under the strain. It is said that aluminium shafts are better to use on short distances because they are faster to reach a stable flight. The carbon-based arrows weight about the same, but are thinner, have far less surface, thus experience less drag and are faster to reach the target.