Once you have achieved reasonable grouping, it is easy to measure an arrow's launch speed and drag with only a tape-measure. For this you need to shoot at a target at three or preferably more ranges while using the same sight pin location. For best accuracy the tests should be conducted on a windless day and the ranges should be as different as reasonably possible - 20, 30 40 and 50 meters or yards would be a good start. For safety and practical reasons one needs to bear in mind the arrow drops - at 50 meters the drop is likely to typically be 2-3 meters. Longer range aiming points may need to be raised on a pole behind the actual target. The objective is to accurately measure the drop for each range and feed the results into the reverse ballistics calculator.
If there were no drag, only two ranges would be required to measures the arrow speed, however drag is real and is approximately proportional to the square of the arrow's speed. So, from launch, an arrows speed is constantly decreasing This is the reason for a minimum of three ranges being required.
The ballistics calculator applies numeric techniques to solve a very complex set of equations that takes into account arrow aerodynamics, arrow mass, atmospheric conditions and gravity. With good groupings and accurate measurements, the results can achieve better than 2% speed accuracy - comparable to a chronograph, and 5% accuracy for the drag coefficient.
Once this calibration is done, an arrow's impact speed and direction can be calculated for any range. Also a sighting scale can be generated, complete with "sweet-spots". As you progress as an archer with ever decreasing group sizes, re-calibrating occasionally may be rewarding.
The mathematics of ballistics is reasonably well covered on Wikipedia - Trajectory of a Projectile.