We currently know the formula behind projectile energy: E=(m*v^2)/2 according to which speed is more important than weight hence why gunpowder weapon was more effective vs armor. How could people measure the projectile speed in up to 10m/s increments in a time frame from Ancient Rome(transition era) to Late Medieval?
The tried-and-true low-technology method of measuring the initial speed of a projectile is the ballistic pendulum. Basically, the projectile is fired into a heavy pendulum; the pendulum will raise to a height which depends on the speed of the projectile; see the Wikipedia article for details. The ballistic pendulum was invented in the 18th century by Benjamin Robins; before that one could only measure average speed by measuring the distance travelled by the projectile and the time it took to reach the target.
On the other hand, before the late 16th century (which is well into Renaissance territory) there was no particular reason to attempt to measure the initial speed of a projectile accurately. Ancient and medieval physics had purely qualitative theories of dynamics; the only parts of mechanics which had actual numerical formulas were statics and kinematics; with no quantitative theory of dynamics no particular use could be made of the knowledge that a specific projectile was shot with 266 or 276 meters / second.
+1 to the Ballistic pendulum idea. Here is a method requiring less mathematical insight, and is more of a brute force method. We actually use a variant of this method in many tests of speed on computers today.
The idea is to measure a large number of events, back to back. Say I am firing arrows; I can line up 20 archers with the instruction that when I say "GO", the first archer fires an arrow; the next has drawn his bow but does not fire until he sees the first arrow land, the third waits for the second arrow to land, and so on. The entire operation of all 20 arrows being fired and landed is timed, by however they may time such things. I'd suggest ticks of something like a metronome or weight-driven clock, or the amount of time it takes a ball to roll down a spiral chute (move it to the top the moment it rolls into your hand at the bottom), or any other standard that measures a consistent amount of time; within 1% or so. Say it is measured in 'tocks'.
Next: Measure the distance of all arrows fired; and add them up. That can be done with any standard measuring stick. Divide that distance by the total tocks, and you have the average distance traveled per tock for the arrow.
The error is minor; expert archers (or slingers) will release their missiles within about 1/10th of a second after seeing the previous hit the ground. The timing is likewise going to be very consistent; even with the spiral chute, the time it takes a practiced ball catcher to move the ball will be so consistent it just becomes part of the timing and varies by less than 1/20th of a second.
This approach probably does not work for gunpowder weapons; but for things in the Roman era, trebuchets, slings, arrows, spears, etc that are too fast to count time accurately, but slow enough to watch, this aggregate measurement approach will work.