A bow is a remarkably elegant and deceptively simple device for storing up muscle power in the tension of the bow limbs. It releases that energy much faster than human muscles can twitch, launching projectiles with great speed and accuracy, almost faster than the eye can follow and considerably faster than one can click a stopwatch with any precision. You can calculate an arrow's speed by relying on basic physics.
Measure off 20 yards from your target, and stand with your bow and arrows facing it.
Fire a few practice arrows at the center of the target until you feel you have the range adjustment just right. You will find that you are aiming somewhere above the bull's-eye, and the arrows drop somewhat as they arc toward it. Memorize this point, and place a piece of masking tape on it to mark it as you collect your arrows.
Fire all of your arrows at the target from the 20-yard distance, aiming carefully at the spot of masking tape. Your arrows may not all hit the bull's-eye, but they should average out around it. Keep trying until you get a satisfactory grouping around the bullseye. You may find you need to adjust the position of the masking tape marker to do this.
Measure out 40 yards from the target. From here, shoot at the target, aiming at the same masking tape marker as before. Don't worry if your arrows are falling below the bull's-eye; they're supposed to. Just aim for the masking tape, and let the arrows land where they may.
Find the center of the grouping fired from 40 yards, and measure how far below the masking tape it is. Also measure how far below the masking tape the bull's-eye is; this distance should be smaller than the distance between the 40 yard grouping and the bull's-eye.
Double the distance from the masking tape to the bull's-eye, and divide the result by 32 feet per second per second. Take the square root of this number; the result is how many seconds it took your arrow to travel the first 20 yards. 60 feet divided by this number should be the arrow's speed in feet per second.
Double the distance from the masking tape to the center of the 40 yard grouping, and divide by 32 feet per second per second. Again, take the square root of this number, which should be how many seconds it took your arrow to travel 40 yards. 120 feet divided by this number is the arrow's speed in feet per second, and should be the approximately same as the result in step 6, allowing for errors in measurement.