"I think you would need to start by determining the muzzle velocity of your cannon in order to do what you're wanting."But would that be necessary ? Given that the acceleration of objects in free fall is a well-known constant (see picture), and I think the upward deceleration is at the same rate (but the inverse of the downward path), then from the total time of flight, the maximum vertical altitude of the ball can be calculated.
I don't think the muzzle velocity needs to be measured, although it can be calculated from the above time & height measurement.The problem with this thought is that you are neglecting air resistance in your assumptions but AR is almost two orders of magnitude greater than gravitational resistance. Consider the following:
When your projectile is fired upward, it starts with the muzzle velocity (say 500 ft/sec.) While moving upward, it loses velocity at the rate of 32 ft/sec
2 from gravity plus some unknown loss (greater at higher velocities than lower ones) due to air resistance. During its last second of vertical flight, air resistance is virtually zero and the shot moves 16 ft upward. The initial velocity downward is zero and it is accelerated by gravity at 32 ft/sec
2 and slowed down by air resistance, again by an unknown but variable amount. At some point in the downward flight, these values will become equal and the shot will cease to accelerate--it has reached its terminal velocity (it will actually slow down a little from this point due to air density becoming greater at lower elevations but this can be ignored for this discussion.)
My comments above are paraphrasing those of General Julian Hatcher in his discussion of vertical bullet flight; see
Hatchers's Notebook, Section XX, page 511.
Am I correct in thinking this would be the same "flying time" as a lower trajectory, so a distance at any elevation can be extrapolated from it?This is an incorrect assumption. The height of the trajectory will be greater when fired vertically than at any other elevation, so the time of flight will be longer for a vertical shot than one fired at any other elevation. Since the goal here was to determine how far a shot went when fired at some relatively horizontal trajectory, the time of flight will be much shorter than a vertical shot.
It seems that either muzzle velocity needs to be approximated or actual distance determined. If one could have two observers with transits spot the landing point of the shot, the distance could be determined from trigonometry. You might also fire the shot on land and add a tail of red yarn or a narrow red strap to make finding the landed shot easier. Range without the tail would certainly be greater than with but, again, we would have a verified minimum range.
One could approximate the muzzle velocity by setting a couple of paper target sheets down range at a measured separation and time the interval between the shot hitting the first sheet and hitting the second sheet. These techniques are pretty crude but at least we would get some reasonable numbers to play with.
