The formulas for computing arc length are useless until you know what dy/dx is.
There are a couple of things that determine the trajectory. Gravity, air resistance and the Magnus effect. Which is the cross product of the speed with the spin so the arc due to the Magnus force alone is a function of the spin to speed ratio.
Wrote a simulation years ago using Mathcad that used 5 differential equations.
1 x position
2 x velocity
3 y position
4 y velocity
5 rotation speed decay.
The initial values for the above are given and the differential equation will determine the trajectory by integrating them.
Slow balls with no spin will still have an arc due to gravity.
So how of the downward force is due to gravity and how much is due to the Magnus force?
I know.
I can calculate exact times and distances assuming I know the correct air density and gravity at that point.
What is interesting is the video that showed up years ago called "The fastest serve ever".
The Magnus force was strong in that one.
BTW, William Henzel provided me with the high speed 240 fps, version of that video so I could make the calculations.
Strangely enough I could never figure out where to put the "deceptive, disruptive, or wobble" forces.
This is no big deal. The engineers that write the software Dan's nemesis, the Omron robot, have this figured out.