What is a good equation to calculate the effective and max firing range of a turret?

A 2 metric ton shell fired from a 45° elevation with a muzzle velocity of 870m/s. What would that be?

I'm very rusty on the physics, but here we go.

sin 45° = 0.707 vertical velocity vector

870 m/s x 0.707 = 615.183 m/s upwards

Gravity = 9.81 m/s/s

615.183 m/s / 9.81 m/s/s = 62.71 seconds to reach apogee

62.71 x 2 = 125.42 seconds to fly the full distance

870 m/s x 0.707 x 125.42 s = 77,144.588 meters total distance

This is not the full story, however. Drag will slow down the shell and result in a shorter distance, but we don't have information for how much it will affect your theoretical shell. Neither do we know what the wind is, how high the gun is above the surface of the planet, or how much the curvature of the planet will affect things.

The only example I have that is close to your specifications is this: The Yamato battleships could fire a 1.460 metric ton shell at 780 m/s and achieve 46,000 meters. Given Yamato's specs, calculating its distance without drag gives a theoretical 62,009 meters range. So, real world testing shows about 75% of theoretical distance.

So if I had to take a wild guess, your example gun will shoot about 57,228 meters.

Somebody please check my math.

7583681
Spot on.