projectile - How can a shooter improve estimating how much to deflect/lead a moving target?

Abbreviate Moving Target to MT, and lead the target/deflect to LTT. Suppose:

4. 
   
   

   that you know the distance to MT that I name $d$, where $ 0 < d \le 50 \text{ km}$.

   
   


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   that your reticle and its ticks are defined for some distance to MT: name this $r$. $r$ can $\neq d$, but $ 0 < r \le 50 \text{ km}$. 2 and 3 are for rifles, but I'm interested in naval and tank artillery too.

   
   


6. 
   

   absence of internal (e.g. primer, powder type, powder burn rate) and external forces (e.g. environmental variables like wind speed, humidity, and air density)..

   
   

How can a shooter apply math to improve LTT, by better estimating at which tick to aim?

Answer

The great thing about reticle ticks is that they're actually measurements of angle. Because of this, a lot of our aiming has very little to do with the calibration of the reticle, because we'll be measuring positions and velocities in terms of reticle ticks (assuming that the reticle's ticks are evenly spaced).

Let $\vec{v}(t)=(v_x,v_y)$ be the apparent velocity of the target (in reticle ticks/sec). The $z$ axis is parallel to the line of sight (positive direction points away from you), while the $x$ and $y$ axes are the horizontal and vertical crosshairs. Let the apparent position (in the reticle) of the object at the moment of firing be $(0,0)$.

We'll assume:

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  The target is far enough away that its distance from you basically doesn't change as it moves (also assumes target has very small $z$-velocity)

  
  


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  Velocity of the target is essentially constant during aiming time

  
  
  

There are still three forces you need to worry about: recoil, gravity, and air resistance. First, let's ignore all three.

No air resistance, gravity, or recoil

Then the position of the target (in reticle ticks) at time $t$ is $(v_xt,v_yt)$. The time it takes for the bullet to reach the plane of the target is $\Delta t = \frac{d}{v_b}$ where $v_b$ is the bullet velocity in m/s. So you should aim at the location

$$\left(\frac{v_xd}{v_b},\frac{v_yd}{v_b}\right)$$

where the units are reticle ticks.

Now let's add in recoil.

No air resistance or gravity, with recoil

There is some randomness to recoil, so you'll always have a bit of inaccuracy thrown in here. But suppose you measure that the average deviaton (in units of reticle ticks) of a shot is $(\Delta x,\Delta y)$. Then you should aim at

$$\left(\frac{v_xd}{v_b}-\Delta x,\frac{v_yd}{v_b}-\Delta y\right)$$

Now let's add in gravity.

No air resistance, with gravity and recoil

The bullet accelerates downward over time, producing a deflection of $\frac{1}{2}gt^2$, so its deflection at the target is $\frac{1}{2}g(\Delta t)^2 = \frac{gd^2}{2v_b^2}$. This means you should aim at

$$\left(\frac{v_xd}{v_b}-\Delta x,\frac{v_y d}{v_b} - \Delta y + \frac{gd^2}{2v_b^2}\right)$$

Now let's add in air resistance.

With air resistance, gravity, and recoil

Air resistance screws everything up, because the bullet is moving supersonically. Supersonic drag is, at best, extremely difficult to numerically calculate and, at worst, chaotic and random so you can't even simulate it well. The use of ballistics tables is recommended at this point.