ANALYSIS OF THE RIFLEMAN'S RULE
It is common for a hunter to find his quarry to be at a different altitude than he. This means that the hunter must shoot either uphill or downhill. This situation requires compensation of the zeroing of the weapon.
There is a general rule that shooter's use called the "Rifleman's Rule." It is stated as follows (
In other words, pretend that the inclined target is at a horizontal distance equal to the slant range distance multiplied by the cosine of the inclination angle, and aim as if the target were really at that horizontal position. The rule is so well established that there are devices to assist shooters with implementing it
Measure the inclination angle of the target above or below the horizontal direction.
Measure the slant range distance to the target.
Multiply the slant range distance by the trigonometric cosine of the inclination angle (this gives the horizontal projection of the slant range).
Use the bullet path (or come-up or come-down) from the level trajectory at this horizontal projection distance to adjust the aim for the inclined target.
This paper examines this rule and its theoretical origins. Note that this paper uses the Mathsoft ODE solver. This routine has issues with units, so I have not used them. All units are implied in this paper. I have assumed the MKS system.
Important Utility Functions
An important part of this discussion is the calibration of the weapon to hit a point at a specified distance. This operation is referred to as "zeroing." A function is developed below, called dZero, that determines angle, dq, that the bore must make with horizontal to hit a specific distance, rZero. Figure 1 illustrates this process.
Figure 1: Variables in the Zeroing Operation.
This simulation provides a check on the theoretical results that we obtain later in the paper.
Basic Ballistic Simulation
Because there is so much algebraic manipulation here, it is useful to have a numerical simulation that will allow the results to be tested against an independent reference. Figure 2 shows the definition of variables that are used in the simulation.
Figure 2: Variables in the Uphill Trajectory Analysis.
Corrected Wikipedia Equations (For Comparison)
Figure 3: Agreement Between Theoretical and Numerical Simulations
Figure 3 shows that the theoretical and numerical models agree. However, we still have not derived the actual rule. We have only shown that we have a good model that can be used as a starting point and a test vehicle.
Equation from Reference 2
Here we see an interesting effect. The bullet will intersect the ground slightly further from the point of zero(rHill>rFlat). This is the key qualitative result of the rifleman's rule.
Derivation of Rifleman's Rule (Using a Taylor Approximation)
Equivalence of Wikipedia Solution and Reference Solution
Verify the Equation from Reference 1
Prove the Equivalence of the Wikipedia and Reference 1 Derivations
This is the original reference work. All discussions stem from this book.
This is a web version of Reference 1 with regard to the "Rifleman's Rule."
This shows that hunting equipment is being built with the "Rifleman's Rule" in mind.
This is the errata for "Modern Exterior Ballistics." It is a useful reference.