Rifleman's rule - AbsoluteAstronomy.com

Rifleman's rule is a "rule of thumb" that allows a rifleman to accurately fire a rifle that has been calibrated for horizontal targets at uphill or downhill targets. The rule provides an equivalent horizontal range setting for engaging a target at a known uphill or downhill distance from the rifle (called the slant range). For a bullet to strike a target at a slant range of

and an incline of

, the rifle sight must be adjusted as if the shooter were aiming at a horizontal target at a range of

. Figure 1 illustrates the shooting scenario. The rule holds for inclined and declined shooting (all angles measured with respect to horizontal).

Strictly speaking, the rifleman's rule is an approximation and it holds generally only for the small angles typically involved in shooting. The rule is derived assuming that the bullet travels in a vacuum. However, empirical evidence suggests that the rule does appear to work with reasonable accuracy in air and with both bullets and arrows.

Definitions

There is a device that is mounted on the rifle called a sight. While there are many forms of rifle sight

Sight (device)

A sight is a device used to assist aligning or aim weapons, surveying instruments, or other items by eye. Sights can be a simple set or system of markers that have to be aligned together as well as aligned with the target...

, they all permit the rifleman to set the angle between the bore of the rifle and the line of sight (LOS) to the target. Figure 2 illustrates the relationship between the LOS and bore angle.

This relationship between the LOS to the target and the bore angle is determined through a process called "zeroing." The bore angle is set to ensure that a bullet on a parabolic trajectory will intersect the LOS to the target at a specific range. A properly adjusted rifle barrel and sight are said to be "zeroed." Figure 3 illustrates how the LOS, bullet trajectory, and range (

) are related.

Procedure

In general, the rifleman will have a table of bullet heights with respect to the LOS versus horizontal distance. Historically, this table has been referred to as a "drop table." The drop table can be generated empirically using data taken by the rifleman at a rifle range; calculated using a ballistic simulator; or is provided by the rifle/cartridge manufacturer. The drop values are measured or calculated assuming the rifle has been zeroed at a specific range. The bullet will have a drop value of zero at the zero range. Table 1 gives a typical example of a drop table for a rifle zeroed at 100 meters.

Table 1: Example Bullet Drop Table

Range (meters)	0	100	200	300	400	500
Bullet Height (cm)	-1.50	0.0	-2.9	-11.0	-25.2	-46.4

If the rifleman is engaging a target on an incline and has a properly zeroed rifle, the rifleman goes through the following procedure:

Determine the slant range to the target (measurement can be performed using various forms of range finders, e.g. laser rangefinder)
Determine the elevation angle of the target (measurement can be made using various devices, e.g. sight attached unit)
Apply the rifleman's rule to determine the equivalent horizontal range ()
Use the bullet drop table to determine the bullet drop over that equivalent horizontal range (interpolation is likely to be required)
Compute the bore angle correction that is to be applied to the sight. The correction is computed using the equation (in radians).
Adjust the bore angle by the angle correction.

Example

Assume a rifle is being fired that shoots with the bullet drop table given in Table 1. This means that the rifle sight setting for any range from 0 to 500 meters is available. The sight adjustment procedure can be followed step-by-step.

1. Determine the slant range to the target.

Assume that a range finder is available that determines that the target is exactly 300 meters distance.

2. Determine the elevation angle of the target.

Assume that an angle measurement tool is used that measures the target to be at an angle of

with respect to horizontal.

3. Apply the rifleman's rule to determine the equivalent horizontal range.

4. Use the bullet drop table to determine the bullet drop over that equivalent horizontal range.

Linear interpolation can be used to estimate the bullet drop as follows:

5. Compute the bore angle correction that is to be applied to the sight.

6. Adjust the bore angle by the angle correction.

The gun sight is adjusted up by 3.2' in order to compensate for the bullet drop. The gunsights are usually adjustable in unit of minutes, half minutes, or quarter minutes of angle.

Zeroing the rifle

Let

be the bore angle required to compensate for the bullet drop caused by gravity. Standard practice is for a rifleman to zero his rifle at a standard range, such as 100 or 200 meters. Once the rifle is zeroed, adjustments to

are made for other ranges relative to this zero setting. One can calculate

using standard Newtonian dynamics as follows (for more details on this topic, see Trajectory

Trajectory

A trajectory is the path that a moving object follows through space as a function of time. The object might be a projectile or a satellite, for example. It thus includes the meaning of orbit—the path of a planet, an asteroid or a comet as it travels around a central mass...

).

Two equations can be set up that describe the bullet's flight in a vacuum.

(Equation 1)

(Equation 2)

Solving Equation 1 for t yields Equation 3.

(Equation 3)

Equation 3 can be substituted in Equation 2. The resulting equation can then be solved for x assuming that

and

, which produces Equation 4.

(Equation 4)

where

is the speed of the bullet, x is the horizontal distance, y is the vertical distance, g is the Earth's gravitational acceleration, and t is time.

When the bullet hits the target (i.e. crosses the LOS),

and

. Equation 4 can be simplified assuming

to obtain Equation 5.

(Equation 5)

The zero range,

, is important because corrections due to elevation differences will be expressed in terms of changes to the horizontal zero range.

For most rifles,

is quite small. For example, the standard 7.62 mm (0.308 in) NATO bullet is fired with a muzzle velocity of 853 m/s (2800 ft/s). For a rifle zeroed at 100 meters, this means that

.

While this definition of

is useful in theoretical discussions, in practice

must also account for the fact that the rifle sight is actually mounted above the barrel by several centimeters. This fact is important in practice, but is not required to understand the rifleman's rule.

Inclined trajectory analysis

The situation of shooting on an incline is illustrated in Figure 4.

Figure 4 illustrates both the horizontal shooting situation and the inclined shooting situation. When shooting on an incline with a rifle that has been zeroed at

, the bullet will impact along the incline as if it were zeroed at a longer range

. Observe that if the rifleman does not make a range adjustment, his rifle will appear to hit above its intended aim point. In fact, riflemen often report their rifle "shoots high" when they engage a target on an incline and they have not applied the rifleman's rule.

Equation 6 is the exact form of the rifleman's equation. It is derived from Equation 11 in Trajectory

Trajectory