
Rifleman's rule
Encyclopedia




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


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 (

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

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


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.


Solving Equation 1 for t yields Equation 3.

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






where

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




The zero range,

For most rifles,


While this definition of


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


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

The complete derivation of Equation 6 is given below. Equation 6 is valid for all








Since the








This completes the demonstration of the rifleman's rule that is seen in routine practice. Slight variations in the rule do exist.
Derivation
Equation 6 can be obtained from the following equation, which was named equation 11 in the article TrajectoryTrajectory
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...
.

This expression can be expanded using the double-angle formula for the sine (see Trigonometric identity) and the definitions of tangent and cosine.

Multiply the expression in the parentheses by the front trigonometric term.

Extract the factor


The expression inside the parentheses is in the form of a sine difference formula. Also, multiply the resulting expression by the factor


Factor the expression



Let


Let


Expand


Distribute the factor


Factor out the



Factor out


Substitute


Substitute the definitions of




This completes the derivation of the exact form of the rifleman's rule.