Pejsa Bullet Height Versus Distance Formula For a Zeroed Rifle

I'm desperately trying to figure out why kamikaze pilots wore helmets.

- I heard a historian ask this question on the History Channel. I cannot think of a good reason for them to wear a helmet, either.

Introduction

Figure 1: Trajectory of Bullet Relative to the
Rifle's Line of Sight (Source: Me, done for the
Wikipedia). Note how the rifle is canted slightly
up for a flat line of sight.

In this post, I will review Pejsa's development of a formula for the height of a bullet relative to the shooter's line of sight, assuming that the rifle is adjusted to have zero height at a known range (referred to as the rifle's "zeroed" range). Figure 1 illustrates the trajectory of a bullet fired from a rifle zeroed at a given range. If you're not familiar with how to zero a rifle scope at 100 yards, visit a website like riflescopescenter.com to learn more.

This formula is a straightforward extension of Pejsa's drop formula for a bullet fired horizontally (post). I will provide a derivation of the formula and show that I can duplicate some of Pejsa's published examples.

Background

Pejsa's Objective

Pejsa showed that you could determine the trajectory given the following information:

• bullet's ballistic coefficient
• bullet's initial velocity
• sighting approach (two methods covered)
  • zero the rifle at a specific range, or
  • specify maximum excursion of the bullet above the line of sight

For this post, I will assume a known zero range. Pejsa goes into some detail on how to use the zero range to determine the maximum bullet height and vice versa. This is useful to shooters who want to know their placement error for a given zero range.

Drop Formula For Horizontally-Fired Projectile

Our goal in this post is to derive Equation 1, which is Pejsa's approximate solution to his projectile drop equation that we developed in Part 1.

Eq. 1

where

• D is the projectile drop [inches].
• v₀ is the initial projectile velocity [ft/sec].
• R is the projectile horizontal travel distance [yards].
• G is a constant with value 41.697 [ft[sup]0.5[/sup]/sec].
• F_m (R)= F₀-3 n R/4 (I call this the "standard form").

Analysis

Range Equation

Pejsa uses Equation 2 to model the bullet's height above the LOS, which is based on the bullet drop formula for a horizontally-fired projectile.

Eq. 2

where

• H is the bullet's height above the line of sight.
• R is range at which we want the bullet height relative to the line of sight.
• Z is range at which the rifle is zeroed.
• D=D(R) is the bullet drop at range R given by Equation 1.
• D_Z is the drop of a horizontally fired bullet at the Z range.
• S is the height of the scope above the barrel.

Pejsa is able to use Equation 1 because, for calculation purposes, he assumes a slightly different shooting scenario than is illustrated in Figure 1. Figure 2 shows that model for the flight path assumed in deriving Equation 2. Observe that it is a rotation of the situation shown in Figure 1. Pejsa assumes that the rotation is so small that an errors introduced are minimal, which is a good assumption for the flat trajectory situation

Figure 2: Model for Pejsa's Range Table Formula.

A bit of geometry is required to derive the formula, which I show in Figure 3.

Figure 3: Deriving the Height Versus Distance Formula.

Mathcad Model

Figure 4 shows how I computed a range table using Equations 1 and 2.

Figure 4: Calculation of Range Table Example.

Results

Table 1 shows my results versus the output from Pejsa's software for a projectile with

• BC=0.4
• V0=2900 feet/sec
• n=1/2 (i.e. data tabulated at ranges where velocity is greater than 1400 feet/sec)

The agreement is nearly perfect to the first decimal place, which I consider good considering that we are interpolating the F function differently.

Conclusion

I am nearing the end of my review of Pejsa's "Modern Practical Ballistics". The last items to cover will be

• deriving an algebraic expression for the trajectory mid-range (i.e point of maximum bullet height),
• deriving an algebraic expression for the zero point as a function of the maximum bullet height,
• One worked example showing how to deal with a projectile moving from supersonic to subsonic.