Derivation of Pejsa Point-Blank Range Formula

Quote of the Day

Almost everything you do will seem insignificant, but it is important that you do it.

— Mahatma Gandhi. I think about this statement frequently. I often encounter situations where I wonder if I can make a difference, but I still keep trying.


Introduction

Figure 1: Illustration of Point-Blank Range.

Figure 1: Illustration of Point-Blank Range.

A reader asked me if I could work through Pejsa's formulas related to calculating Point-Blank Range (PBR). During my earlier review of Pejsa's work, I chose not to cover this material just because it was not some of my favorite material – taste is definitely a part of mathematics. However, it is not difficult material to work through, and I like to answer questions when I can.

Background

Definitions

Figure 1 illustrates the key variables involved in computing the zero characteristics for a rifle setup for a specific point-blank range.

Line of Sight (LOS)
The line passing through the viewing axis of the rifle sight.
Near Zero (ZN)
The point nearest the shooter at which the bullet intersects the LOS.
Far Zero (ZR)
The point farthest from the shooter at the which the bullet intersects the LOS.
Midpoint (ZR)
The range along the LOS at the bullet attains it maximum height (Hm).
Point-Blank Range (ZPBR )
The maximum distance where the trajectory of the bullet does not exceed a predetermined amount of rise or drop, also the maximum distance a specific round can be fired and hit a given target without any compensation for bullet drop (Source).

Shortcomings of Pejsa's Approach

The primary advantage of Pejsa's approach is that it is algebraic – no iteration is required. However, the model does have shortcomings:

  • The formulas are focused on projectiles that are similar in shape to the US military's M2 30 caliber bullet.

    The M2 bullet shape has a drag coefficient as shown in Figure M by the line labeled GP or Pejsa. The Pejsa formulas scale the G1 ballistic coefficients so that it will properly fit the GP shape. Always use the G1 ballistic coefficient for modeling using Pejsa formulas.

  • The formulas are not directly usable with other coefficients, e.g. G7.

    Fortunately, G1 coefficients are available for nearly every bullet. However, G1 is considered very "old-school" compared to more modern shapes. To compare the drag coefficients of the various shapes, see Appendix A.

  • The formulas could be modified to work with other bullet shapes, e.g. G7.

    I will not be the person to take that task on. See this web site for some work in this area.

Objectives

I will derive Equations 1 -4 from Pejsa's Modern Practical Ballistics.

Eq. 1 \displaystyle M=\frac{1}{{\frac{2}{{F0}}+\frac{G}{{{{V}_{0}}\cdot \sqrt{{{{H}_{m}}+S}}}}}}
Eq. 2 \displaystyle {{Z}_{N}}=\frac{{1-SH}}{{\frac{1}{{F0}}-\frac{1}{M}}}
Eq. 3 \displaystyle {{Z}_{F}}=\frac{{1+SH}}{{\frac{1}{{F0}}+\frac{1}{M}}}
Eq. 4 \displaystyle {{Z}_{{PBR}}}=\frac{{1+SQ}}{{\frac{1}{{F0}}+\frac{{SQ}}{M}}}

where

  • S is the sight height above the LOS.
  • Hm is the maximum height achieved by the bullet above the LOS.
  • \displaystyle SH=\sqrt{{1+\frac{S}{{{{H}_{m}}}}}}
  • \displaystyle SQ=\frac{{SH}}{{\sqrt{2}}}
  • M is the point along the trajectory at which the bullet has the high positive deviation from the LOS. M is shorthand for "Midpoint," which is a bit of a misnomer – it is not the geometric midpoint of the trajectory.
  • G =41.67, a constant that appears throughout Pejsa's work. It is a scaled version of the gravitational constant, g. I derive its value here.

I will compare the results produced by these four formulas against a web-based calculator in my analysis.

Analysis

Midpoint Formula

I covered the derivation of the midpoint formula in this post, and I will not discuss it further here.

Near Zero Formula

Figure 2 shows the derivation of the near-zero formula.

Figure M: Derivation of Near Zero Formula.

Figure 2: Derivation of Near Zero Formula.

Far Zero Formula

Figure 3 shows the derivation of the far-zero formula.

Figure M: Far Zero Formula Derivation.

Figure 3: Far Zero Formula Derivation.

Point-Blank Range Formula

Figure 4 shows the derivation of the point-blank range formula.

Figure M: Derivation of Point-Blank Range Formula.

Figure 4: Derivation of Point-Blank Range Formula.

Excel Implementation

I always appreciate when an author includes an implementation of their results so that I can experiment with them. I have created an Excel version of Equations 1 – 4, which you can download here.

In the worksheet, I include a comparison between the results from the Pejsa formulas and my reference web-based calculator (Figure 5).

Figure M: Comparison Between Pejsa and Web Site Values.

Figure 5: Comparison Between Pejsa and Web Site Values.

Conclusion

This post shows that the Pejsa formula provides simple, algebraic results that compare well with results from a often-used web site calculator. I provided a tool (i.e. Excel workbook) that will allow others to experiment with Pejsa formulas.

Appendix A: Comparison of Drag Coefficients.

Figure 6 the drag coefficients for various projectile shapes. The A shape is for an air gun projectile.

Figure M: Drag Coefficients For Different Ballistic Shapes.

Figure 6: Drag Coefficients For Different Ballistic Shapes (Source).

 
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2 Responses to Derivation of Pejsa Point-Blank Range Formula

  1. Chris says:

    This is a spectacular contribution! Calculating maximum point blank range (and the 25, 50 and 100 yd. sight-in heights) allows quick and accurate rifle setup for hunting. Many times, having the trajectory, drop, etc. for every 25 yards is not too important. Having the maximum point blank range: priceless.

    Thanks so much!

     
    • mathscinotes says:

      If you end up coding this stuff up in Javascript, send me a note. I would like to check it out.

      mathscinotes

      P.S. If you want to completely duplicate what the web site model does, you need to add compensation for air density. I did cover the modeling of air density as a function of altitude, temperature, and humidity on this post. Ask another question if it is confusing.

       

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