Parameter Determination for Pejsa Velocity Model

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Introduction

Figure 1: Picture of Pejsa Book Cover (Source).

I have had several people ask me questions about the Pejsa ballistic model (previous post) and I thought it would be useful to include some additional posts on the topic. In this post, I will discuss how the formula and parameters were determined for the velocity versus range formula for the range of velocities from 1400 feet per second to 4000 feet per second – sorry about the use of US customary units. Pejsa's formula were setup specifically to used ranges in yard, velocities in feet per second, and projectile drops in inches.

I do recommend that folks read through Pejsa's book (Figure 1) for themselves. Unfortunately, it is not an easy read and the formulas derived are not as general as I would like. The main issue with the formulas are that they are quite specific to US customary units – e.g. the fact that one yard equals three feet is used in final formulas. Also, some of the approximations assume a specific range of ballistic coefficients. However, the formulas are very useful because they provide accurate answers to common ballistic situations using simple algebraic formulas.

Background

For background, see this post. Pejsa provides different formulas for the variation of projectile velocities with respect to range based on the projectile's velocity. In this post, I will only address his formula for projectile velocities greater than 1400 feet per second. In later posts, I will work through the lower velocity ranges.

Analysis

Derivation

Figure 2 shows how one can derive Pejsa's velocity versus range formula for the velocity range of 1400 ft/s to 4000 ft/s.

Figure 2: Derivation of Pejsa Velocity Versus Range Equation.

Parameter Determination

Figure 3 shows how to determine the single constant factor, K₁, in the Pejsa equation.

Figure 3: Determine K₁ Using Empirical Data.

Verification

One question involved how to generate a plot of the rate of velocity change with respect to distance -- this is a graph that appears in Hatcher's Notebook, a commonly cited ballistics reference. Figure 4 shows how I derived an expression for this curve. I will use modern data from Berger Bullets to make my comparison. I used Berger as a data reference rather than Hatcher because the Berger web site gives me raw numbers instead of a graph (i.e. quicker to work with).

Figure 4: Derive Rate of Velocity Change with Range.

Graph of Results

Figure 5 shows how I setup my graph.

Figure 5: Graph Setup.

Figure 6 shows a comparison of Pejsa's projectile formula (blue line) with the Berger web simulator (orange line).  As Pejsa states, his  formula provides good accuracy until the projectile velocity approaches 1400 feet per second, which occurs at a range of 1858 feet. Figure 6 also compares the rate of change of velocity with respect to distance and the data from Hatcher's Notebook for a similar projectile. Again, the results are similar until the projectile velocity nears 1400 feet per second.

Figure 6: Graph of Empirical and Theoretical Results.

Conclusion

As Pejsa states, the agreement of the velocity versus distance is pretty good for velocities above 1400 ft/s.

Appendix A: Hatcher's Deceleration per Foot Graph.

Figure 7 shows the projectile rate of deceleration per foot of travel from Hatcher's Notebook.

Figure 7: Deceleration Graph from Hatcher's Notebook.