Category Archives: Ballistics

I thought it might be interesting show how you can approximate the drag coefficient for a bullet given a standard bullet velocity versus distance table. The folks at Barnes have put together a very nice discussion of how they go about characterizing a projectile using Doppler radar data. They also created an excellent infographic for a common hunting round showing the velocity versus distance table, actual drag coefficient, and G1/G7 reference drag coefficients (Figure 1). I thought I would take their table data and use that data to generate the drag coefficient chart. Since my data is limited, I do not expect a perfect reconstruction, but it should be close. Continue reading →

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. Continue reading →

I have had several people ask me to review how Pejsa generated his F function. Recall that Pejsa's approach is based on using a parameter called the ballistic coefficient (BC) to scale the performance of a reference projectile – Pejsa used the US military's 30 caliber M2 bullet, which dates back to the Springfield rifle. This effort involves basic curve fitting, and I will illustrate the process for the velocity interval from 1400 feet per second (fps) to 4000 fps. This velocity range is the most important to most folks and it illustrates the basic curve fitting process well. Continue reading →

In the main body of this post, I work through a detailed example from "Modern Practical Ballistics" on how to apply Pejsa's formulas to determine a range table for a projectile moving through a wide range of velocities. In Appendix A, I work through a second example from the text as additional validation of my implementation. In Appendix B, I show how to use an Ordinary Differential Equation (ODE) solver to generate a range table for comparison with Pejsa's algebraic approximations. Continue reading →

This post will cover Pejsa's formula for the trajectory midpoint as a function of the rifle's zero range. Shooters often have a preferred zero range, like 100 yards or 200 yards. This formula allows the shooter to determine his midpoint range directly from the zero range. The midpoint range can then be used to determine the maximum bullet height above the line of sight, which can be used to determine the maximum bullet placement error. Continue reading →

Pejsa defines the trajectory midpoint as the range at which the projectile height reaches its maximum (Figure 1). Pejsa's midpoint formula allows you to compute the midpoint given a specific maximum height (Hm). The derivation is straightforward and I will not provide much additional commentary beyond the mathematics itself. Continue reading →

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 drop 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. Continue reading →