Category Archives: Ballistics

Recoil Calculation Example

I have been reading about the US Air Force's battle to retire the A-10 Warthog (Figure 1). The USAF has never cared for the A-10 and has made a number of attempts to replace it with either the F-16 or the F-35. During my reading, I saw the following statement about the recoil of it 30 mm Gatling gun, and the impact of this recoil on the A-10's speed. Continue reading

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Drag Coefficient From A Ballistic Drop Table

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

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Derivation of Pejsa Point-Blank Range Formula

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

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WW2 Reticles

I have watched a lot of old WW2 combat footage, and I have noticed that many of the machine guns and fighter planes had similar reticles. A reticle is a fine-grid of lines used in conjunction with an eyepiece to assist in taking measurements or with accurately pointing an instrument. Figure 1 shows a reticle similar to what I have seen in numerous combat scenes. Continue reading

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Stadiametric Rangefinding Approaches

Recently, I was reading about stadiametric range finding methods being used by hunters and their telescopic sights – I was surprised to find a lot of writing on the topic. As I researched the topic, I saw that there are three common approaches used in telescopic sights: milliradian (mil), Minute Of Angle (MOA), and Inch Of Angle (IOA). I will review these methods here. Continue reading

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Generating Pejsa's F Function

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

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Range Table Construction Using Pejsa's Formulas

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

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Pejsa Formula for Midpoint as a Function of Zero Range

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

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Pejsa Trajectory Midpoint Formula Given a Maximum Projectile Height

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

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Pejsa Bullet Height Versus Distance Formula For a Zeroed Rifle

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

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