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.


Figure 1: Trajectory of Bullet Relative to the Rifle's Line of Sight (Source: Me, done for the Wikipedia).

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.

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.


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 \displaystyle \sqrt{D}=\frac{{\frac{G}{{{{v}_{0}}}}}}{{\frac{1}{R}-\frac{1}{{{{F}_{m}}\left(R\right)}}}}


  • 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].
  • Fm (R)= F0-3·n·R/4 (I call this the "standard form").


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 \displaystyle H=-\left( {S+D} \right)+R\cdot \frac{{S+{{D}_{Z}}}}{Z}


  • 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.
  • DZ 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 M: Model for Pejsa's Range Table Formula.

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.

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.

Figure 4: Calculation of Range Table Example.


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.

Table 1: Bullet Height Computed Using Pejsa Software and My Mathcad Model.
Range (yd) Pejsa Software (in) Mathcad Version (in)
0 -1.5 -1.5
50 0.3 0.3
100 1.0 1.0
150 0.5 0.5
200 -1.4 -1.4
250 -4.8 -4.8
300 -9.8 -9.8
350 -16.5 -16.5
400 -25.2 -25.2
450 -36.1 -36.0
500 -49.3 -49.2


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.
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5 Responses to Pejsa Bullet Height Versus Distance Formula For a Zeroed Rifle

  1. Chris says:

    There is a point blank range (PBR) calculator here:

    This is useful because, instead of calculating trajectory, it "fits" the best trajectory so that a given maximum ordinate (say, +2" from LOS) and -2" from LOS (which represents a 4" vital kill zone) fit a range . With a sight height of 1.5", minimim PBR is at the muzzle and maximum PBR is 238 yards (using the default settings), when zeroed 1.89" high at 100 yds. Since ranges often have 50 and 200 yd ranges too, having the rise/drop at 50, 100 and 200 yds. allows one to "check" trajectory. I have found the calculations you have provided to be very accurate, and I am excited to implement similar functionality using the Pejsa model.

    I am trying to understand if this can be directly calculated (sorry if it obvious) or if this is iterated in some way using a looping structure in code (I can code).

    I can also answer the kamikaze helmet question definitively, if that is a carrot!


    • mathscinotes says:

      I will jump at the carrot by putting out a post this week on the subject. I tried to give a pretty thorough review of the Pejsa book, but I did not cover what he had to say about point-blank range calculations.


  2. Chris says:

    Thank you!

    I think your coverage was very thorough, though I don't have the math skills you have, and jumped to the solutions ... :-).

    I think solving point-blank range calculations with the Pejsa model would be really interesting. Again, if it can solved directly, then I guess getting to an equation would be tough for me. I am sure I can solve this with Javascript (and happy to share, if it interests you) using brute-force loop iteration.

    I thought the kamikaze question might be a joke (and, the answer is not too interesting), but the first kamikaze pilots were pilots which conducted less treacherous missions before. They were issued standard leather caps (not really helmets, and not that it would matter). They wore them for final ceremonies and photographs before their last missions, and in fact, because they had been given what might be known as "last rites" during these ceremonies, it was very dishonorable to return. Unsuccessful pilots would ditch their planes at sea rather than face the ridicule of returning. The pilots wore these caps to prevent head injuries (and loss of consciousness) during dogfight maneuvers.

    More on this:

    Thanks Mark,

    • mathscinotes says:

      I put out a post on your question today. I have included an Excel spreadsheet (no macros) that should allow you to experiment with the formulas a bit. I also checked the agreement between Pejsa's formulas and the web page you reference. The agreement is very good.

  3. Chris says:

    I cannot thank you enough! I get a lot out of your blog, and I appreciate your willingness to look at this. You have some serious math skills!


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