Pejsa Formula for Midpoint as a Function of Zero Range

You can't work a Biegert too hard.

— Statement by a football coach to my son, who thought he was working too hard during practice. The same football coach had coached my brothers. You do not want to hear a football coach make this statement.


Introduction

Figure 1: Lots of Algebra in this Post.

Figure 1: Lots of Algebra in this Post (Source).

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.

Of all Pejsa's formulas, this one is the most algebraically challenging to derive, but the process was worthwhile to go through. For example, it was the first time that I had an actual application for the Cardano cubic equation solution.

Equation 1 shows Pejsa's midpoint formula as a function of the rifle's zero range.

Eq. 1 \displaystyle M=F\cdot \left[ {1+C{{M}^{{\frac{1}{3}}}}\cdot \left( {{{{\left( {CZ-1} \right)}}^{{\frac{1}{3}}}}-{{{\left( {CZ+1} \right)}}^{{\frac{1}{3}}}}} \right)} \right]

where

  • M is midpoint range (yards).
  • CM=\frac{{Z\cdot F}}{{\left( {Dz+SH} \right)}}\cdot {{\left( {\frac{G}{{{{V}_{0}}}}} \right)}^{2}} is a temporary variable used to make writing Equation 1 simpler.
  • CZ=\sqrt{{\frac{8}{{27}}\cdot CM+1}} is a temporary variable used to make writing Equation 1 simpler.
  • Z is the zero range (yards).
  • Dz is the projectile drop at the zero range when fired horizontally (inches).
  • SH is the height of the scope above the bore of the rifle (inches).
  • V0 is the initial velocity (ft/s).
  • G is a constant (41.68).

Background

All the required background was supplied as part of this three-part series.

Analysis

Equation Setup

Figure 2 shows how we can use Pejsa's drop formula for a horizontal projectile to generate a cubic polynomial with one real solution.

Figure 2: Derivation of a Cubic Equation Used to Compute the Trajectory Midpoint Given the Zero Range.

Figure 2: Derivation of a Cubic Equation Used to Compute the Trajectory Midpoint Given the Zero Range.

Solution

In Figure 3, I solve the Cardano cubic equation for the real root. This is where the serious algebra occurs.

Figure 3: Solution to the Cardano Cubic.

Figure 3: Solution to the Cardano Cubic.

Example

Here is an example of how I would use Equation 1 in a real-world application. In this case, given a zero range, I can compute the midpoint of the trajectory. Given the midpoint, I can compute the maximum height of the bullet along its trajectory.

Figure 4: Working an Example From the Back of Pejsas' Book.

Figure 4: Working an Example From the Back of Pejsa's Book.

Conclusion

With this post, I have now reviewed all the major formulas in Pejsa's work. The last exercise will be computing the range table for a projectile moving from supersonic to subsonic speed. I view this calculation as more of a bookkeeping challenge than anything else, but it is a bit confusing.

 
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One Response to Pejsa Formula for Midpoint as a Function of Zero Range

  1. Tom Lawrence says:

    Here is my take on Pejsa's 'time of flight'.

    t = t*[2R/(2X* - R)]

    t* = F0/Vx0

    X* = F0

    R = the range

     

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