Pejsa Trajectory Midpoint Formula Given a Maximum Projectile Height

The most amazing achievement of the computer software industry is its continuing cancellation of the steady and staggering gains made by the computer hardware industry.

— Henry Petroski


Introduction

Figure 1: Critical Trajectory Points.

Figure 1: Critical Trajectory Points (Source:Me).

Pejsa defines the trajectory midpoint as the range at which the projectile height reaches its maximum (Figure 1). Pejsa's midpoint formula (Equation 1) allows you to compute the midpoint given a specific maximum height (Hm).

Eq. 1 \displaystyle M=\frac{1}{{\frac{G}{{{{V}_{0}}\cdot \sqrt{{{{H}_{m}}+SH}}}}+\frac{2}{F}}}

where

  • V0 is the projectile's initial velocity (ft/s),
  • Hm is the maximum projectile height above the line of sight (in),
  • SH is the height of the scope above the projectile's trajectory (in),
  • G is a constant (41.68),
  • F is what Pejsa calls the coefficient of retardation (ft).

The derivation is straightforward and I will not provide much additional commentary beyond the mathematics itself.

Background

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

It is worth commenting that this is the only derivation that I recall where Pejsa uses a horizontal line of sight. He is able to use a horizontal line of sight because he makes the observation that his horizontal drop formula actually works for a projectile going forward (drag inhibiting its forward motion) and backwards (drag enhancing its forward motion). It is an interesting aspect of the symmetry of the formula.

This means that the derivation assumes  that the projectile is moving BACKWARDs from the midpoint and is dropping down to the muzzle under the force of gravity with drag accelerating it.

Analysis

Derivation

Figure 2 shows the derivation of the midpoint formula. Observe that the derivation makes an approximation based on a truncating a Taylor series expansion.

Figure 2: Derivation of Formula for the Trajectory Midpoint.

Figure 2: Derivation of Formula for the Trajectory Midpoint.

Example

Figure 3 shows a worked example from the tables in the back of Pejsa's book. The agreement was excellent.

Figure 3: Example Taken from Pejsa's Tables in the Back of His Book.

Figure 3: Example Taken from Pejsa's Tables in the Back of His Book.

Conclusion

In this post, I derived and provided an example of Pejsa's midpoint formula. I have two more posts left in my Pejsa review odyssey:

  • Derive the maximum projectile height for a rifle zeroed at a range of Z.
  • Provide an example of the projectile height relative to the line of sight for a projectile moving from supersonic to subsonic speeds.
 
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2 Responses to Pejsa Trajectory Midpoint Formula Given a Maximum Projectile Height

  1. Tom Lawrence says:

    I notice that in previous posts, V was a function of F0 while D was a function of Fm. Here, they are both functions of F(undefined). Are you assuming that F0 = Fm? I notice that this would be approximately true for sufficiently small X.

     
  2. Pingback: Derivation of Pejsa Point-Blank Range Formula | Math Encounters Blog

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