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