Talent is God given. Be humble. Fame is man-given. Be grateful. Conceit is self-given. Be careful.
— John Wooden
I recently have had a number of readers ask me to continue my review of Pejsa's "Modern Practical Ballistics". The last major topic I have left to cover is his formula for the drop of a horizontally‑fired projectile as a function of distance. My plan is to derive the formula and present an example of its use. The derivation is not difficult, but it is a bit long and I will divide my presentation into a couple of posts.
This post will examine Pejsa's derivation of the vertical drop differential equation. In my second post, I will examine how he generates both an exact solution and a useful approximate solution that is commonly used in practice. My third post will contain a worked example.
I do not view Pejsa's work as the "state of the art", but it can play a useful role for people who want to develop simple applications for their ballistic work. Pejsa was working at a time when computers had limited capability and people were desperate for simple algebraic methods for getting answers. I consider his work the ultimate expression of the original empirical work done by Mayevski and Ingalls back in the 1870s. This work was important to the development of modern gunnery and I have spent a fair amount of time reading through their papers as part of my interest in battleship gunnery.
For those who want a glimpse into modern ballistics methods, please see the work by McCoy.
I am going to work hard to use Pejsa's notation, which I do not like and he applies inconsistently. However, using his notation will make referring back to his book much less confusing. I will do my best to carefully comment on the role of each equation and how I am using it.
- Atmospheric drag at velocities below the speed of sound is often modeled as varying by the square of the projectile velocity. n is a correction term that is used to "fix" this square-law model for transonic and supersonic velocities. Pejsa defines a different n for four common projectile velocity intervals (ft/s):
- n = 0.5 : 1400 ≤ V < 4000
- n = 0.0 : 1200 ≤ V < 1400
- n = -3.0 : 900 ≤ V < 1200
- n = 0.0 : 0 ≤ V < 900
For this 3-part series, I will be focus my examples on the 1400 ft/s to 4000 ft/s interval, but the work applies equally well to the other intervals.
- A is parameter is a proportionality constant that is specific to the projectile in question and relates acceleration to projectile velocity, i.e. . It will be used in deriving the projectile drop differential equation, but will not play a significant role in its solution.In his book, Pejsa sometimes uses A to represent acceleration. This created some confusion for me as I read the book. In these posts, I will make sure that A is only used to represent the proportionality constant.
- Pejsa refers to F(x) this as the retardation coefficient and has units of distance. He describes his interpretation of the coefficient in the following quote.
One percent of F is distance in which a projectile loses 1% of its speed to air drag.
For a more detailed discussion of physical meaning of F(x), see this post. F(x) and A are related by the formula , which I will use in the mathematical development to follow.
Here are the key conventions and assumptions in Pejsa's analysis.
- lateral distance is measured along the x-axis, which is positive to the right.
- projectile drop is measured along the y-axis, which is positive toward the ground.
This means that projectile drop is positive in the direction of the ground.
- the force of drag (FDrag) as a function of projectile velocity (v) can modeled using powers of velocity (2-n, where n varies with the velocity of the projectile), i.e. .
Newton showed that the drag on a slow-moving projectile (i.e. less than transonic speed) varies with the square of it velocity. The drag on projectiles moving at transonic and supersonic speeds can be modeled using other powers of velocity. The convention as been to define n as the correction required to the low-velocity, square-law model.
- I use a yellow highlight for key results from Pejsa's book and a green highlight for significant intermediate results.
All the work done here was in Mathcad, which for this post was primarily used as a mathematical editor. I will use its numerical capabilities in my second post on this topic.
Pejsa's Vertical Drop Differential Equation
We are going to derive the following differential equation for the projectile's velocity in the y-direction as a function of distance.
- g is the acceleration due to gravity.
- F0 is F(0).
- n is an exponent determined by the projectiles' velocity. n is a constant in certain velocity intervals.
- v0 is the initial projectile velocity.
- A is the retardation coefficient, which is a function of the ballistic coefficient.
- y' is projectile velocity in the y-direction.
Application of Newton's Laws of Motion
Figure 2 shows the Free Body Diagram (FBD) of the projectile and the associated differential equations.
Equation 5 in Figure 2 is the differential equation for the projectile drop as a function of time. We want to modify this differential equation so that we can solve for the projectile drop as a function of horizontal distance.
Velocity Versus Horizontal Distance
In Figure 3, I show how Pejsa computes the projectile velocity as a function of distance.
Derivation of Flat Trajectory, Projectile Drop Differential Equation
Figure 4 shows how to derive Pejsa's differential equation for the drop of a projectile fired on a flat trajectory as a function of horizontal distance x.
Equation 11 in Figure 4 is the key result. This form of the differential equation is useful because:
- It gives us projectile drop as a function of distance.
- It will be shown to have a closed-form solution.
- All parameters can be determined using a projectile's ballistic coefficient, which is readily available for different projectiles.
- It can be used over a wide range of projectile velocities by changing the value of n in the equation.
Given the key differential equation, we can generate a closed-form solution in our next post. If you want to know a bit about Arthur Pejsa, here is a Youtube interview with him.