People of mediocre ability sometimes achieve outstanding success because they don't know when to quit. Most men succeed because they are determined to.
— George Allen
Introduction
Figure 1: Bicycle as Projectile (Braden Hanna).
In our previous post, we developed an expression for y' (=dy/dx, Newton's notation) expressed as differential equation in terms of x. We will now solve this equation through the use of an integrating factor. Having solved for y' in terms of x, we can integrate that expression to obtain y(x).
The exact expression for y(x) is a bit complex and Pejsa spent a quite a bit of book space developing a good approximation for y(x) that is both accurate and simple. In this post, we will derive both the exact and approximate solutions.
Background
Pejsa's Approximate Solution
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 |  |
where
- 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")
Pejsa uses the subscript "m" to stand for "mean". I should mention that while Pejsa's derivation uses this formula, his actual software uses the following modified form
(I call this the modified form). He has a worked example on page 94 that also uses the results from the modified Fm , with no prior warning of a change. I spent many hours trying to find the discrepancy. I assume that he made this change to improve his approximation a bit, which I demonstrate in Appendix A.
Analysis
Solving For y'(x)
Figure 2 shows how we can use an integrating factor to solve the differential equation for y' as a function of x.
Figure 2: Solving The Differential Equation for y'.
Solving For y(x)
Figure 3 show how we can integrate y'(x) to obtain an exact solution for y(x).
Figure 2: Integrating y' to Obtain y.
From my standpoint, the exact solution is reasonable for implementation using either software and spreadsheets. I agree that it would be painful if all you had was a 1970s–era calculator.
Approximation
Figures 4 and 5 show how Pejsa used a Taylor series approximation of the exact solution to obtain a computationally-easier result that provides good agreement for typical projectiles (i.e. ballistic coefficients from 0.3 to 0.5 [Pejsa, page 143]).
Figure 4: Developing a Taylor Series Approximation.
Figure 5: Approximate Solution With No Unit Assumptions.
Standard Form of Pejsa Approximate Solution Assuming US Units
Figure 6 derives the most commonly seen form of Pejsa's approximate solution using US customary units.
Figure 6: Approximate Solution Using US Customary Units.
Conclusion
Now that we have developed both the exact and approximate solutions, I will work an example in part 3. An example is definitely needed.
Appendix A: Error Between Exact and Approximate Solutions
Figure 7 shows a plot of the percentage error between the exact solution and the approximate solutions (standard and modified Fm) for a projectile with a ballistic coefficient of the 1 (I will discuss ballistic coefficients in part 3). Observe that the errors are small for both forms of Fm, but the modified Fm is distinctly better.
Figure 7: Plot of Errors Between Exact and Approximate Solutions.
Excellent encyclopedic post!
As I mentioned in a previous comment it's possible to derive the drop formula by using (positive sign of g conforming to your notation):
(v_y/v_x)'=g/(v_x)^2 --> y''=g/(v_x)^2
Here "_" is used for subscripts/indices.
Using Pejsa's velocity relation v_x=v_0(1-nx/F_0)^(1/n) the drop formula y(x) follows after two subsequent integrations over x.
The above relationship is implied from McCoy's treatment of flat-fire approximation in his book.
I agree with your comment and I intend to eventually cover McCoy's work as well – I consider his work the modern standard. This series of posts are really my notes taken when I read Pejsa's book five years ago and I thought they were worth making available. Eventually, all my notes end up as a blog post of some sort.
I found Pejsa's book confusing to read and I wanted to get things into a more consistent form.
mathscinotes
I think other readers of Pejsa's work (known for being hard to read) will value your effort.
Personally, I like Pejsa's approach because of its more analytical nature compared to the classic use of drag functions.
In your Final Integration block, you mention the assume that n>1 which may not be true for all n, reference your earlier work on stating the drag function, -A*v^(2-n) as a power of n. I believe you meant to state that n is not equal to 1.
Thanks Ronan! I was going pretty fast when I wrote this up and I appreciate when people help me weed these errors out.
mathscinotes
In appendix A you plot the errors between the exact solution and the approximate solutions using a standard Fm and modified Fm.
I only see a reference in the text to the standard approximate solution. "Fm=Fo-1/4.n.x " I assume this is the "standard Fm". What is the "modified Fm"?
The terms standard Fm and modified Fm are mine and defined here. Here is how I defined the terms. Note that the subscript m stands for "mean".
Pejsa used the standard form in the text of his book. However, he used the modified form to work his examples and in his BASIC source included in the back of the book. This was an irritating aspect of his book. I could not duplicate the answers in his examples until I discovered the modified form of Fm.
mathscinotes
Please disregard the previous comment. I finnally saw the modified Fm right at the top of the post. I blame the cold in my part o the world 🙂
Some confusion with different terms used in "Solving For y'(x)"
I assume
"dt" - difference in time
"dy" - difference in y-axis distance
"dx" - difference in x-axis distance
What is "d"?
I assume you are referring to my use of the equation
. The d is just part of the operator notation that I use when solving differential equations. An alternative representation would be 
.
mathscinotes
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