Last night I laid there in bed trying to somehow quantify the seemingly unquantifiable, known as "Felt Recoil". It wasn't long before some ideas spun about, I climbed out of bed, grabbed a Gatorade and sat down at the kitchen table. With paper, pen, and calculator, I started attempting to create a formula that would derive a value which utilized rifle recoiling velocity in combination with recoil energy, to paint a more realistic picture of what one must actually endure to their body upon firing.
Now, for all intents and purposes, this is not actually a felt recoil "score", if you will. I perceive the idea of "felt" recoil as somewhat infinitely variable; it's anatomy-based, perception-based, buttpad-relative in so many ways, hold-relative, and etc. I will make my own generalization that no two people will ever witness the same "felt" recoil with the same gun and same load--which I firmly do believe to be true.
My attempt then becomes to summate a total of known factors which can be held as actual true conditions (conditions not perceived but real), while leaving the rest out of the equation and up to the shooter to analyze based on their own variables between gun and shoulder.
I initially began with equations containing far too much waste material, which was unneeded, as some of the new variables resulted from existing variables that were already well-formed Newtonian truths--from the free recoil energy equation itself. It was important to note first, that muzzle energy has nothing to do with recoil.
If it's attempted to use muzzle energy in any way to generate a total recoil "score", it could be derived until ending up at a point where the calculation can become weighted by muzzle energy while essentially producing no recoil at all.
The inverse of this would be deriving a value with extreme recoil, yet no muzzle energy much at all, like 20Ft-Lbs. However, this situation can be entirely valid and true, unlike the inverse previous situation.
Here is why:
Every action has an equal and opposite reaction, as someone once said.
Imagine that the action is a burst of explosive pressure within a confined space such as the middle of a tube. If a bullet of equal weight exists in each end of that tube, at equal distance from the explosion pressure epicenter, the expanding gases will yield the same force on both bullets, propelling them opposite directions with the same velocity. Thus they will have the same total energy as each other, but traveling opposite directions. Now, if the experiment is repeated, yet this time one bullet is made of a denser metal containing more mass than the other original bullet, the same explosive pressure force will need to act for a longer duration on the heavier bullet for it to reach the original speed of a bullet in the first test. However, during this event it must be kept in mind where the center of average volume would exist, while the original weight bullet accelerates away from the starting point more rapidly. As it is moving more rapidly than the other bullet, volume behind it increases more quickly. Therefore, the average volume-center moves in the direction of the lighter bullet, following it that direction until exiting the tube. Once it leaves the tube, pressure falls as confinement disappears. The heavier bullet ceases to incur any more reactive force as the gas vents, leaving it stalled at a velocity value that did not utilize the available tube length. Because of this, the heavier bullet is pushed on for a shorter time duration, and it's energy becomes less than the bullet which already left and utilized the full available tube length. The lighter bullet is said to gain energy through this simple idea. As the opposite bullet gains weight, the original bullet gains energy, so on and so forth.
Take these common sense ideas now and imagine the extreme version; one bullet weighs as much as an average rifle, while the other stays at original, fixed light weight. Put a stock on the lighter fixed-weight bullet--for imagination purposes it's a gun stock that weighs nothing and is considered the rifle itself. The "stocked" (call it a miniature rifle backwards in a tube) bullet has more energy than the actual opposing bullet; in other words, what we would normally call the projectile has less energy than the gun!
Anyways, off on a bit of a tangent there I went, but I think it's critical for some readers to understand why the rifle recoil energy isn't equal to the bullet energy, it's merely an issue of time duration under pressure. Both ARE pushed on equally, under the same opposing force, but that force is able to escape more quickly on the side with lighter "plug", thus it appears to bias energy travel also in that direction.
In conclusion to this, it summarizes why muzzle energy cannot be definitively tied to recoil energy, in any way, or the formula is then corrupt.
Traveling backwards a bit now, I attempted to chain any other variables together, in any way to produce a final "recoil severity score" that could not be fooled at any level. What I arrived at became stripped of complexity, short, and beautiful to me. That's not to say it's a flawless prediction, but through countless efforts I found it to essentially mimic what one would feel from the same rifle as recoil increases, or from the same butt pad as the load and/or rifle weight changes.
I call it,
Marshall's Recoil Severity.
The formula is as follows:
(Recoiling velocity²)*(Free recoil energy in FT-LBS)/1000
Or
Vᵣ² * Eᵣ
1000 = Sᵣ (Recoil Severity)
I believe two reasons for this equation form to hold a high representative value of recoil severity. 1. The credibility of output from the original FRE equation--that being classic Newtonian derived--at least to a high degree, yields an acceptable basis of recoil formation law. 2. FRE is raised to an exponential value of a square of recoiling velocity, from which the square tumbles right out of the energy equation itself--meaning a given energy containing mass which is raised 2x in velocity yields a square representative (4x) energy, and is self-reducing or inherently checked upon by the variables needed to form the FRE value itself. With lowered FRE the end value is inversely reduced upon exponentially in calculated output by the FRE equation factors themselves, which include mass. Finally, a denominator of 1000 keeps the output more organized or easily generalized into I.E., 2 or 3 whole numbers of output using most existing rifle weight/cartridge combinations known to regularly exist.
Please feel free to critique any errors or problems you can find with this method for predicting recoil severity.