That method of calculating wind deflection was first published by a French artillery officer named Didion in, IIRC, 1868. He authored a book on the basics of ballistics for military academy students of his time, and it was included in it.
For those unfamiliar with the principles, even though commonly called "wind drift," bullets aren't actually drifting in the wind due to it blowing sideways on them. Indeed, they are so fast that any air molecule that glances off the side of the bullet will do so at an extremely acute angle, meaning there is no direct sideways blowing of air on the bullet. For that to happen, the air mass would have to be both moving forward at the speed of the bullet plus be moving sideways. Instead, what happens is the side wind and the headwind from the speed of the bullet combine into a vector that is a little off straight ahead in the windward direction. Precession tilts the bullet nose into this slightly angled headwind so that drag is no longer straight back toward the gun but rather is angled like the headwind. It is the sideways component of that drag vector, due to that angle, that moves the bullet sideways. So, if the headwind is at 1250 fps, and the sidewind is at 14.7 fps, 1250/14.7 is about 85, and thus a side-directed force equal to about 1/85 of the drag on the bullet will be moving the bullet to the side.
Knowing that much, you next want to know how much drag is on the bullet. The difference in actual TOF in the air from what it would be in a vacuum tells you how much force has decelerated the bullet on its way to the target. If you know the mass of the bullet, the calculation of the average decelerating force is trivial. You would then calculate how far 1/85th of that force would move the bullet during its TOF to learn the wind deflection. Fortunately, there's the shortcut observed by Didion, which is simply to multiply the wind speed by that difference in TOF. It is all really doing the same thing, but it gives your brain a fun puzzle to see why that has to be so.
This bit of reality bedeviled F. W. Mann in his 1907 book, The Bullet's Flight…. He apparently never saw Didion's work. The poor fellow never did understand how wind drift works, and was clearly much vexed by fact that if he dropped a bullet in a crosswind from a height that gave it the same time of fall to a surface as its TOF to a target, it was blown only a fraction of an inch instead of the the several inches the bullet was deflected by wind at the target.
A number of years ago I wrote a point mass solver in Excel because I wanted to be able to see intermediate calculation results that aren't displayed in a commercial ballistics programs. One of the interesting things I hadn't considered when I started the project was that the height of apogee above the gun muzzle when firing at a positive angle of departure, plus the fall from apogee to the target would not add up the the total drop. Total drop is how far a bullet falls below a bore-sighted target when the bullet is fired truly horizontally. This is because, when you fire angled up, drag on the bullet can be divided into horizontal and vertical drag vectors. At the start, the vertical component of drag is pointed downward, so it assists gravity in slowing the bullet's climb. But when the bullet passes apogee, it starts turning downward, so the vertical drag component is now pointed up and fighting the pull of gravity. You might think the two halves of the trajectory, adding to and subtracting from gravity would simply cancel each other out, but the fact the bullet is slowing and having its total drag decrease as it flies, and that this decrease is non-linear, plus the fact the arc of the trajectory bends down faster and faster as the bullet goes downrange, increasing the vertical portion of the total drag, all adds up to long range drop becoming significantly smaller than the gravitational constant and time of flight predict. You really need to shoot in a vacuum to make that work out.
For those unfamiliar with the principles, even though commonly called "wind drift," bullets aren't actually drifting in the wind due to it blowing sideways on them. Indeed, they are so fast that any air molecule that glances off the side of the bullet will do so at an extremely acute angle, meaning there is no direct sideways blowing of air on the bullet. For that to happen, the air mass would have to be both moving forward at the speed of the bullet plus be moving sideways. Instead, what happens is the side wind and the headwind from the speed of the bullet combine into a vector that is a little off straight ahead in the windward direction. Precession tilts the bullet nose into this slightly angled headwind so that drag is no longer straight back toward the gun but rather is angled like the headwind. It is the sideways component of that drag vector, due to that angle, that moves the bullet sideways. So, if the headwind is at 1250 fps, and the sidewind is at 14.7 fps, 1250/14.7 is about 85, and thus a side-directed force equal to about 1/85 of the drag on the bullet will be moving the bullet to the side.
Knowing that much, you next want to know how much drag is on the bullet. The difference in actual TOF in the air from what it would be in a vacuum tells you how much force has decelerated the bullet on its way to the target. If you know the mass of the bullet, the calculation of the average decelerating force is trivial. You would then calculate how far 1/85th of that force would move the bullet during its TOF to learn the wind deflection. Fortunately, there's the shortcut observed by Didion, which is simply to multiply the wind speed by that difference in TOF. It is all really doing the same thing, but it gives your brain a fun puzzle to see why that has to be so.
This bit of reality bedeviled F. W. Mann in his 1907 book, The Bullet's Flight…. He apparently never saw Didion's work. The poor fellow never did understand how wind drift works, and was clearly much vexed by fact that if he dropped a bullet in a crosswind from a height that gave it the same time of fall to a surface as its TOF to a target, it was blown only a fraction of an inch instead of the the several inches the bullet was deflected by wind at the target.
A number of years ago I wrote a point mass solver in Excel because I wanted to be able to see intermediate calculation results that aren't displayed in a commercial ballistics programs. One of the interesting things I hadn't considered when I started the project was that the height of apogee above the gun muzzle when firing at a positive angle of departure, plus the fall from apogee to the target would not add up the the total drop. Total drop is how far a bullet falls below a bore-sighted target when the bullet is fired truly horizontally. This is because, when you fire angled up, drag on the bullet can be divided into horizontal and vertical drag vectors. At the start, the vertical component of drag is pointed downward, so it assists gravity in slowing the bullet's climb. But when the bullet passes apogee, it starts turning downward, so the vertical drag component is now pointed up and fighting the pull of gravity. You might think the two halves of the trajectory, adding to and subtracting from gravity would simply cancel each other out, but the fact the bullet is slowing and having its total drag decrease as it flies, and that this decrease is non-linear, plus the fact the arc of the trajectory bends down faster and faster as the bullet goes downrange, increasing the vertical portion of the total drag, all adds up to long range drop becoming significantly smaller than the gravitational constant and time of flight predict. You really need to shoot in a vacuum to make that work out.
