Analysis of Variance of Firing Angles and Hit Percentages on the Battlefield

[Hancock the Superb]

Introduction:
In the American Civil War, infantry soldiers fought with rifled muskets, reasonably accurate projectile-launching weapons used to wound or kill enemy soldiers. However, in order to hit the chosen target, soldiers need to choose a correct firing angle. The primary question this study answers is at certain distances, what is the likeliness that the mean soldier can choose a correct firing angle.

Basic Assumptions:

• A male human is roughly 70.2 inches tall (5 feet 10.2 inches) and has a head that is 10 inches in height.
• A rifleman fires from about the height of his chin (as he compresses his body before firing).
• The target is an infinitely long and 70.2 inches in height, so as to isolate the vertical angle of firing.
• The bullet is an accelerating reference frame (according to Einstein’s Theory of General Relativity), thus the projectile can be assumed to travel in a straight line instead of an arc influenced by gravity.
• The chosen image supplies the firing angles of rifles chosen at random (that each angle is randomly chosen and independent of the others is a stretch, but I will make do all the same).
• The GIMP Measure Tool can approximate the firing angles (again, a stretch, but I will make do).

Firing Angles for Specific Distances:

The attachment TrigonometryforRifleAngle.JPG is no longer available

Based off simple trigonometry, the angle of error that still hits the target can be approximated.

Table of yards and degrees of error:

The attachment TableofRangesandAngles.JPG is no longer available

Variance of Firing Angle of Reenactors:

The attachment RifleAngles.jpg is no longer available

The GIMP Measure Tool was used to approximate the firing angles of each musket to the right of the smoke. Unfortunately, the Measure Tool does not indicate a positive or negative slope, and thus negative and positive values were randomly assigned to the angles. It should be understood that this may exaggerate the variance in firing angles. Running a 99% confidence interval with 13 degrees of freedom indicates that the mean firing angle is 99% likely to occur between -4.60 degrees and 2.51 degrees. Thus, it is likely that the mean infantryman will aim his rifle within 7.11 vertical degrees of the needed value to hit the target.

Chance of Correct Angle:
The chance of the mean rifleman will choose the appropriate angle to hit the target can be approximated by the needed angle divided by 7.11 degrees (the mean variance of the firing angle).

This formula suggests that a rifleman has the following probabilities of firing at the correct angle at the given distances:
10 yards – 100%
50 yards – 31.3%
100 yards – 15.7%
160 yards – 9.80%

Conclusion:
The average Civil War infantryman would be lucky to hit a target in battle conditions. At the estimated average engagement range of 100 yards (Paddy Griffith), a soldier could be expected to aim correctly only 15.7% of the time. Under battle conditions, it is not only likely for this percentage to decrease, but other factors, such as fouling of the barrel and smoke, would likely contribute to a very small chance of hitting a chosen target. Outside of 160 yards, chances of hitting a target would be very slim.

It should be noted here that this study is merely an overview of the chances of hitting a target, and it should be verified by other sources or studies which do significantly more research to determine those chances. In addition, sharpshooters with sights are not accounted for, and their accuracy should not be inferred from any equations or data derived.

[Marching Thru Georgia]

Hancock The Superb wrote:

The earth is an accelerating reference frame (according to Einstein’s Theory of General Relativity), thus the projectile can be assumed to travel in a straight line instead of an arc influenced by gravity.

Incorrect. The bullet begins to move in and arc as viewed by anyone attached to the accelerating reference frame since the bullet becomes unattached as soon as it leaves the gun. The bullet's arc is the primary reason that engagement ranges were ~110 yards.

[Hancock the Superb]

I researched this and wrote this to shed light on the game mechanics, and how regardless of range, regiments will score impressive scores fighting from 150 yards away. It seems to me that although these ranges are somewhat accounted for in the weapons.csv file (the "quality" ranges of the rifles), the curve is not significant enough. As I demonstrated, the accuracy of fire curve is quite steep, and can be approximated by a function of a/x. With all other battle factors mixed in, at 100 yards, it would still be unlikely for a soldier to hit his target, even if he shoots of 60 rounds.

Fortunately, this may be fixed by a) updating the "quality" range curve (which I believe is hardcoded into the game), or b) playing around with the elevation table in statetables.csv file to attempt to approximate the curve in combination with elevation bonuses/penalties.

[Hancock the Superb]

Hancock The Superb wrote:

The earth is an accelerating reference frame (according to Einstein’s Theory of General Relativity), thus the projectile can be assumed to travel in a straight line instead of an arc influenced by gravity.

Incorrect. The bullet begins to move in and arc as viewed by anyone attached to the accelerating reference frame since the bullet becomes unattached as soon as it leaves the gun. The bullet's arc is the primary reason that engagement ranges were ~110 yards.

The accelerating reference frame is the bullet, sorry, I miswrote. The idea is that I do not need to account for the fall of the bullet aimed at the top of the head or the bottom of the feet because the bullet will fall the same distance over a specific time regardless, which does not change the angle of error. Of course, at longer ranges the bullet will need to travel slightly farther, but those distances would need to be significant in relation to the earth, not being merely several hundred yards.

[Mayonaise]

Very interesting post. I am always trying to reconcile how a thousand men lined up with rifles don't obliterate each other in short order. I think it also helps explain why you often hear of an entire volley passing overhead. You would think there is some unconscious group think going on regarding at what elevation muskets were aimed, and/or deceiving topography causing all/most to aim high or low.

I'd also like to point out, your math assumes that the person shooting is horizontally lined up with a target. Of course, that's half the battle, and the margin of error is much higher. (your angle has to be correct for say, a few inches inches to 2 feet (the width of a person depending on if its the head, leg, chest), and not 6 feet, the height. Even a line of men has many gaps where a bullet would harmlessly pass through.

Of course countering all that is simple skill. It is not a random calculation- a skilled marksmen would far exceed your calculations.

[Marching Thru Georgia]

Hancock The Superb wrote:

The idea is that I do not need to account for the fall of the bullet aimed at the top of the head or the bottom of the feet because the bullet will fall the same distance over a specific time regardless, which does not change the angle of error. Of course, at longer ranges the bullet will need to travel slightly farther, but those distances would need to be significant in relation to the earth, not being merely several hundred yards.

Incorrect again. Your homework for today is to calculate the drop of a bullet traveling 950 ft/sec over the distances of 50, 100 and 150 yd neglecting air friction. Then calculate dTheta from this result. Tomorrow we will add air friction back in. Note, you can save yourself some work by reorienting your frame of reference so you only have to solve for one triangle.

[NY Cavalry]

If the rifle is held at the horizontal the bullets will fall 4.79 in. in 50 yards, 19.17 in. in 100 yards, and 43.13 in. in 150 yards.

We are trying to hit a 70.2 in target and looking for a angle?


Again, with the rifle at the horizontal, the bullet will hit the ground in .604 seconds at a distance of 573.8 feet with the given velocity.


If these numbers are correct why were they always told to low? Doesn't the bullet fall enough already.

[Marching Thru Georgia]

NY Cavalry wrote:

If the rifle is held at the horizontal the bullets will fall 4.79 in. in 50 yards, 19.17 in. in 100 yards, and 43.13 in. in 150 yards.

Now I need Hancock to show his work so I can see he did not copy off your paper. :laugh:
I am trying to show Hancock that one cannot neglect gravity. The angle narrows not linearly with distance but as a function of time**2.

[Blaugrana]

Again, with the rifle at the horizontal, [...] If these numbers are correct why were they always told to [aim] low? Doesn't the bullet fall enough already.

The more typical error must have been to aim rifles above the horizontal, and by too much.

[Marching Thru Georgia]

NY Cavalry wrote:

If these numbers are correct why were they always told to low? Doesn't the bullet fall enough already.

It was due to a person jerking the gun back and up just before firing. It's a very common reaction to the anticipated recoil of the gun. I use to shoot blackpowder competitively and also teach it to interested people. I would usually allow the newbie to fire a couple of rounds and then substitute a cap that I had removed the fulminate from. You could watch what happened to the end of the gun when the trigger was pulled. If they were new to shooting, the barrel inevitably came up. It was a good teaching device.