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

9 years 7 months ago - 9 years 7 months ago #1 by Hancock the Superb

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  • 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:



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

    Table of yards and degrees of error:



    Variance of Firing Angle of Reenactors:



    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.

    Hancock the Superb
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    Last edit: 9 years 7 months ago by Hancock the Superb.
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    9 years 7 months ago #2 by Marching Thru Georgia

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  • 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.

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    9 years 7 months ago - 9 years 7 months ago #3 by Hancock the Superb

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  • 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
    Last edit: 9 years 7 months ago by Hancock the Superb.

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    9 years 7 months ago - 9 years 7 months ago #4 by Hancock the Superb

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  • Marching Thru Georgia wrote: 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.

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    Last edit: 9 years 7 months ago by Hancock the Superb.

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    9 years 7 months ago #5 by Mayonaise

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  • 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.

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    9 years 7 months ago #6 by Marching Thru Georgia

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  • 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.

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    9 years 7 months ago - 9 years 7 months ago #7 by NY Cavalry

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  • 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.
    Last edit: 9 years 7 months ago by NY Cavalry.
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    9 years 7 months ago #8 by Marching Thru Georgia

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  • 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.

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    9 years 7 months ago #9 by Blaugrana

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  • NY Cavalry wrote: 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.

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    9 years 7 months ago #10 by Marching Thru Georgia

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  • 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.

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    9 years 7 months ago #11 by Jim

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  • For some additional information: CW Ballistics blog entry

    -Jim

    "My God, if we've not got a cool brain and a big one too, to manage this affair, the nation is ruined forever." Unknown private, 14th Vermont, 2 July 1863
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    9 years 7 months ago - 9 years 7 months ago #12 by born2see

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  • This is the equation that Norb uses:






    :P


    B

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    9 years 7 months ago #13 by Marching Thru Georgia

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  • So Norb is only doing a 0th order calculation. Sloppy. :laugh:

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    9 years 7 months ago #14 by Bloody 8th

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  • what is the purpose of getting this detailed. I love that you guys are this detailed but couldn't you just put at 150yds so many guys will get hit and so on... Don't kill me on this caus I am very appreciative of your attention to detail and that is what makes this game the best game ever.

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    9 years 7 months ago #15 by Willard

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


    The answer is very simple and has nothing to do with mathmatics and everything to do with human nature. Inexperienced shooters have a tendancy to shoot too high and basic marksmanship was something not worked on often enough in the ACW. More time was spent on drill and march than actually shooting rounds down range. To overcompensate with green troops, the command was to fire low minimize the problem.
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    9 years 7 months ago #16 by Garnier

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  • born2see wrote: This is the equation that Norb uses:


    Obviously the game doesn't calculate ballistics, because if it did, shots would be able to fly past the target and hit someone 500 meters away.

    Better to keep it simple and just have a factor for different ranges. But we would love to know what that factor is, since it seems to have an extremely small effect. Troops firing at 200 yards kill almost as many as at 20 yards.

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    9 years 7 months ago #17 by Hancock the Superb

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  • Marching Thru Georgia wrote: 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.


    First off, I would appreciate it if you would kindly refrain yourself from belittling my credibility; I understand that you may have significant experience in the area of projectile motion, but that does not give you permission to "teach".

    Second, I checked my work with one of my physics professors, and as I expected, my physics were sound; it is perfectly reasonable to omit gravity as long as we assume that each bullet will fall approximately the same distance (which I will proceed to show why this works). Remember, I am trying to estimate the angle of error needed to just hit the target, not find any exact angle. The statistical analysis can be compared to that of an Analysis of Variance Test (ANOVA).

    Since each of the three hypothetical bullets will travel roughly the same distance; at 10 yards the distances are different by 4 orders of magnitude, which will not influence my work with 3 significant figures. Thus, since the distances, velocities and times can be considered the same, gravity will affect each bullet the same (in terms of vertical displacement). Taking advantage of the accelerating reference frame of the bullet, each bullet can be assumed to travel in a straight line. Remember, I am not trying to find specific angles. The amount I add due to gravity to the lowest angle, I would have to add to the upper angle. So when I "subtract" the two angles (I actually bypassed the step using triangles and the law of cosines), the angle difference due to gravity is cancelled out, further indicating that using the accelerating reference frame of the bullet is a reasonable approach, considering I am merely exact to 3 significant figures.

    Hopefully this will sway readers as to why my approach can be considered accurate.

    Hancock the Superb

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    9 years 7 months ago #18 by Hancock the Superb

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  • Garnier wrote:

    born2see wrote: This is the equation that Norb uses:


    Obviously the game doesn't calculate ballistics, because if it did, shots would be able to fly past the target and hit someone 500 meters away.

    Better to keep it simple and just have a factor for different ranges. But we would love to know what that factor is, since it seems to have an extremely small effect. Troops firing at 200 yards kill almost as many as at 20 yards.


    This is what I am hoping to get at: the game models ballistics in a fashion that is potentially inconsistent with reality. Hopefully we will gain access to these factors.

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    9 years 7 months ago #19 by Marching Thru Georgia

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  • Hancock The Superb wrote:

    First off, I would appreciate it if you would kindly refrain yourself from belittling my credibility; I understand that you may have significant experience in the area of projectile motion, but that does not give you permission to "teach".

    Second, I checked my work with one of my physics professors, and as I expected, my physics were sound; it is perfectly reasonable to omit gravity as long as we assume that each bullet will fall approximately the same distance (which I will proceed to show why this works). Remember, I am trying to estimate the angle of error needed to just hit the target, not find any exact angle. The statistical analysis can be compared to that of an Analysis of Variance Test (ANOVA).

    Since each of the three hypothetical bullets will travel roughly the same distance; at 10 yards the distances are different by 4 orders of magnitude, which will not influence my work with 3 significant figures. Thus, since the distances, velocities and times can be considered the same, gravity will affect each bullet the same (in terms of vertical displacement). Taking advantage of the accelerating reference frame of the bullet, each bullet can be assumed to travel in a straight line. Remember, I am not trying to find specific angles. The amount I add due to gravity to the lowest angle, I would have to add to the upper angle. So when I "subtract" the two angles (I actually bypassed the step using triangles and the law of cosines), the angle difference due to gravity is cancelled out, further indicating that using the accelerating reference frame of the bullet is a reasonable approach, considering I am merely exact to 3 significant figures.

    Hopefully this will sway readers as to why my approach can be considered accurate.

    As an actual physicist, I can tell you that what you have been writing is shear nonsense. Gravity does not act linearly as you are trying to assume, but as the square of the time the projectile has been in flight. It cannot be omitted. Until you grasp this fact, you are simply digging the hole you are in deeper. I must say that trying to invoke general relativity, incorrectly, was original but nonetheless wrong.

    If you want to make arguments as to why very few people were hit at distances of 100 yd. or more while in battle using qualitative arguments such as smoke, lack of training, nerves, etc. that's fine. Historians do those sort of things all the time. But when you make quantitative calculations, you need to know what you are doing. No one can check what effect smoke had on accuracy 150 years ago. But someone can always check your arithmetic.

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    9 years 7 months ago - 9 years 7 months ago #20 by Hancock the Superb

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  • Marching Thru Georgia wrote: As an actual physicist, I can tell you that what you have been writing is shear nonsense. Gravity does not act linearly as you are trying to assume, but as the square of the time the projectile has been in flight. It cannot be omitted. Until you grasp this fact, you are simply digging the hole you are in deeper. I must say that trying to invoke general relativity, incorrectly, was original but nonetheless wrong.


    As I have evidently been unable to communicate: I have abandoned gravity all together. The downward trend of the bullet in the screen is from firing at below a horizontal angle. The effect of gravity is not accounted for because as I showed in the previous post, the time it takes for each of the three bullets to hit the target is the same to three significant figures, thus the effect of gravity (verticle displacement) is the same on each bullet to three significant figures. So when I find the difference of the angles, the angle added due to gravity (as the projectile takes an parabola instead of a line) is cancelled on both sides of the equation.

    The accelerating reference frame is merely another justification as to why each bullet travels in a straight line in relation to the other bullets. I may be wrong in invoking this justification, but in terms of what I have learned, it is used correctly as long as it is to only evaluate the displacement of the other bullets.

    Look, I am not doubting that you are correct: gravity plays an important role in finding the angle of error. I am trying to explain why my way of calculating the angle accounts for gravity in a different fashion than yours.

    You are extremely well qualified as a working physicist; however, my physics professor is just as qualified and believes my model accurately captures the angle of error that my study is focused on.

    1/2at^2 = 1/2at^2 regardless of initial velocities as long as the times are the same. Total displacement = v0t + 1/2at^2. I am subtracting the 1/2at^2 from each equation because it affects the displacement the same. So for a downward fired bullet, dx<0. For a horizontal bullet, dx=0. For a upward fired bullet, dx>0. In my several years of experience in the realms of physics and calculus, the equations used and the math applied are correct. If anyone has different equations and demonstrates a different answer than mine, I am an avid learner. I am the first to say that bullheadedness hinders learning.

    Hancock the Superb
    Last edit: 9 years 7 months ago by Hancock the Superb.
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    9 years 7 months ago - 9 years 7 months ago #21 by KG_Soldier

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  • Hancock the Superb: "SCOREBOARD"

    Get him Hancock.

    This debate is way more interesting that listening to Obama and Romney spout talking points.

    In the red corner. . . weighing 135 pounds, High School Prodigy, cross country runner, HANCOCK THE SUPERB!

    In the blue corner. . . weighing god knows what, the Old Physicist from Western Colorado, where the altitude affects how quickly water boils, MTG!

    And for the record. . . I have a BA in History and English and a MFA in fiction, so I have no idea what the hell you guys are talking about.

    BUT at one point in my life, I did pass College Algebra, and I vaguely remember something about crossing out the same stuff on both sides of the equation, so I have the High School Prodigy up 1-0.
    Last edit: 9 years 7 months ago by KG_Soldier. Reason: remembered the cross country running bit
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    9 years 7 months ago #22 by Hancock the Superb

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  • Humorous post, KG! :laugh: About guessed my weight too. 130 lbs and 6 ft tall. Prodigy, likely not. But if I ever win a Nobel Prize in any field, I'll let you consider me one (a rather unlikely event).

    Perhaps a little biased for a scorekeeper? :)

    MTG deserves credit (his way definitely works); I am probably not communicating my mathematics in a coherent fashion.

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    9 years 7 months ago #23 by JC Edwards

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  • Hancock the Superb wrote: I am probably not communicating ..... in a coherent fashion.

    Since when HAVE you EVER communicated in a coherent fashion?! :P

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    9 years 7 months ago #24 by Barrow

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  • >1/2at^2 = 1/2at^2 regardless of initial velocities as long as the times are the same. Total
    >displacement = v0t + 1/2at^2. I am subtracting the 1/2at^2 from each equation because it
    >affects the displacement the same. So for a downward fired bullet, dx<0. For a horizontal
    >bullet, dx=0. For a upward fired bullet, dx>0. In my several years of experience in the
    >realms of physics and calculus, the equations used and the math applied are correct. If
    >anyone has different equations and demonstrates a different answer than mine, I am an avid
    >learner. I am the first to say that bullheadedness hinders learning.

    If your target were hanging from a tree limb, and you knew he would let go of the tree limb the precise
    moment you pulled the trigger, then gravity would cancel out for both your projectile and his fall. The dx of your bullet would match the dx of the falling target. So in that case the best option is to aim directly at the target. If you are aiming at a target that is standing on the ground gravity does not cancel out. The ground is negating the dx of the target while the projectile is in motion.

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    9 years 7 months ago #25 by born2see

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  • B,

    THAT is an example even I can understand! Thanks for that.

    B

    :P

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    9 years 7 months ago - 9 years 7 months ago #26 by Garnier

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  • So in that case the best option is to aim directly at the target


    Only if your gun is level with the tree limb relative to the earth's surface.

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    Last edit: 9 years 7 months ago by Garnier.

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    9 years 7 months ago #27 by Marching Thru Georgia

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  • Hancock The Superb wrote:

    the angle added due to gravity (as the projectile takes an parabola instead of a line) is cancelled on both sides of the equation.

    And therein lies your problem. There is nothing on the other side of the equation to cancel. The target is motionless, it is not growing into the ground while the bullet is in flight. From the perspective of the bullet, the target is actually accelerating upward. At 100 yd. if you aim between the target's knees and feet, you will miss.

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    9 years 7 months ago #28 by Barrow

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  • Garnier wrote:

    So in that case the best option is to aim directly at the target


    Only if your gun is level with the tree limb relative to the earth's surface.



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    9 years 7 months ago #29 by Garnier

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  • What seemed to me like common sense was wrong. Good show.

    Play Scourge of War Multiplayer! www.sowmp.com
    Also try the singleplayer carryover campaign

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    9 years 7 months ago #30 by born2see

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  • "I observed, that though we are satisfied his doctrine is not true, it is impossible to refute it. I never shall forget the alacrity with which Johnson answered,... -- "I refute it thus.""


    James Boswell - Life of Samuel Johnson


    B

    "Those in whose judgment I rely, tell me that I fought the battle splendidly and that it was a masterpiece of art.” - George McClellan to his wife describing the battle of Antietam

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