Shooting On Slopes or, The Legend of Cosine-Range Mike Brown, MSIT (Mad Scientist In Training) 9 April 2001 DRAFT for REVIEW "...but many men who used to have a high reputation are no longer taken seriously now that their works have been printed. Some people put their ignorance in print, passing it off as wisdom..." Lope De Vega 1562-1635 The above quote, taken from the Sniper s Paradise web site (http://www.snipersparadise.com/) is a serious caution to those who would write as experts on any topic. I acknowledge my risk in challenging the knowledge of men who are vastly more experienced than I. Fools go in where angels fear to tread. may be an appropriate quote. The Problem It is well-known that shooting uphill or downhill may require a correction to the aiming point or to the sight elevation to insure a hit. All knowledgeable shooters acknowledge that these shots will overshoot the target unless corrected. My study of this topic indicates that the nature of this correction is poorly understood by some who use it professionally. Worse, it appears that an imprecise method is being taught to tactical operators whose lives may be at risk in using it. Where a first-shot hit defines mission success, no shortcuts should be permitted which compromise accuracy. Bad training doctrine must be rooted out and destroyed. The imprecise method for slope correction is to apply the cosine of the slope angle to the true range to the target and put elevation for that corrected range on your sights. I dub this the cosinerange method. It is at best an approximation of the exact correction. Unfortunately, it seems to be taken as gospel in some circles. It is the basis of the slope-doping function in the Mil-Dot master, and has been published as gospel by several authors. In a business where MOA accuracy is called for, the use of approximations that introduce perhaps several MOA errors does not seem appropriate. I should note here that I criticize this method from the comfort of an armchair, not being a tactical shooter of any nature. I am merely a devotee of the art and science of precision shooting. To those who would say It works, I ask Under what conditions?. I shot a gnat on a 50 degree slope does not constitute an analysis. What assumptions and limitations must you accept before deeming the method close enough? How many Maxwell Smart shots are acceptable to you? Would you rely on something which only works in a limited set of circumstances? I think I can demonstrate cases where the method fails, and fails badly, even if everything else is ideal.
It seems to me that having two methods to correct for the same problem, one of which is an approximation and the other quite precise, is an invitation for the shooter to use the imprecise one at the wrong time. Learn the correct method and master it and never be in doubt. One printed source will be cited which advocates the imprecise method, and an internet source referenced. The printed source is Michaelis s The Complete.50 Caliber Sniper Course - Hard Target Interdiction. The SniperCountry.com website has a piece by K. ooch which also states the imprecise method. http://www.snipercountry.com/hottips/slope_shooting.htm is the particular URL I found. I infer from ooch s remarks that the book The Military and Police Sniper by Lau may also state this imprecise method. I admit to not having read Lau s book. This critique is not directed at that work - unless, of course, it is wrong! I ve seen other web sites espousing the cosine-range correction, but can t recall them at this time. The SniperCountry review of the Mil-Dot Master has a photo clearly showing the cosine-range relationship. The Legend I think that the legend of cosine-range traces back to one common origin, which had a set of constraints over which the cosine-range approximation is reasonable. Those constraints have become lost in the retelling, and the legend of cosine-range as exact was born. I infer from Plaster s The Ultimate Sniper that the original source may have been an FBI course. I speculate that one of the lost constraints was the use of a range-finding scope where elevation is put on, not by elevation knob, but by calibrated reticle, where the reticle is calibrated with holdovers for various ranges for a particular load. In that case, the only (or the fastest) way to correct for elevation is to hold to a different range crosshair. This is a situation which justifies an approximation, if nothing else is available. The other lost constraint, or fact, is the fact that this is an approximation! This method will get the shooter more-or-less close, if the range and/ or slope is not extreme. It swaps overshoot (uncorrected) for undershoot (cosine-range corrected). Plaster also states that it is not to be used for hostage-involved situations. In other words, it isn t accurate enough if someone s life depends on it! This admonition is repeated in the Mil-Dot Master instructions. Plaster s work has tables which have correct drop and compensation information in them. The compensation coefficients are extracted from the Sierra Reloading Manual. He does not get analytical as to their derivation, but it would seem, from reverse engineering the values, that the theoretical derivation of the tables is sound. The correction factors for slope printed there are the values of (1- Cos θ) for slope angles θ. These corrections may be applied to the bullet drop table values, or may be applied directly to the elevation derived from the drop tables. The values derived are to be taken off (comedowns) the elevation for the range involved. A Picture is Worth... The sketch below (Figure 1) depicts the slope problem, and implies the proper solution. The proper elevation for a slope shot is shown to be the level elevation times cos θ. Alternatively, you can view it as a comeoff problem and use (1 - cos θ) as a correction factor. I dub the method
X Line of bore Drop (t) Line of Sight Range, R X(t) At some time t the bullet would go through point X(t), except for the influence of gravity. If point X(t) is directly above the intended point of impact, by exactly the amount drop(t), a hit will be scored. Note that the angle of elevation - the comeup is greatly exaggerated in these sketches. Drop(t) * cos 45 X(t) Missed it by *that* much! - Maxwell Smart Line of bore Drop (t) Line of Sight Range, R horizontal On the 45 uphill shot, with nothing changed, bullet goes through point X(t), and a little later through X. The drop, however, is now not at a near right angle with respect to the LOB. Only a fraction of the drop is in a direction which brings the bullet to the LOS. That fraction is Cos θ. The delay between X and X is due to that fraction of the drop which opposes the velocity of the bullet along the LOB. The time of flight to X is at most a small fraction larger than the TOF to X. This sketch is dimensionally identical to the horizontal case, merely rotated through 45
cosine-drop in contrast to cosine- range The approximate solution is to use R*cos θ as the range for determining elevation. I take no credit for the derivation of the cosine-drop correction. Plaster, and apparently Sierra, among others, have published this information (Plaster without analysis). Michaelis is explicit in using R*cos θ as the Slant Range, and carrying out the calculation to 5 decimal places. According to his method, the elevation for this slant range are interpolated from the ballistics data tables, corrections for barometric pressure and temperature are applied, the wind drift is corrected for altitude and temperature, and then - perhaps a miss! That is, if the target is at sufficient range, or the slope is steep enough. I use quotes on slant range because what he s actually describing is the horizontal component of the true range. The problem here is that both Michaelis and ooch (and many others, apparently) are applying a linear correction at a point in the problem which is quadratic in nature. This approximation is possibly adequate for shorter ranges, or mild slopes, but will overcorrect for the slope problem. As with any approximation technique, it holds over a limited set of conditions, and degrades, possibly rapidly, when the conditions are not within the prescribed envelope. The range-based correction appears to be based more on coincidence or convenience than on science. The Analysis Let me explain. What we are correcting for in this problem is the fact that the drop of the bullet is not at right angles to the line of sight when we are shooting uphill or downhill. Drop is drop and it is always straight down. The drop will be identical whether or not the target is level with the shooter, if the target is at the same true range. Now, drop is a function of t 2, (time, t, since the shot was fired). Neglecting air resistance in the vertical dimension, Drop = -16.1 *t 2, where Drop is in feet and t is in seconds. This is merely the free fall drop under the acceleration of gravity. The - sign indicates that drop is always downward. This means, e.g., the drop at t= 400 milliseconds is 4 times as great as the drop at t=200 milliseconds. Further, since range is, to a first approximation (neglecting wind drag), directly proportional to time, the drop may be asserted to be proportional to the square of the range. Drop is actually proportional to range (2+x), because it takes longer to get to any particular range than the initial velocity would indicate. The correction published by Michaelis and others, and believed by ooch and others, attempts to make the following relationship true: elevation(r) * cos θ = elevation (R * cos θ), and it just ain t so! It isn t true because the time of flight to the two ranges is different, and with a shorter TOF involved in the slant range elevation, drop which really does occur is not being accounted for, or at least not properly accounted for. The drop which must be corrected on a slope shot is Drop (Range, slope) = Drop (Range, no slope) * cos θ
This is the drop at some slope theta, and range R. It is a function of the drop at that same range, R when the shot is horizontal, and the cosine of the elevation angle. If the elevation is proportional to the drop - and it is - then the correct elevation for a slope shot at range R is elevation (Range, Slope) = elevation (Range, no slope) * cos θ Note that what must be corrected is a drop, not a range! There is some range shorter than R which will give precisely the correct elevation for (R, θ)-butthe commonly published one is not it! It is much easier to put the right elevation on the sight than it is to screw around with finding the elevation for a made-up range which has no real bearing on the problem at hand. In defense of my analysis, Art Pejsa s Modern Practical Ballistics states explicitly that the cosine-range method is wrong. His analysis proceeds along a slightly different tack than mine, but his conclusion and mine are compatible, I think. The Analysis for an Airless World Drop = --- R ----- 2 2 V 0 In an airless world, the time of flight to range R is R/Vo, where Vo is the muzzle velocity. The drop to a target at range R is given as. This is simply the gravitational drop at the time that the projectile reaches the target. If we take the ratio of the drop at the cosine-range (drop(rcos θ)) to the cosine-drop (drop(r)*cos θ), we have D1 ------ D2 --- --------------- Rcosθ 2 2 V 0 = ------------------------------- R --- ----- 2 = cosθ cosθ 2 V 0 This demonstrates that the approximate drop is always less than the actual drop unless theta is equal to zero. In other words, you aren t going to put enough el on your sight using the approximation. Whoa! This analysis is for an airless world. What about the real world? A World With Air Art Pejsa derives the equation for drop as ----- V 0 Drop = ------------- 1 1 -- --- R F
,where is gravity combined with a scaling coefficient (not the plain acceleration of gravity), Vo is muzzle velocity as before, R is range in yards and F is a retardation coefficient related to air density and the ballistic coefficient, value on the order of 3000 for common bullet. Taking the ratio again (this time of the square roots), we get ----- Drop1 ---------------- Drop2 = V 0 ------------------------- 1 1 --------------- --- Rcosθ F ------------------------------ ----- V ------------- 0 cosθ 1 1 -- --- R F After some tedious algebraic simplification the fraction can be shown to be (this is the part where the proof is left as an exercise to the student! I m not going to type all that stuff!): Drop1 ---------------- Drop2 = F 2 Rcosθ FR 2 cosθ ------------------------------------------------------ F 2 Rcosθ FR 2 cosθ cos 2 θ When the above equation is squared, you get an expression for the relative error in terms of F, R, and θ. Without going through that drudgery, I will remark that this solution has the satisfying notion (to me, at least) of being of the same general form as the ideal case solution. The drop will be some fraction of cos theta, the fraction being a function of R, theta, BC, and air density. In the fraction above, the numerator is always smaller than the denominator, except in the case of theta = 0, where they are equal. Thus the fraction is never greater than 1. Thus the approximation drop is always less than the actual drop, as in the airless case. We can t analyze how much less than 1 unless we know F, but the point is that in the real world, with air drag effects included, there s an error in the cosine-range approximation. For Example: As an example of the difference between the methods, let s look at the situation where the target is at 1000 yards, at 25.8 inclination. I pick that inclination because the cosine is an even 0.9 Lets use the drop table data for M118 ammo from Plaster s work (page 122). The 1000 yard drop is 395 inches, the 900 yard drop is 293 inches, or 31.1 MOA at 900 yards. The choice of 25.8 permits direct use of the 900 yard range data in the cosine-range method for ease of comparison. 90% of the 1000 yard drop is 355.5 inches, or 34 MOA at 1000 yards. That s a long way from 31 MOA - about 32 inches low at 1000 yards. A shot at a man using the cosine-range method would result at worst in a leg wound.
Other Factors Now, what about MET(meteorological) and ENV (environmental) (as Michaelis calls them) corrections? When do you put these on? It seems to me that the horizontal range elevation must be adjusted for MET and ENV conditions before the slope corrections are applied. Air density corrections are effectively adjusting for different time of flight to the target, and that altered time of flight applies on slope shots as well as level ones. The correct calculation sequence is elevation (R, θ, MET) = (elevation(r,0) MET adjustments) * Cos θ Does This Mean I Have to Throw Away My Mil-Dot Master? To the large community out there who have invested in the Mil-Dot Master, there is good news! It is still usable for the cosine-drop correction. There s a bit or reordering of the ranging procedure required. I m winging it here, because I don t have a Mil-Dot master in my hands. From the target size and Mil subtension, find the true range (slope = 0) to the target. From your data tables, find the drop at that range. Apply any MET and ENV factors to that drop. Enter the drop on the calculator as a range, at the 0 slope, and read the corrected drop (on the range scale) at the appropriate target slope. Take that corrected drop to the MIL-Minute conversion window and put the correct (true) range back on the 100% mark. Read the MOA/MIL correction opposite the corrected drop value. Conclusion I ve shown that the drop correction for a slope shot is a rather simple trigonometric function based on the drop to a horizontal target. The complex calculation involved is the drop calculation, not the correction calculation, and Pejsa s work grossly simplifies the drop calculation. I ve shown that this cosine-drop corrected drop is always larger than the drop that is arrived at from the cosine-range correction. This means that a rifle corrected to the cosine-range drop will shoot low. I ve shown how to use the Mil-Dot Master to apply the cosine-drop correction. I think the above constitutes a better mousetrap. Your opinion may differ.