INCLINED FIRE

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INCLINED FIRE William T. McDonald

June 2003

"Inclined fire" is a shooting situation in which a gun is first sighted in at a shooting range that islevel or nearly level, and later must be fired at a target located either upward or downward atsome inclination angle and some slant range distance between the shooter and the target. Anupward inclination angle is called an elevation angle, and a downward inclination angle is calleda depression angle. This is the sort of situation that sometimes confronts a hunter inmountainous terrain with steep hillsides and deep ravines. It is a frequent situation for a soldier ormarine in urban combat, or a sniper shooting at distant targets. It also occurs for a lawenforcement officer or a SWAT (Special Weapons and Tactics) team member in an urban venue.

In such a situation a gun will always shoot high compared to where it shoots on a level trajectory.How high the gun will shoot varies with both inclination angle and slant range distance to thetarget. The problem facing the shooter is how to determine how much the impact point of thebullet will change at the inclination angle and slant range distance, and then to adjust his or heraim so that the target is hit. Adjusting the aim can take place in either one of two ways. A hunterwill usually aim a little lower for an inclined shot than he or she would aim for a horizontal shot, inorder to compensate for the gun shooting higher. A soldier or marine in urban combat wouldlikely do the same, because shots at moving or partially concealed targets must happen quickly.But a military or SWAT team sniper in a situation where there is more time to prepare for the shotwould probably dial a come-up or come-down on his telescope sight, after calculating the correctaiming adjustment.

To use either of these techniques the shooter must know where the bullet will be relative to theline of sight through the sights on the gun at the slant range distance to the target. In otherwords, the shooter must know what we call the Bullet Path, that is, the distance between the lineof sight and the position of the bullet at the slant range distance of the target. Bullet Path isalways measured PERPENDICULAR TO THE LINE OF SIGHT . It is positivewhen the bullet is above the line of sight, and negative when the bullet is belowthe line of sight. A convenient mental picture of Bullet Path is where the shooter would "see"the rear end of the bullet, if he or she could watch the bullet through the gun sight as it flies thetrajectory. We make a careful distinction between Bullet Path and Bullet Drop. Bullet Drop isthe VERTICAL distance between the extended bore line of the gun and theposition of the bullet, at the slant range distance of the target. It is very important to

keep in mind that Bullet Path and Drop are not the same, and the shooter needs to know BulletPath, not Drop, in order to aim the gun.

Of course, in any shooting situation the first step is to measure or estimate the inclination angleand the slant range distance to the target. There are instruments to measure both theseparameters, if the shooter (or a team mate) has time to use the instruments in the shootingsituation. Otherwise estimates must suffice. But, knowing these parameters, the shooter alsomust have some means of computing the correct aiming adjustment. And unfortunately, acomputation is absolutely necessary for accurate fire when the angle is steep and/or the rangedistance is long.


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The Sierra Infinity exterior ballistics computer program will calculate the trajectory of the bullet forany inclination angle. Given the slant range distance to the target, the software then provides thecorrect aiming adjustment (the come-up or come-down) to place the bullet on the target. Thistrajectory calculation is made considering that the gun has been sighted in on a level range.However, in all but a very few field situations it is just not practical for the hunter, military person,SWAT team, or sniper team to carry a computer for this purpose. So, a method is necessary tocalculate approximate but reasonably accurate aiming adjustments on something no larger thana hand-held engineering calculator, that is, a calculator that performs arithmetic and trigonometriccalculations.

Any error made in calculating or estimating the aiming adjustment for inclined fire will besystematic rather than random in nature. What this means is that, for any given situation (a giveninclination angle and slant range distance to the target), the aiming error will be the same forevery round fired, unless the shooter makes some further aiming adjustments between shotsbased on observed misses. But in most shooting situations for hunters, military, or lawenforcement personnel, the first shot must count; there is little chance of a second shot to correcta miss observed on the first shot.

Of course, some aiming error is tolerable, depending on the shooting scenario. For a hunter after,say, deer in mountainous terrain, inclination angles can be quite steep, but generally not morethan about 45o, and slant range distances can be up to around 400 yards. The hunter typicallywould like to place his bullet within about an 8-inch circle just behind the front leg of the deer tohit the heart or lung and put the deer down. This limits the tolerable error in bullet placement toabout 4 inches vertically on the skin of the deer. That error is multiplied by the cosine of theinclination angle when viewed by the shooter, so that the aiming error after the aiming adjustmentcan be no more than about 3 inches at the slant range distance from the hunter to the animal.

In an urban warfare venue a soldier can be on a street engaging enemies in windows in highbuildings or on rooftops, or vice versa. In such cases inclination angles can be very steep andpartially concealed targets very small, but ranges are usually short, often less than 100 yards.The Bullet Path inaccuracy cannot be more than about 2 inches in such situations. Lawenforcement SWAT teams typically have similar situations in urban environments, but sometimeswith longer slant range distances and inclination angles not quite so steep. However, accuracyrequirements are even tighter because of hostages or bystanders. The error in the calculatedaiming adjustment cannot be more than about 1 inch in such situations.

There are at least three methods of calculating approximate aiming adjustments in use presently.The purpose of this article is to evaluate the accuracy of each of these methods by comparingthe aiming adjustment (come-up, come-down, holdover, holdunder) resulting from each methodwith the adjustment calculated by Sierra's Infinity program. We have designed the trajectorycalculations in Infinity to be highly accurate, so we believe it is a suitable reference to evaluatethe accuracy of each approximate method. We begin first by explaining what Infinity does, whichalso will serve to explain the problem of inclined fire graphically. Then, we will proceed to theaccuracy evaluation for a specific rifle and cartridge combination and some specific shootingscenarios.

A shooter begins by sighting in his or her gun on a nearly level (horizontal) shooting range, asshown in Figure 1 below. Sighting in simply makes the bullet trajectory cross the Line of Sight(LOS) at the zero range, which is the position of the target. The barrel of the gun is tilted upwardby a small superelevation angle to make this happen. This angle is typically less than 1.0 degree


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for modern cartridges and practical range distances. Infinity very accurately calculates thesuperelevation angle and all the trajectory parameters for this level fire situation. Bear in mindthat all the figures below are not to scale and greatly exaggerated to illustrate the points ofdiscussion.

Figure 1. Level Fire Trajectory for Sighting In

For inclined fire, the Infinity user points and clicks on the "Up Hill - Down Hill" operation in the"Operations" drop-down menu and enters an inclination angle. Suppose the user enters apositive inclination (elevation) angle. Infinity will then calculate a trajectory for the situationillustrated in Figure 2.

The horizontal axis is rotated upward by the elevation angle, and the vertical axis is also rotatedby the same angle. The line of sight follows accordingly; the Sight Height is constant, becausewhen the shooter aims uphill the gun sights are the same distance above the bore line as theywere in the horizontal situation. And with no changes in the gun sights the LOS crosses therotated horizontal axis at the same zero range distance from the shooter, although the elevated

trajectory is no longer zeroed at this point.


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Figure 2. Inclined Trajectory Situation

For this situation Infinity uses the same superelevation angle that was calculated for the level firetrajectory. Infinity very accurately calculates the elevated trajectory, including Drop and BulletPath. Both Figure 2 and Figure 3 show the Drop, Bullet Path, and height of the LOS at a givenslant range distance R. Figure 3 is a close-up view of these trajectory parameters at R to more

clearly illustrate the relationships. Again bear in mind that these figures are greatly exaggeratedfor illustrative purposes. Drop, Bullet Path, and LOS Height are measured in inches orcentimeters, while range distances are measured in hundreds of yards or meters.

According to the conventions that we use, Drop is always measured in the VERTICAL direction,because almost all targets stand vertically; Bullet Path is measured PERPENDICULAR TO THELOS; and the LOS Height is measured perpendicular to the Rotated Horizontal Axis, as moreclearly depicted in Figure 3. The figures exaggerate the relatively small differences in the slantrange distances from the shooter to the intersection points on the Inclined Trajectory and theother rotated lines in Figures 2 and 3. For example, if a shooter wishes to place a bulletaccurately on a target at a slant range distance R along the line of sight, from Figure 3 the

shooter would simply use a come-down equivalent to the Bullet Path calculated by Infinity. At aslant range

Figure 3. Close-up View of Inclined Trajectory at Slant Range R

distance longer than the point at which the trajectory crosses the line of sight, Infinity would givea negative Bullet Path and the shooter would use the equivalent come-up.

As stated earlier, Infinity is very accurate but impractical for use in the field. So, a practicalmethod of calculating aiming corrections is needed. Three such methods are currently used.Each method will be described and its accuracy will be evaluated below. First, however, thiswriter would like to especially acknowledge and thank three individuals who have motivated andcontributed to the information in this article. The first is Mr. Lou Schweibert, President ofSchweibert Precision (Ballisticard SystemsTM www.ballisticards.com) of Atascadero, CA, U. S.


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A. Mr. Schweibert has corresponded with this writer about several topics in ballistics and haspermitted this writer to review a paper of his concerning the first method. The second person is

Mr. Ruben Nasser of Asuncion, Paraguay, who has corresponded with this writer about severalballistics topics including inclined fire, initially described the second method, and introduced thiswriter to the third individual. This third person is Mr. Mike Brown of Austin, TX, U. S. A. Mr. Brownhas written a very interesting paper (as yet unpublished in June 2003) analyzing both the firstand second methods, and also has corresponded with this writer. All three of these individualshave made substantial contributions to the knowledge base of this writer concerning inclined fire,and this writer is very grateful.

First Method for Aiming Adjustment: Rifleman's Rule

No one seems to know who invented the first practical method of calculating aiming corrections

for inclined fire. It has been known for a long time by names such as "Rifleman's Rule," "QuickFix," and perhaps others. Analyses of this method appear in the book Modern ExteriorBallistics by Robert L. McCoy, and also in the unpublished paper "The Rifleman's Rule -Revisited" by Mike Brown. These analyses are based on bullet trajectories in a vacuum (noaerodynamic drag), because in a vacuum the trajectory variables of a bullet can be expressed byclosed form mathematical equations. This makes the problem amenable to pencil and paperanalysis, but substantial errors can be expected because a vacuum trajectory is really a grossapproximation to the actual trajectory of a bullet, which cannot be expressed in closed formequations.

The Rifleman's Rule (RR) Method derived by these analyses is the following:

1. Measure the inclination angle of the target above or below the horizontaldirection.

2. Measure the slant range distance to the target.3. Multiply the slant range distance by the trigonometric cosine of the

inclination angle (this gives the horizontal projection of the slant range).4. Use the Bullet Path (or come-up or come-down) from the level trajectory at

this horizontal projection distance to adjust the aim for the inclined target.

In other words, pretend that the inclined target is at a horizontal distance equal to the slant rangedistance multiplied by the cosine of the inclination angle, and aim as if the target were really atthat horizontal position.

Second Method for Aiming Adjustment:Improved Rifleman's Rule

The second method of calculating aiming corrections for inclined fire has been called the"Improved Rifleman's Rule" in the paper by Mike Brown. The inventor of this method also is


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unknown, although Mr. Brown and Mr. Nasser cite Mr. Tiro Fijo of Paraguay for suggesting themethod. In his paper Mr. Brown has analyzed the method, again based on bullet trajectories in avacuum.

This Improved Rifleman's Rule (IRR) Method is the following:

1. Measure the inclination angle of the target above or below the horizontaldirection.

2. Measure the slant range distance to the target.3. Take the Bullet Path (or come-up or come-down) from the level trajectory

at a horizontal distance equal to the slant range distance, and multiply thatparameter by the cosine of the inclination angle.

4. Use the result of this calculation to adjust the aim of the gun at theinclined target.

Third Method of Aiming Adjustment: Sierra's Approach

The third method of calculating aiming corrections for inclined fire was discovered by this writerand has been published in the Sierra Bullets Reloading Manuals since about 1985. It is notknown with certainty that this method originated with this writer, but it was discoveredindependently. It was based on an observation made with the aid of the exterior ballisticssoftware programs that Ted Almgren and this writer created at that time. The observation was(and is) that bullet Drop at any given slant range distance on inclined trajectories changes verylittle from the Drop at the equivalent horizontal range distance on a level trajectory.

This observation in turn led to the third method of calculating aiming corrections for inclined fire:1. Measure the inclination angle of the target.2. Measure the slant range distance to the target.3. From the level trajectory at a horizontal distance equal to the slant range distance, take

both the Drop and Bullet Path.4. Change the algebraic sign on the Drop number (because Drop is always negative in

Infinity change it to a positive number). Then, multiply this positive number by the quantity[1.0 - cosine (inclination angle)].

5. Algebraically add the result of step (4) to the Bullet Path (remembering that Bullet Pathcan be either a positive or negative number in Infinity) to obtain an adjusted Bullet Path.Then use this result to adjust the aim of the gun at the inclined target.

Accuracy Evaluation of the Methods

The accuracy of each of the three methods will be evaluated by comparing the aiming adjustmentproduced by the method with the aiming adjustment calculated by Infinity, which will be taken asthe "true" aiming adjustment. A particular rifle, cartridge, and shooting situation have beenchosen to facilitate the accuracy evaluation. The rifle is a bolt action chambered for the 7.62 x 51mm NATO (308 Winchester) cartridge loaded with Sierra's 30 caliber 168 grain Hollow Point BoatTail Match bullet at 2650 fps muzzle velocity. The telescope sight will be assumed to have a


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large objective lens, so that the Sight Height (distance between the centerline of the telescopeand centerline of the bore) will be taken as 1.75 inches. The firing point altitude will be set at3000 ft above sea level, because we will evaluate accuracy for depressed fire as well as elevatedfire. The atmospheric conditions will be the Standard Metro conditions adjusted to the firing pointaltitude, and with no winds. It will be assumed also that the rifle is sighted in at a zero rangedistance of 100 yards on a level range at 3000 ft altitude and with the same atmosphericconditions, so that the comparisons for inclined fire will not need to be adjusted for changes inaltitude or weather conditions. The accuracy evaluation will be performed to a maximum slantrange distance of 1000 yds from shooter to target, although this range is very improbable atsteep angles.

This evaluation will be carried out in English units rather than metric units, with apologies toreaders accustomed to metric units. Table 1 lists Drop and Bullet Path parameters for the levelfire trajectory. These parameters are used for the computations in the three methods of aimingadjustments. A high degree of precision is used for all the parameters discussed in thisevaluation (tenths of an fps, hundredths of an inch, hundredths of an MOA). The reason is thatwe will be comparing large numbers to arrive at small differences. Of course, we are well awarethat the value of one "click" on a telescope sight is 1/4 MOA, sometimes 1/8 MOA, but the moreprecise numbers allow more precise comparisons.

Remaining Velocity, while not required for the evaluation, is listed for reference purposes in Table1. Actually, Remaining Velocity at any given range distance changes very little with inclinationangle.

For example, at an elevation angle of 60o the Remaining Velocity at 1000 yds slant rangedistance is 1175.1 fps, just 2.5 fps faster than the 1172.6 fps at 1000 yds horizontal rangedistance for the level fire case. For a depression angle of 60o the Remaining Velocity at 1000 ydsslant range distance is 1170.6 fps, just 2.0 fps slower than for level fire. These differences arecaused primarily by the changes in air density as the bullet climbs or falls. The 60o inclinationangle is a worst case (among the four inclination angles considered) for the terminal velocitydifferences; the differences are smaller at any given range distance for lower inclination angles.

As Tables 1 through 8 show, the range distances chosen for the evaluation are grouped nearerto the firing point and then farther from the firing point, with a single range distance in the middle


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at 500 yds. This is because the aiming errors are most important at shorter ranges for someshooting situations, while in other situations they are very important at longer ranges. Thesechoices also show how the aiming errors for each method vary with range distance. Note that wehave chosen one range distance, 875 yds, for the tables because the maximum effective rangefor the sniper rifles presently used by the U.S. Army (M24) and U.S. Marine Corps (M40A3), bothusing the 7.62 x 51 mm NATO cartridge, is 800 meters (875 yds).

Concerning the Bullet Path parameter in Tables 1 through 5, a positive valuedenotes that the bullet is above the LOS, while a negative value shows that thebullet is below the LOS. If the Bullet Path is positive at some given range distance, a come-down is necessary to lower the bullet impact point to the LOS. If the Bullet Path is negative atsome range distance, a come-up is required to raise the bullet impact point to the LOS. In all thetables a come-down is shown as a negative come-up, and a notation (Dn) is placed beside thenumerical value to signify that it is a come-down. To the right of each Bullet Path column in eachtable is a column listing the corresponding come-up in MOA (Minutes of Angle). If the Bullet Pathvalue is negative for some given value of range distance, then with no change in the rifle sightsthe bullet would pass below the LOS by the listed number of inches. A come-up adjustment tothe rifle sights by the number of MOA listed in the column to the right of the Bullet Path then is

necessary to move the bullet impact point upward to the LOS at the given range distance.


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Tables 2 through 5 compare the trajectories computed by Infinity with the aiming adjustmentscalculated using each of the three methods described above for the inclination angles 15o, 30o,45o, and 60o, respectively. Then, Tables 6, 7, and 8 show the errors in the calculated aimingadjustments versus slant range distance and versus inclination angle for all three methods.

These are the aiming errors that remain AFTER the aiming adjustments calculated by eachmethod are made. To understand more clearly, consider the RR Method used for a 15oinclination angle and a target at 500 yds slant range distance from the shooter. From Table 2 theRR Method calls for a come-up (10.35 MOA) to move the bullet impact point upward 54.21inches to be on the LOS at the 500 yds slant range distance. Table 2 also shows that Infinityrequires a true come-up (10.87 MOA) to move the bullet impact point 56.89 inches upward at thesame slant range distance for a trajectory elevated at +15o. If the gun sights are adjusted tomove the bullet up 54.21 inches, or if the shooter holds over the target by 54.21 inches, thenwhen the round is fired the bullet will strike 2.68 inches below and perpendicular to the shooter'sLOS. This aiming error (or miss in case of a round fired) is shown in the second column of Table6 for 500 yds slant range and +15o inclination angle. This same calculation performed for a


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trajectory depressed at -15o results in a 2.46 inch aiming error (or miss) for the RR Method at500 yds slant range distance, and this is shown in column 3 of Table 6.


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In the remainder of this article we wish to focus attention on some key observations from thesedata. The first is that for any given inclination angle value the trajectory for elevated fire is verylittle different from the trajectory for depressed fire. This may be seen in Tables 2 through 5 bycomparing values in column 2 (Bullet Path for elevated fire) with corresponding values in column4 (Bullet Path for depressed fire). For example, the worst case is for an inclination angle of 45oand at 1000 yds slant range distance (see Table 4). The difference in the Bullet Path valuesbetween the elevated (+ 45o) trajectory and the depressed (- 45o) trajectory is 3.68 inchescompared to a total Bullet Path magnitude of approximately 270 inches. This is a very smalldifference at the 1000 yds slant range distance. And this difference is much smaller at shorterrange distances.

It is also true in all cases that the depressed trajectory is a little flatter (Bullet Path magnitude is alittle less) than the elevated trajectory. This is caused mainly by a component of the drag force onthe bullet acting in the upward vertical direction for a depressed trajectory. This upward-directed

force component keeps the bullet on the downward trajectory from falling quite as fast comparedto the bullet on the upward trajectory. On the upward trajectory the component of drag force inthe vertical direction is directed downward, causing the bullet to fall a little faster as it flies.

Another observation is that the errors in the aiming adjustments calculated by all three methodsgrow rapidly as the inclination angle grows steeper and the slant range distance grows longer.This can be seen in Tables 6, 7, and 8. This has been known for a long time; the numbers in thetables simply confirm this fact.

Let us imagine a few practical shooting situations. As mentioned above, in an urban warfare or


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SWAT team situation very accurate bullet placement on targets is necessary. Furthermore,inclination angles can be very steep, but then the slant ranges are quite short. For example, if thefiring point is on top of a 20-story building, approximately 200 feet tall (10 feet per story), shootingdownward at a -45o angle involves a slant range distance to a target on the ground of only about94 yds. Shooting downward at -60o shortens the slant range distance to 77 yds. Tables 6 and 7show that the RR and IRR Methods will have aiming errors under 1.0 inch at -45o, and no morethan 1.5 inches at -60o, at these range distances. Table 8 shows that Sierra's Method produceszero aiming errors for both angles at the short slant range distances.

In a hunting situation in mountainous terrain inclination angles larger that 30o are seldomencountered, but an angle as large as 45o might be encountered when hunting game such asmountain goats, mountain sheep, or mouflon in certain parts of the world. In such a situation ahunter needs an aiming error no larger than 3 inches at slant ranges up to 400 yds. Interpolatingfrom Table 6 shows that the RR Method is not accurate enough for 400 yds at 30o, but would beappropriate for a little more than 300 yds at that angle. From Table 7 we conclude about thesame accuracy performance for the IRR Method. Table 8 shows that Sierra's Method wouldsuffice for slant ranges well beyond 400 yds and for inclinations even greater than 45o, shouldsuch conditions be encountered.

For long range military sniping situations, inclination angles are not large; 30o is likely the limit.But slant range distances can be up to 875 yds. Tolerable aiming error is probably about 8 inchesfor targets at these limits for inclination angle and range distance. Table 6 shows that the RRMethod does not meet this accuracy requirement. Table 7 shows that the IRR Method ismarginally accurate, but probably acceptable. Table 8 shows that Sierra's Method exceeds theaccuracy requirement by a substantial margin.

In fact, the tables show that of the three methods considered, Sierra's Method appears toproduce the most accurate aiming adjustments for all range distances and inclination angles. It istrue, though, that the accuracy evaluation must be performed for the intended shooting scenario.

Accuracy is not the sole criterion for selection of the method to be used for a specific scenario. Ifmore than one method is sufficiently accurate, then the selected method should be the easiestone to use in the field situation. Sierra's Method is not difficult to use, but it may not be theeasiest.

Many other comparisons and observations can be made from the data in the tables, but theseare best left to the interested reader. It is important to mention once again that this analysis hasbeen carried out for just one cartridge in several possible scenarios. This cartridge is widely usedfor most of the shooting scenarios addressed. An exception is the hunting scenario. Match bullettypes are not recommended for hunting. But the analytical approach in this article is valid for aninclined fire accuracy evaluation of any cartridge combination.

Taken from Sierra's Website.

INCLINED FIRE

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