3.3 Effects of Shooting Uphill or Downhill
When a gun is sighted in on a level or nearly level range and then is fired either uphill or downhill, the gun will always shoot high. This effect is well known among shooters, particularly hunters, but how high the gun will shoot is a subject of considerable controversy in the shooting literature. In fact, at the present time some literature has information that is simply erroneous. In this subsection, we will try to explain the physical situation carefully so that it can be understood clearly, and then provide some examples using Infinity to perform precise calculations.
Throughout this subsection the terms “bullet drop” and “bullet path” will be used frequently, so we will review the definitions of those terms before we begin to explain the physical situation. One may refer back to Figure 3.0-1 concerning these definitions. Bullet drop is always measured in a vertical direction regardless of the elevation angle of the trajectory. At any range distance measured along either a level range or a slant range, drop is then the vertical distance between the extended bore line and the point where the bullet passes. Drop is expressed as a negative number, denoting that the bullet falls away from the extended bore line as the bullet travels.
Bullet path, on the other hand, is always measured perpendicular to the shooter’s line of sight through the sights on the gun. Thus, it would be where the shooter would “see” the bullet pass at any instant of time while looking through the gun sights, if that were possible. At the gun’s muzzle, the bullet path is negative because the bullet starts out below the line of sight of the shooter. Somewhere near the muzzle, the bullet will follow a path that rises and crosses the line of sight, then travel above the line of sight until the target is reached. The bullet path is then positive throughout this portion of the trajectory. The bullet will arc over and cross the line of sight at the zero range. So, the bullet path is zero at the zero range, and then becomes negative at distances greater than the zero range.
The explanation of the physical situation for uphill/downhill shooting begins with a simple observational fact — that bullet drop at any given range from the muzzle is almost independent of firing elevation angle. What this means is that if the drop of a bullet trajectory at, say, 150 yards is measured when the gun is fired on a level range, then the drop at a slant range distance of 150 yards will be almost the same value when the gun barrel is elevated at +45 degrees, – 15 degrees, – 60 degrees, or any other positive or negative elevation angle. It is very important to remember that we use “start range” because that is the range that the bullet must actually travel to reach the target. This is true for all range distances practical for small arms fire.
To illustrate this point, Table 3.3-1 has been prepared for a group of five cartridges, three for rifles and two for handguns. The table shows drop numbers at a specific range distance for each cartridge, as a function of the bore elevation angle of the gun at the firing event. These drop numbers have been computed with Infinity. These trajectories have been computed for a firing point altitude of 2500 feet above sea level. The selected cartridges in Table 3.3-1 illustrate typical behavior of drop at a specific (and relatively long) range distance versus the bore elevation or depression angle. The 338 Winchester Magnum cartridge exhibits the worst case in the table. At a range distance of 600 yards, there is only about 0.5 inch difference in drop value between a level trajectory and a trajectory elevated 60 degrees or depressed 60 degrees. This is because the major driving cause of bullet drop is gravity acting over the bullet’s time of flight. There are two other smaller effects on drop as the bullet travels. When a bullet is traveling upward on an elevated trajectory, there is a component of gravity that adds to the drag deceleration of the bullet, but the bullet is traveling into less dense atmosphere that reduces the aerodynamic drag. So, these small effects tend to offset one another. The opposite small effects occur when the bullet is traveling downward along a depressed trajectory.
This result is true in general. At practical range distances for small arms fire the change in vertical drop with firing elevation or depression angle is very small, even for very steep angles. However, the bullet path can change dramatically, particularly at steep angles.
Figure 3.3-1 shows how this happens. Ordinarily, a shooter will sight his gun in on a target range that is level or nearly level. Figure 3.3-1 (a) shows this situation. When sighting in, the shooter adjusts his sights so that the line of sight intersects the trajectory at the range (Ro in the figure), which is the range where he wants his gun zeroed in. Ro is called the zero range for level fire. The vertical distance between the line of departure (extended bore line) of the bullet and the point where the bullet passes is the drop (do). This symbol is used to denote the drop at the range where the gun is zeroed in.
Note that the angle between the bullet’s line of departure (extended bore line) and the line of sight is very small. This angle is greatly exaggerated in Figure 3.3-1 for purposes of illustration. Even for very long-range target shooting (1000 yards or more), the angle A is much less than 1.0 degree, and it is typically less than 10 minutes of arc for sporting rifles and handguns.
Table 3.3-1 Bullet Drop at a Specific Range Distance versus Bore Elevation Angle for a Selection of Cartridges
|Cartridge and Load Range Distance||Elevation Angle||Bullet Drop|
|22 Hornet, Sierra’s 200 yds||0 deg (level)||– 13.39 in|
|45 gr. Hornet bullet,||20||– 13.38|
|2700 fps Mzl Vel||45||– 13.36|
|– 20||– 13.40|
|– 45||– 13.41|
|270 Winchester 400 yds||0 deg (level)||– 39.98 in|
|Sierra’s 140 gr.||20||– 39.94|
|SBT GameKing,||45||– 39.90|
|2900 fps Mzl Vel||60||– 39.89|
|– 20||– 40.01|
|– 45||– 40.05|
|– 60||– 40.06|
|338 Winchester 600 yds||0 deg. (level)||– 109.05 in|
|Magnum, Sierra’s||45||– 108.66|
|250 gr. SBT GameKing,||60||– 108.57|
|2700 fps Mzl Vel||– 45||– 109.44|
|– 60||– 109.53|
|44 Magnum, Sierra’s 150 yds||0 deg (level)||– 28.34 in|
|240 gr. JHC bullet,||20||– 28.33|
|1300 fps Mzl Vel||45||– 28.32|
|– 20||– 28.36|
|– 45||– 28.37|
|38 S&W Special, 100 yds||0 deg (level)||– 16.04 in|
|Sierra’s 125 gr.||20||– 16.03|
|JSP bullet,||45||– 16.03|
|1100 fps Mzl Vel||– 20||– 16.04|
|– 45||– 16.05|
Now consider the situation where the shooter fires his gun uphill at a steep angle, as shown in Figure 3.3-1 (b), with no changes in the sights. Since the true bullet drop changes very little, at a slant range distance Ro from the muzzle the bullet has a vertical drop nearly equal to do, as shown in the figure. However, the line of sight at slant range distance Ro still is located a distance do in a perpendicular direction away from the line of departure. Because of the firing elevation angle, the bullet trajectory no longer intersects the line of sight at the slant range Ro. In fact, the bullet passes well above the line of sight at that point, as Figure 3.3-1 (b) shows. In other words, the bullet
shoots high from the shooter’s viewpoint as he or she aims the gun, and at steep angles it may shoot high by a considerable amount at longer ranges.
Figure 3.3-1 (c) depicts the situation when the shooter fires the gun downhill. Again the vertical drop at the slant range distance Ro changes a very small amount from the value do for level fire, but the line of sight and line of departure are still separated by the perpendicular distance do at that range point. Compared to the case of level fire, the bullet again shoots high from the shooter’s viewpoint as he or she aims the gun. Furthermore, if the gun is fired uphill at some elevation angle, and then fired downhill at an equivalent depression angle, the two bullets will shoot high by nearly the same amount at the same slant range distances.
A careful look at Figure 3.3-1 (a) or (b) shows us that the amount by which the bullet shoots high at the slant range distance Ro is equal (approximately) to the perpendicular distance do from the line of sight to the extended bore line minus the projection of the drop do on that same perpendicular line. From plane trigonometry, the distance by which the bullet shoots high at Ro is:
Amount by which the bullet shoots high = do [1.0 – cosine A]
where A is the elevation angle (or depression angle). Now, if you have forgotten or never studied trigonometry in school, don’t worry. The Infinity program will make exact calculations for you, and two examples of these calculations will be shown below.
First though, let us point out that this explanation of the physics of uphill or downhill shooting has been given specifically for a slant range distance equal to the zero range distance for level fire, and this has been done just for convenience. The sketches are easier to draw and to understand for that situation. The result, however, applies for all slant range distances. At any range distance from the muzzle, the amount by which the bullet will shoot high at any elevation or depression angle A is very nearly equal to the drop for level fire at that range distance multiplied by the quantity [1.0 – cosine A].
Two examples for uphill or downhill shooting have been prepared using Infinity, and they are shown in Tables 3.3-2 and 3.3-3. The first example is for a 7 mm Remington Magnum, a flat-shooting rifle cartridge. The second example is for a 44 Remington Magnum handgun cartridge that has a trajectory with much more arc. It is presumed that both the rifle and the handgun have telescope sights and are sighted in at an altitude of 2500 feet. Then, they are fired uphill or downhill while at the same altitude. The tables show the reference bullet path for level fire together with the changes in bullet path depending on the elevation angle and slant range distance. When reviewing Tables 3.3-2 and 3.3-3, keep in mind that a depression angle is a negative elevation angle.
Two conclusions are evident from these examples. First, shooting uphill or downhill can have a strong effect on the trajectory of any bullet, always causing the bullet to shoot high relative to the bullet path for level fire. This effect grows larger as the slant range distance grows longer and the elevation angle grows steeper. The second conclusion is that a bullet always shoots slightly higher when it is fired downhill than when it is fired uphill at the same angle. The reason for this, as explained above, is that when the bullet travels upward, there is a component of gravity acting as drag on the bullet that increases the drop slightly. When the bullet travels downward, on the other hand, there is a component of gravity acting as drag on the bullet that decreases the drop slightly.
Table 3.3-2 Example of Bullet Path Changes for a Rifle Bullet Fired Uphill or Downhill
Cartridge: 7 mm Remington Magnum with Sierra’s 140 grain Spitzer Boat Tail bullet at 3000 fps muzzle velocity Zero range: 300 yds for level fire Shooting environment: 2500 ft altitude with standard atmospheric conditions
|Elevation||Parameter||Slant Range Distance (yds.)|
|0||Bullet Path (in)||3.71||4.45||0.0||– 10.49||– 28.06|
|+ 15||Bullet Path Change (in)||0.07||0.29||0.68||1.26||2.08|
|– 15||Bullet Path Change (in)||0.07||0.29||0.70||1.32||2.19|
|+ 30||Bullet Path Change (in)||0.27||1.13||2.68||5.03||8.31|
|– 30||Bullet Path Change (in)||0.27||1.15||2.73||5.13||8.50|
|+ 45||Bullet Path Change (in)||0.59||2.49||5.89||11.05||18.27|
|– 45||Bullet Path Change (in)||0.59||2.50||5.94||11.17||18.48|
Table 3.3-3 Example of Bullet Path Changes for a Handgun Bullet Fired Uphill or Downhill
Cartridge: 44 Remington Magnum with Sierra’s 240 grain Jacketed Hollow Cavity bullet at 1300 fps muzzle velocity Zero range: 100 yds for level fire Shooting environment: 2500 ft altitude with standard atmospheric conditions
|Elevation||Parameter||Slant Range Distance (yds.)|
|0||Bullet Path (in)||2.40||0.0||– 9.88||– 28.35|
|+ 15||Bullet Path Change (in)||0.09||0.38||0.89||1.63|
|– 15||Bullet Path Change (in)||0.10||0.42||1.04||2.01|
|+ 30||Bullet Path Change (in)||0.37||1.55||3.67||6.84|
|– 30||Bullet Path Change (in)||0.37||1.61||3.92||7.49|
|+ 45||Bullet Path Change (in)||0.80||3.42||8.16||15.28|
|– 45||Bullet Path Change (in)||0.81||3.50||8.44||16.03|