5.2 Effects of Altitude and Uphill/Downhill Shooting
Sierra’s computer program for exterior ballistics calculates precise bullet trajectories at any altitude and for any firing angle, uphill or downhill. The program not only accounts for the altitude of the shooter, but it also calculates the change in drag that occurs as the bullet changes altitude after it leaves the muzzle. With this capability we can see exactly what effects altitude and firing angle have on the trajectory of any bullet.
Table 5.2-1(rifle) Table 5.2-1(handgun) shows the effects of altitude change on the drop of three different bullets. This table is for level fire only; it has no uphill or downhill shooting effects in it. The results listed for the .30-30 bullet are fairly typical of a class of hunting cartridges firing round nose or flat nose bullets at moderate velocity, like the .35 Remington, .30-40 Krag, .300 Savage, .45-70 and even the .30-06 with a 810 or 220 grain round nose bullet. At 300 yards the .30-30 bullet shoots flatter by almost 9 inches at 10,000 feet altitude than at sea level.
The data shown for the 117 grain .257 spitzer flat base are typical of another class of hunting cartridges shooting bullets with medium ballistic coefficients at somewhat higher velocities. typical cartridges in this class are the .257 Roberts, 7mm Mauser, 8mm Mauser, and .30-06 with medium weight spitzer bullets. Cartridges in this class are considerably less sensitive to altitude change, until range exceeds 400 yards or altitude exceeds 10,000 feet.
The performance of the third bullet in Table 5.2-1(rifle) Table 5.2-1(handgun) is typical of the big magnum cartridges in 7mm, .30, and .338 calibers used for both hunting and extra long range target shooting. If you hunt with a rifle in this class, you can just about forget altitude effects on your bullet trajectory, at least for most hunting in North America. If you are a long range target shooter, altitude changes will affect your shooting, and you should be aware of the altitude corrections you will have to make if you change altitudes.
Now, let us turn to the subject of uphill and downhill shooting. Table 5.2-2(rifle) Table 5.2-2(handgun) and Figure 5.2-1 have been prepared to aid this discussion. First, Table 5.2-2(rifle) Table 5.2-2(handgun) shows how bullet drop changes with uphill and downhill firing angle for the three example bullets in Table 5.2-1(rifle) Table 5.2-1(handgun) . Again, the drop changes shown in Table 5.2-2(rifle) Table 5.2-2(handgun) for the three bullets are typical of the three classes of hunting cartridges described above. Drop, as we use the term at Sierra, is always measured in thevertical direction at the target. It is the vertical distance between the line of departure (direction of the rifle bore axis) and the bullet trajectory. Figure 5.2-1 shows how true bullet drop is measured for level, uphill, and downhill shooting situations.
(It may be helpful to refer ahead a few pages to the Introduction to the Ballistics Tables for definitions of the terms used to describe bullet trajectories, such as line of departure, line of sight, bullet path, etc.)
The data in Table 5.2-2(rifle) Table 5.2-2(handgun) show clearly that drop changes very little with firing elevation angle at all practical ranges for the three bullets. For example, the .257 spitzer flat base bullet fired at 2600 fps at an altitude of 5000 feet above sea level would have a drop of 131.75 inches at 600 yards for the case of level fire. The same bullet fired upward at an angle of 45 degrees would have a drop of 131.00 inches at a slant range of 600 yards. The difference is only 0.75 inch, which is negligible compared to the total drop.
This result is true in general. At practical range for hunting and target shooting, the change in vertical bullet drop with firing elevation angle is negligible even for very steep angles. However, the bullet path height (distance of the bullet above or below the shooter’s lien of sight) can change a good deal, particularly at steep angles. Figure 5.2-1 shows how this happens. Ordinarily, a shooter will sight his rifle in on a level shooting range, and Figure 5.2-1 (a) shows this situation. When sighting in, the shooter adjusts his sights so that the line of sight intersects the bullet trajectory at the range called R o in Figure 5.2-1 , which is the range where he wants the rifle zeroed in. The distance between the line of departure and the line of sight at the range R o is the drop d o . We use this symbol to denote the amount of drip at the range where the rifle is zeroed in.
Note that the angle between the line of departure and the line of sight is actually very small. This angle is greatly exaggerated in Figure 5.2-1 for purposes of illustration. Even for long range (1000 yard) target shooting, it is less than one degree, and it is typically 5 to 10 minutes of arc for hunting rifles.
Now, consider the situation when the shooter fires his rifle uphill at a steep angle, as shown inFigure 5.2-1 (b) . Since the true bullet drop changes very little, at a slant range distance R o from the shooter the bullet has a vertical drop essentially equal to d o , as shown in the figure. However, the line of sight at R o still is located a distance d o 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 range R o . In fact, the bullet passes well above the line of sight at that point, asFigure 5.2-1 (b) shows. In other words, the bullet shoots high from the shooter’s viewpoint as he aims the rifle, and at steep angles it may shoot high by a considerable amount, as we will show shortly.
Figure 5.2-1 (c) shows the situation when the shooter fires his rifle downhill. Again, the vertical drop at the slant range distance R o changes negligibly from the value d o for level fire, but the line of sight and line of departure still are separated by the perpendicular distance d o . The bullet again passes above the line of sight, instead of intersecting it, at the range R o . Compared to the case of level fire, the bullet again shoots high from the shooter’s viewpoint as he aims the rifle. Furthermore, if the rifle is fired uphill at some elevation angle, and then fired downhill at the same elevation angle, the two bullets will shoot high by almost exactly the same amount.
Figure 5.2-1 (c) shows the situation when the shooter fires his rifle downhill. Again, the vertical drop at the slant range distance R o changes negligibly from the value d o or level fire, but the line of sight and line of departure still are separated by the perpendicular distance d o . The bullet again passes above the line of sight, instead of intersecting it, at the range R o . Compared to the case of level fire, the bullet again shoots high from the uphill at some elevation angle, and then fired downhill at the same elevation angle, the two bullets will shoot high by almost exactly the same amount.
It is reasonably easy to calculate how much higher a rifle will shoot for any given elevation angle. To do this, we need to know the bullet drop versus range for the load used, and we can find this in the Ballistics Tables for any Sierra bullet. Let d be the bullet drop at any range R from the muzzle. (In this calculation we are only concerned with the amount of the drop; the minus sign in front of the drop numbers in the Ballistics Tables can be neglected.) The following table shows how much higher the bullet will shoot when aimed either uphill or downhill than it will when fired on a level:
Elevation Angle Bullet Path Height
+/- 5 degrees .004 d inches
+/- 10 .015 d
+/- 15 .034 d
+/- 20 .060 d
+/- 25 .094 d
+/- 30 .134 d
+/- 35 .181 d
+/- 40 .234 d
+/- 45 .293 d
+/- 50 .357 d
+/- 55 .426 d
+/- 60 .500 d
To use this table, we first look up the drop d for each value of range for our load in the Ballistics Tables. Then, we calculate the increase in bullet path height for each elevation angle of interest using the multiplying factors in the table above. This tells us how much higher the bullet will shoot than it will for level fire at each value of range and for each elevation angle we wish to consider. Finally, we may add this increase to the bullet path height for level fire (from the Ballistics Tables) to get a new bullet path height at each elevated firing angle.
As an example of this procedure, and also to show how much even a relatively flat-shooting cartridge can be affected by elevated firing angles, the calculations in Table 5.2-3(rifle) Table 5.2-3(handgun) have been performed for a .270 Winchester firing the 130 grain spitzer boat tail bullet at 3000 fps muzzle velocity. The bullet drop data at the top of Table 5.2-3(rifle) Table 5.2-3(handgun) are taken from the Ballistics Tables for the 130 grain .270 spitzer boat tail bullet. Bullet path data for the rifle zeroed in at 200 yards are also listed so the new bullet path can be calculated for each elevation angle.
To understand how the numbers in Table 5.2-3(rifle) Table 5.2-3(handgun) are calculated, let’s look at the case where the firing elevation is +/- 30 degrees and the slant range is 400 yards. At this elevation angle the increase in bullet path height is 0.134 times the drop at each value of range. At 400 yards the drop is 37.56 inches, so the increase in bullet path height is 0.134 x 37.56 = 5.03 inches. So, if the shooter aims at a game animal 400 yards away or up or down a 30 degree slope, he has to remember that his bullet will shoot about 5 inches higher than it would on the level. With his rifle sighted in for 200 yards, his bullet path would be 19.47 inches below his line of sight at 400 yards for level fire. So, if he corrects the bullet path for the 30 degree angle, his new bullet path will be 14.44 inches below his line of sight at 400 yards, instead of 19.47.
It is clear from Table 5.2-3(rifle) Table 5.2-3(handgun) that steep elevation angles have important effects for this .270 Winchester load, and the steeper the angle the shorter the range where the effect becomes important. It turns out that steep elevation angles are important for nearly all hunting rifles. To cite just a few others, when fired uphill or downhill at a 45 degree angle, a .22-250 with the 55 grain spitzer bullet at 3700 fps muzzle velocity will shoot 4.2 inches high at 300 yards; a .300 Winchester Magnum with the 180 grain spitzer boat tail at 3000 fps muzzle velocity will shoot 5.6 inches high at 300 yards; and a .30-30 with the 150 grain flat nose at 2200 fps will shoot 5.9 inches high at 200 yards. The importance of understanding and compensating for firing elevation angle effects is pretty clear from these figures.