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 3.01 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 3.31
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 3.31
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 3.31 shows how this happens. Ordinarily,
a shooter will sight his gun in on a target range that is level
or nearly level. Figure 3.31 (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 3.31 for
purposes of illustration. Even for very longrange 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 3.31 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 3.31
(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 3.31 (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 3.31 (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 3.31 (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 3.32 and 3.33. The first example
is for a 7 mm Remington Magnum, a flatshooting 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 3.32 and 3.33, 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 3.32 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.) 



Angle (deg.) 

100 

200 

300 

400 

500 













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 3.33 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.) 

Angle (deg.) 

50 

100 

150 

200 











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 

