3.0
Exterior Ballistic Effects on Bullet Flight
When a bullet flies through the air, two
types of forces act on the bullet to determine its path (trajectory)
through the air. The first is gravitational force; the other is
aerodynamics. Several kinds of aerodynamic forces act on a bullet:
drag, lift, side forces, Magnus force, spin damping force, pitch
damping force, and Magnus cross force. The most important of these
aerodynamic forces is drag. All the others are very small in comparison
when the bullet is spinstabilized. To a very good approximation,
the drag force and the gravitational force together determine the
trajectory of any spinstabilized bullet. The other small aerodynamic
forces cause only small variations from the trajectory that is determined
by drag and gravity acting alone on the bullet. For trajectory computation
purposes, this allows the bullet to be modeled as a point mass with
a ballistic coefficient. In other words, the bullet motion is modeled
as a three degrees of freedom (3 DOF) physics problem, that is,
as a point mass with three translational degrees of freedom. The
bullet trajectory calculated with this approach is almost exactly
correct. If we wished to treat the small variations caused by the
other aerodynamic forces, it would be necessary to model the bullet
as a spinning body with six degrees of freedom (6 DOF), three translational
and three rotational degrees of freedom. This approach, while more
exact, is very complex in both a physics and a mathematical sense,
and it is neither practical nor necessary to use this approach for
sporting purposes. In Section 4.0 we will discuss a few 6 DOF effects
that can be observed on long range sporting bullet trajectories.
However, it will be obvious that we can treat these effects only
qualitatively. The only quantitative observation we will be able
to make is that these effects are small. The 3 DOF model of bullet flight is used
in all exterior ballistics programs available to handloaders, including
Infinity.
In the 3 DOF model, the gravitational force always acts vertically
downward at the location of the bullet, regardless of the bullet’s
orientation relative to the vertical direction.
Aerodynamic drag always acts opposite
to the bullet’s direction of travel through the air. That is,
drag always tends to slow the bullet down. As the direction of bullet
travel changes during the trajectory, so does the direction of the
drag force. Drag is a very complex function of bullet velocity relative
to the air, and it depends critically on the density of the air
and on the speed of sound in the air through which the bullet is
moving. Air density, in turn, depends on true barometric pressure,
temperature of the air, and relative humidity at the location of
the bullet as it flies. These atmospheric parameters depend critically
on the altitude of the bullet above sea level. The speed of sound
in the air depends primarily on temperature of the air, and so it
also depends on altitude at the bullet location.
In addition to gravity and drag, there
are other strong effects on the path of a flying bullet. Wind, which
is any motion of the air mass through which the bullet is flying,
is one such effect. A headwind or tailwind will cause the bullet
to experience more or less drag, respectively, than it would if
it traveled in still air. A crosswind causes the bullet to turn
in the direction that the crosswind is blowing. A vertical wind
causes the bullet to turn upward or downward, following the vertical
wind that blows upward or downward, respectively. And so, the path
of the bullet (trajectory) is changed by winds compared to the path
it would have in still air. The 3 DOF model of bullet flight permits
the wind effects to be computed almost exactly for a spinstabilized
bullet in an exterior ballistics program such as Infinity.
Shooting uphill or downhill also causes
significant changes to the trajectory of a flying bullet compared
with its trajectory on a level firing range. In fact, if a gun is
sighted in on a level range, a bullet fired either uphill or downhill
always shoots high relative to the shooter’s line of sight
through the gun sights. Two effects contribute to the trajectory
changes. One is geometrical, and we will describe this effect in
a later subsection. The other is the fact that when a bullet travels
upward or downward relative to the firing point, the density of
the air changes, affecting the drag. These trajectory changes also
are computed in Sierra’s Infinity
program.
The temperature of the propellant (powder)
in a cartridge at the instant of ignition can have a strong effect
on chamber pressure. If a gun is sighted in on a target range at
some temperature, and then is fired in another environment in which
the propellant temperature is different, the muzzle velocity of
a bullet can be significantly different compared to what it was
when the gun was sighted in. The difference in muzzle velocity,
of course causes a change in bullet trajectory. The temperature
sensitivity of different powders varies from one type to another.
Only the manufacturer of any type of powder can quantify the
temperature sensitivity of that powder to a handloader. But, an
exterior ballistics program can be used to explore the sensitivity
of the trajectory of any bullet to changes in muzzle velocity. Sierra’s
Infinity program
is designed to make this computation especially easy. All the effects mentioned above are described
in more detail in the following subsections of this Section 3.0.
They are treated one at a time, so that the reader can determine
which are most important for his or her particular shooting application.
However, one of the most significant advantages of an exterior ballistics
software program like Infinity is
that any combination of these effects can be explored for any one
or more cartridges. For example, the effect of a wind and altitude
change can be computed, or the effect of a change in atmospheric
conditions can be determined. Likewise, the effect of a change in
muzzle velocity when shooting uphill or downhill can be determined,
or a 44 Magnum handgun can be compared with a 44 Magnum rifle. Any
other imaginable combination of cartridges and shooting conditions
can be explored.
Four more interesting topics are described
later in this section. The first is sighting in, or zeroing, a gun
at a selected range distance under various shooting conditions.
This is a familiar process to almost all shooters, but a number
of interesting questions frequently arise. The second topic is Point
Blank Range (PBR), which is a very interesting concept for hunters
and shooters participating in the silhouette games. PBR is the range
distance out to which a shooter can hold his sights directly on
a target, without holding over or under, and be assured of a hit.
A technique is described to maximize the PBR of any gun/cartridge
combination for a target of a given vertical size. A third topic
which is of frequent interest to shooters is the maximum range to
which a gun/cartridge combination will shoot and the elevation angle
of the bore that must be used to achieve that maximum range. Designers
of public shooting ranges often address this question. The final
topic is the maximum altitude a bullet will reach if a gun is pointed
straight up and fired. This is often a question for safety reasons.
As a reference for discussions in the
following subsections, we must describe the important parameters
of a bullet trajectory. Figure 3.01 has been prepared as an aid
to this description. A bullet exits the muzzle of a gun in the direction
of the extended bore line. After it exits the muzzle, gravity causes
the bullet to begin to fall away from the extended bore line. The
bullet trajectory then arcs downward, as shown by the bold line
in Figure 3.01. “Drop” is a term used to denote the vertical
distance from a point on the extended bore line to a corresponding
point on the trajectory, at any range distance from the muzzle.
It is very important to understand that “drop” is measured
in the vertical direction (direction of gravity), regardless of
whether the barrel of the gun is level (as shown in Figure 3.01)
or tilted upward or downward. In Figure 3.01
the drop (do)
at range (Ro)
is illustrated.
Figure 3.01. Illustrating the Parameters
of a Bullet Trajectory
The shooter, however, aims the weapon
at a target using the line of sight through the gun sights. The
line of sight is not parallel to the extended bore line; there is
a small angle between the two lines, which is greatly exaggerated
in Figure 3.01. From the shooter’s point of view through the
gun sights, the trajectory begins below the line of sight by a distance
equal to the sight height, then rises to cross the line of sight,
and then arcs over to again cross the
line of sight at a second point called the zero range Ro.
The sight height is a very important
parameter. With iron sights, it is the distance from the centerline
of the bore to the tip of the front sight, as shown in the figure.
With telescope sights, the sight height can be taken as the distance
from the centerline of the bore to the axis of the telescope, which
also is the centerline of the objective lens.
“Bullet path” is a term used
to denote the perpendicular distance between a point on the line
of sight and the corresponding point on the bullet trajectory at
any range distance from the muzzle. It is very important to understand
that “bullet path” is measured perpendicular to the line
of sight regardless of whether the barrel of the gun is level or
tilted upward or downward. Bullet path is then a measure of where
the bullet would be as “seen” by the shooter, if that
were possible. At the muzzle, the bullet path is negative by an
amount equal to the sight height, because the bullet starts out
below the line of sight. The bullet then rises to first cross the
line of sight, and then the bullet path is positive, reaching a
maximum value at a distance of about 55% of the zero range
distance Ro.
The bullet path then decreases to a zero value at the zero range
and goes negative at ranges greater than the zero range.
