Air and the Ballistic Coefficient

If you were considering just gravity before, adding air makes the calculation of a firing solution exponentially more complicated.  Not only is it capable of exerting a force on any object, even stationary ones, through wind, but it also exerts a drag force on a flying bullet due to the force the bullet exerts on the air.  This is generically known as Newton's 3rd Law of Motion.

The force of drag takes into account the object's density, cross-sectional area, speed, and a constant specific to the object in question, known as the drag coefficient.  The force is expressed by the following equation:

$F_d=\frac{1}{2}CρAv^2$

As you can see, the force is going to be most effected by the speed due to it's square.  Rifle bullets are typically much faster than other types of bullets, such as from pistols.  Since drag force is undesirable and rifles need to be accurate over long ranges, this equation helps explain why extra attention is paid towards the design of rifle bullets and reducing both the drag coefficient and cross-sectional area.  Though it should be noted that reducing the area too much will lower the bullets effectiveness against it's target.  The design of these projectiles can get quite complicated given all the different performance characteristics a shooter needs.

A comparison showing various pistol bullets against the highly-engineered .408 Chey Tac rifle bullet.  Notice the tapered rear end (known as boat tail) which aerodynamically aids in reducing drag.  Also note that these two images are not to scale; the .408 is much larger.

In ballistics, the drag force and other aerodynamic factors are simplified into a single concept known as the ballistic coefficient (BC).  This term is defined in layman's terms by ballistician Bryan Litz as "the ability of the bullet to maintain velocity, in comparison to a 'standard projectile'."  There are actually two standard projectiles, called G1 and G7.  G7 is considered more accurate due to it being more similar to modern bullet designs.  In either case it is a unitless number.  The higher the number, the better the BC.  This value is calculated by the equations below.  These equations can only be used for small and large arms projectiles.

$BC=\frac{m}{d^2•i}$
Where:
m = mass of bullet
d = cross-sectional diameter
i = coefficient of form

$i=\frac{2}{n}•\sqrt{\frac{4n-1}{n}}$
Where:
n = number of calibers of the bullet's orgive

$n=\frac{4•l^2+1}{4}$
Where:
l = length of the head in number of calibers

Just like bullet drop (and also a contribution to bullet drop), BC is determined and provided by cartridge manufacturers.  BC is especially important as a consideration when dealing with distances that risk the bullet slowing down into the transonic zone, where stability is dramatically decreased.