The physics of shooting can be split into a few important
sub-categories; Internal ballistics (how the bullet is projected from the gun)
this includes all events from the ignition of the propellant and the bullet exiting
the barrel and the major factor here is the properties of the propellant,
external ballistics which is the behaviour once the bullet is in free flight which
is dominated by fluid dynamics and Newton’s laws of motion and thirdly how the
shooter then compensated for these factors on the firearm to get an accurate
shot. But I will be covering the external ballistics which is by far the most
important and most complicated.

However, before I jump into the physics I need to define a
unit of measurement used very commonly in shooting as many adjustments and
calculations are done using this unit. Minute of Arc/Angle or “MOA” is defined
as 1/60th of a degree. Which is the angle subtended by a sphere with
a diameter of 1.047 inches at 100 yards. Which for rough approximations we can
use 1 inch per 100 yards. This approximation only starts to become a problem
when making adjustment of 20+ MOA which is approximately at distances >900
yards

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External Ballistics

Probably one of the easiest and most important factors in external ballistics
is gravity’s effect on the trajectory. This follows Newton’s laws of
gravitation and motion so no matter the bullet it will accelerate towards the
earth at 9.81 ms-2 we can assume this to be true as due to the shape
of a bullet air resistance in the vertical plane is very small. This was tested
by Rhett Allain from “wired.com” and produced this graph 1 showing the time
difference between a falling bullet with and without air resistance. As you can
see the time difference between the two is very small when considering a
vertical drop.

The second effect is air resistance however this is slightly
more complicated. Firstly, I need to define the Ballistic Co-efficient, qualitatively
this is how efficiently the bullet can move through the air. Quantitatively
it’s the sectional density (weight/diameter2) divided by the form
factor (i.e. the shape of the bullet, but for simplicity we can assume this is
1 which is by far the most common). This is just a scale factor you multiply by
the standard drag deceleration (drag of a 1lb bullet with diameter of 1 inch). This
is the red line in this graph. 2

Interestingly there is a large deviation at ~330ms-1 this
is due to the drag increasing very quickly when transitioning from sub to
supersonic speeds.

Then once you have your value for the drag deceleration for
your bullet you can then sub it into Newton’s laws of motion to find the
horizontal component of the trajectory, so you can now plot the full
trajectory. This allows you to extract ranging information, this is a typical
trajectory for a 7.62mm bullet 3

For example, taking rough data from the graph, moving from
100-300 yards the bullet will drop ~ 1ft which is 12 inches which is ~4 MOA at
300 yards allowing you to adjust your sights up by 4 MOA so you’re in the right
ballpark for your first shot. Importantly the ranging information will vary
from weapon to weapon.

The third most important but the hardest to compensate for
as a shooter is the effect of the wind on the bullets trajectory. A common
misconception is that the wind ‘pushes’ the bullet of course however data 4
comparing the horizontal deflection of a bullet flying in a 10mph crosswind for
.269 seconds and a bullet dropped in a 10-mph wind for 0.269 seconds shows a
large discrepancy showing that ‘pushing’ is not the cause of the bullets drift
because the force applied and therefore acceleration would be the same. To
explain how it works I will assume there is a wind travelling left to right
perpendicular to the velocity of the bullet. The reason for the drift is
because the wind provides a force perpendicular to the direction of travel and
the viscous drag of the bullet provides a force in the opposite direction to
the direction of travel. So, the resultant force will be back and to the right
as shown in the diagram below 5. Due to the aerodynamics of the bullet this
will cause it to yaw into the wind, so it is facing the opposite direction to
the resultant force. So now the bullet has an angle of attack 6 so it is not
facing in the direction of travel and this causes a difference between the
forces on each side of the bullet causing it to accelerate in the direction of
the wind.

Another key but commonly overlooked
factor is ambient air density and the three factor that contribute to air
density are; temperature, pressure and humidity. So, if temperature and
humidity are high and pressure is low this will lead to the lowest air density
thus reducing drag causing the bullet to drop less over distance. Initially one
would think a lower humidity would decrease the density however water vapour
has a density of 0.8 g/l whereas dry air has a density of 1.225 g/l so by
displacing more air with water vapour you can decrease the average density.

Probably the last two major factors
that a shooter would take into account will the effects caused by the rotation
of the earth under the projectile in free flight known as Coriolis drift. This
is split into two components the horizontal effect and the vertical effect. The
horizontal component is caused by the projectile flying in a non-inertial frame
so in the bullets reference frame it travels in a straight line however when in
the rotating reference frame the bullet will seem to curved by a force however
this is a fictitious or inertial force. This causes objects in the northern
hemisphere to be deflected to the right and objects in the southern hemisphere
to be deflected to the left this effect goes from a maximum at the poles and 0
at the equator by a cos function.

The vertical component called the Eötvös effect this is again an
inertial force perceived as an increase in gravitational force when shooting to
the west and a decrease when shooting to the east. This is because when the
bullet is in flight the target will either start to fall below the horizon or
come up from the horizon as the earth rotates under the bullet. Opposite to the
horizontal component the effect is maximum at the equator and 0 at the poles.

When travelling down the barrel the
bullets are given a spin this give the bullets a massive amount of stability
which has allowed us to shoot at much longer distances (smoothbore muskets had
an accurate range to about 100yds whereas modern rifles can easily hit 10x that
accurately) however this spin also causes two minor effects which cause the
bullet to drift off course. The first effect is Gyroscopic drift which causes a
clockwise rotating bullet to feel a force to the right and vice versa for
counter clockwise. Secondly the Magnus effect which is due to the instantaneous
velocities at the top and bottom of the bullet are in opposite direction so
when there is a wind perpendicular to the axis of rotation this causes a
pressure differential from above and below the bullet creating a vertical force
(for a clockwise rotation and a left to right wind this creates an upward force
and vice versa for counter-clockwise).

Finally, you can see that what
seems like a simple topic has a hidden layer of complexity that the shooter
must be able to understand, take into account and be able to make adjustments
for these variables every time they take a shot as many of these variables can
change very quickly. E.g. the wind could pick up between shots or some clouds
cover may pass overhead causing the temperature to drop causing you to fall
low. This is where the skill in the sport distinguishes the good from the
great.