Advanced Formulas
and
The Physics Behind the Shot
Index of Formulas
The force of gravity 
- e j -
Unit vector, opposite to the direction of the acceleration of gravity
- F G -
The force of gravity
Explanation: The
force of gravity is proportional to the mass of the projectile and the
local acceleration of gravity. The force is directed towards the center
of the earth and attacks at the CG.
The force of gravity is responsible for the bending of the trajectory.
The centrifugal force
Explanation: The
figure above shows a cut through the globe. The formula gives the components
of the centrifugal force in an xyz - reference frame, the y -axis being
antiparallel to the force of gravity. The y - component of the centrifugal
force can be regarded as a correction of the force of gravity, the other
components are generally neglected in ballistics because of their smallness.
The Coriolis Force
- F
c - Coriolis force
- v - Velocity vector
with respect to xyz - coordinate system
- w - Vector of the angular velocity
of the earth´s
rotation with respect to xyz - coordinate
system.
Explanation
The magnitude of theCoriolis
force is so small that it is usually completely neglected.
The Coriolis effect can be demonstrated by the following
experiment.
Imagine a disk which first does not rotate. A mass m which moves with
constant velocity v from the center of the disk towards the rim
moves on a straight line. No force affects the movement of the mass.
Now let the disk rotate with a constant angular velocity w (see figure
above). Still no external force affects the mass. An observer who rotates
with the disk - and has no indication of this rotation - sees the mass
moving along a bended path. This observer would state that the mass
must have been affected by a "force", which acts perpendicular to the
direction of movement.
This fictitious "force" is called the Coriolis
force F c
The real problems in this type of calculation is that we are shooting
on a sphere and the results will be dependant on where you are on the
earth at the time of the shot and what direction you shoot in. It is
more straight forward if you shoot only perfectly north or south (according
to the earths axis of rotation) and on a flat surface. If you shoot
any other direction other than perfectly north or south it gets much
more complicated very quickly. So as you can see, knowing where you
are on the surface of the earth is extremely important. You would need
to know where you are to a greater degree of accuracy than a standard
civilian GPS unit could place you. The calculation gets more complicated
when you consider the fact that lines of longitude are “Great Circles”
and represent differing amounts of linear measure depending on how far
you are from the equator.
Also you must take into consideration any east/west component of the
bullets path. Then so-called “centripetal acceleration” must also be
factored into the equation with the ultimate end of this being the projectile
being put into orbit. Spherical geometry is not my best subject but
it is an important one when dealing with this subject.
Simple exterior ballistics (more integral calculus) cannot be excluded
either because they determine how long the bullet will stay in the air
before returning to the surface. Keep in mind this is not a flat surface
but one that is curving away from us thus the “centripetal acceleration”
factor in our calculations.
That said, here's a brief explanation of the basics. I am sure that
others reading this with more understanding will find faults with my
logic, math and explanations so have at me guys.
The Coriolis effect can be looked at from one of two perspectives as
all things in physics. All things are relative. The simplest way to
look at it is from the projectiles perspective. It is expressed by the
following equation:
F c = -2m(wv i )
Where F c = Coriolis effect
m = mass of the object
w = Angular velocity of rotating surface
v i = Velocity of rotating surface
* For the earth the angular velocity is 1 rev per day or about 7x10
-5 radians/24 hours
-Sort of the same radians as in our mil-dot reticles but that's a different
issue-
*v i will change depending on where latitudinally you are on the earth.
The v i of 1100mph applies only at the equator. At the poles it is 0mph
**Note that this is only a linear equation so the rate that the projectile
loses its velocity is not considered here
The other consideration is called the Magnus
Force. This is the same
force that causes a curve-ball to curve. It mainly depends on bullet
geometry, spin rate, velocity and angle of yaw. Modern projectiles yaw
very little but it does happen.
This is a long and lengthy discussion that barely scratches the surface.
These effects appear to be negligible when compared to other more significant
factors such as the wind and velocity of the target.
The Drag Force
- c D -
Drag coefficient
- F D -
Drag force
Explanation: The
drag force F D is the component
of the force F W in the direction opposite to that
of the motion of the centre of gravity.
The force F W results from pressure differences at
the bullet's surface, caused by the air, streaming against the moving
body. In the case of the absence of yaw, the drag F D
is the only component of the force F W .
The drag force is the most important aerodynamic force. Given the
atmosphere conditions p,T,h , the reference area A and
the momentary velocity v w , the drag force is completely
determined by the the drag
coefficient c D .
The Drag Coefficient
The drag coefficient c
D is the most important aerodynamic coefficient and generally
depends on - bullet geometry (symbolic variable B ),
- Mach number Ma,
- Reynolds number Re
- The angle of yaw d
The following asssumptions and simplifications are
usually made in ballistics:
1. Re neglection
It can be shown, that with the exception of very low velocities, the Re dependency
of c D can be neglected.
2. d dependency
Depending on the physical ballistic model applied, an angle of yaw is either
completely neglected ( d =0) or only small angles of yaw are considered. Large
angles of yaw are an indication of instability.
For small angles of yaw the following approximation is usually made:
a) c D ( B,Ma, d ) = c Do ( B,Ma ) + c
D d ( B,Ma ) * d 2 /2
Another
theory which accounts for arbitrary angles of yaw is called the "crossflow
analogy prediction method". A discussion of this method is far beyond the scope
of this article, however the general type of equation for the drag coefficient
is as follows:
b) c D ( B,Ma, d ) = c Do ( B,Ma ) + F ( B,Ma,Re, d
)
3. Determination of the zero-yaw drag coefficient
The zero-yaw drag coefficient as a function of the Mach number Ma is
generally determined experimentally either by wind tunnel tests or from Doppler
Radar measurements.
Fig.: Zero-yaw drag coefficient for two military bullets
M80 (cal. 7.62 x 51 Nato)
SS109 (cal. 5.56 x 45)
There is also software available which estimates the zero-yaw drag coefficient
as a function of the Mach number from bullet geometry. The latter method is
mainly applied in the development phase of a new projectile.
4. Standard drag functions
Generally each bullet geometry has its own zero-yaw
drag coefficient as a function of the Mach number. This means, that specific
- time-consuming and expensive - measurements would be required for each
bullet geometry. A widely used simplification makes use of a "standard drag
function" c Do standard which depends
on the Mach number alone and a form factor i D which depends on the
bullet geometry alone according to:
c Do ( B,Ma ) = i D ( B ) * c
Do standard ( Ma )
If this simplification
is applicable, the determination of the drag coefficient of a bullet
as a function of the Mach number is reduced to the determination of a
suitable form factor alone. It will be shown that the concept of the
ballistic coefficient , widely used in the US for small arms projectiles follows
this idea.
- c D -
Drag coefficient; c D (B,Ma,Re, d )
- c Do standard -
Zero-yaw standard drag function
- i D -
Form factor
The
Ballistic Coefficient ( bc )
The 'ballistic coefficient' or bc is a measure for the drag experienced
by a bullet moving through the atmosphere, which is widely used by manufacturers
of reloading components, mainly in the US. Although, from a modern point
of view, bc s are a remainder of the pioneer times of exterior
ballistics, ballistic coefficients have been determined experimentally
for so many handgun bullets, that no treatise on exterior ballistics would
be allowed to neglect it..
The bc of a test bullet bc test moving
at velocity v is
a real number and defined as the deceleration
due to drag of a "standard" bullet
devided by
the deceleration due to drag of the test bullet.
The standard bullet is said to have a mass of 1 lb (0.4536
kg) and a diameter of 1 in (25.4 mm). The drag coefficients of the standard
bullet can be derived from the G1-function given in literature and
will be named c Do G1 ( Ma ) .
Using c Do test ( B,Ma ) = i D test ( B )
* c Do G1 ( Ma )
one finds for the bc (assuming "standard" atmosphere
conditions)
bc test =1 / i D test (B) * m test / d 2 test
This formula also shows that the bc and the
form factor i D of
a "test" bullet are two aspects of the same principal simplification: the substitution
of the (unknown) particular drag function of a bullet by the (given) "standard" drag
function of the standard bullet.
- c Do test - Zero-yaw drag
coefficient of test bullet
- c Do G1 - Zero-yaw G1 standard
drag coefficient
- i D test - Form factor of test bullet
- bc test - Ballistic coefficient of test bullet
- m test - Mass of test bullet in lb
- d test - Diameter of test bullet in inches
The Lift Force
- c L - Lift
coefficient; c L (B,Ma.Re, d )
- e L- Unit
vector
- F L - Lift force
Explanation: The
lift force F L (also
called cross-wind force )
is the component of the wind force F W in the direction perpendicular
to that of the motion of the center of gravity in the plane of the yaw angle.
The lift force vanishes in the absence of yaw and is the reason for the drift
of a spinning projectile even in the absence of wind.
The Spin Damping Moment
Explanation: Skin
friction at the bullet's surface retards its spinning motion. The spin
damping moment (also: roll damping moment) is given by the above formula. The
spin damping coefficient depends on bullet geometry and the flow type
(laminar or turbulent).
The Magnus Force
F m = P (p x T x h)/2 x A x c Mag (B x M a x R e (w x d/v w ) x (w x d/v
w ) x S) x
(w x d/v w ) x v w 2 x e M
Where
- e M = 1/sin S (e c x e t )
- M a = v w /@(p . T . h)
- R e = P(p x Tx H) x l x v w /u x T
- C mag = Magnus force coefficient
- e M = Unit Vector
- F m = Magnus Force
- A = Bullet cross section area
- a = Velocity of sount in air
- B = Symbolic Variable, indicating bullet geometry
- d = bullet diameter
- e c = Unit vector into the direction of the bullet's
longitudinal axis
- e t = Unit vector into the direction of hte tangent ot
the trajectory
- g = Acceleration due to gravity
- h = Relative humidity of air
- l = bullet length
- m = bullet mass
- M a = Mach number
- p = air pressure
- R e = Reynolds number
- T = Absolute air temperature
- u = absolute viscosity of air
- v w = bullet velocity with respect to wind
system
- @ = Azimuth angle
- S = Yaw angle
- P = Air density
- w = spin rate of bullet (angular velocity)
Explanation: The
Magnus force F M
arises from an asymmetry in the flow field, while the air stream against
a rotating and yawing body interacts with its boundary layer and applies
at the CPM.
Depending on the flow field, the CPM may be located ahead or behind the
CG. The Magnus force vanishes in the absence of rotation and in the absence
of a yaw angle.
The Magnus force is usually very small and mainly depends
on bullet geometry, spin rate, velocity and the angle of yaw. In exterior
ballistics, the above expression is used for the Magnus force.
For the whole bullet, the Magnus effect (which arises
from the boundary layer interaction of the inclined and rotating body with
the flowfield) results in the Magnus force F M which
applies at its centre of pressure CPM. The location of the CPM varies as
a function of the flowfield conditions and can be located either behind or
ahead of the CG.
The figure above assumes that the CPM is located behind the CG. Experiments
have shown that this comes true for a 7.62 x 51 FMJ standard Nato bullet
at least close to the muzzle in the high supersonic velocity regime.
The Overturning Moment
- c M - Overturning moment
coefficient, c M ( B, Ma, Re, d )
- e W - Unit vector
- M W - Overturning moment
Explanation: The
point of the longitudinal axis, at which the resulting wind force F 1
appears to attack is called the centre of pressure CPW of the wind force, which,
for spin-stabilized bullets is located ahead of the CG. As the flow field varies,
the location of the CPW varies as a function of the Mach number. Due to the
non-coincidence of the CG and the CPW, a moment is associated with the wind
force. This moment M W is called overturning
moment or yawing moment.
For spin-stabilized projectiles M W tends
to increase the yaw angle and destabilizes the bullet. In the absence of spin,
the moment would cause the bullet to tumble.
The
forces F 1 and F 2
form a free couple, which is said to be the aerodynamic moment of
the wind force or simply overturning moment M w.
This moment tries to rotate the bullet about an axis through the CG, perpendicular
to the axis of symmetry of the bullet. The overturning moment tends to increase
the angle of yaw d .
The force F W , which applies at the CG can be split into
a force, opposite to the direction of the movement of the CG (the direction
of the velocity vector v ), which is called the drag
force F D or
simply drag and a force, perpendicular to this direction,
which is called the lift force F L or
simply lift .
The Magnus Moment
- c Mp - Magnus moment coefficient; c Mp (B,Ma,Re, w
, d )
- e MM - Unit vector
- M M - Magnus moment
Explanation: As the Magnus force applies at the CPM, which
does not necessarily coincide with the CG, a Magnus moment M M is
associated with that force. The location of the centre of pressure of the
Magnus force depends on the flow field and can be located ahead or behind the
CG. The Magnus moment turns out to be very important for the dynamic
stability of spin-stabilized bullets. For the Magnus moment, the above expression
is used in exterior ballistics.
The Magnus force, which applies at its centre of pressure CPM can be substituted
by a force of the same magnitude and direction, which applies at the CG,
plus a moment, which is said to be the Magnus moment M
M This
moment tries to rotate the bullet about an axis, perpendicular to the longitudinal
axis of the bullet.
However, the gyroscopic effect also applies for the
Magnus moment and the bullet´s axis will be shifted into the direction
of the moment. Thus, as far as the conditions of the figure above are valid,
the Magnus moment will have a stabilizing effect as it tends to decrease the
angle of yaw d .
It can be easily shown that this is only true, if the centre of pressure
of the Magnus force CPM is located behind the CG. The Magnus
force destabilizes the bullet and increases the
angle of yaw, if its centre of pressure is located ahead of
the CG, which may come true in a specific velocity regime.
The Gyroscopic Stability Condition
- c M a - Overturning moment
coefficient derivative; c M a (B,Ma )
- s g - Gyroscopic stability
factor
Explanation: A spin-stabilized projectile is said to be gyroscopically
stable ,
if, in the presence of a yaw angle d , it responds to an external wind force F 1
with the general motion of nutation and precession. In this case the longitudinal
axis of the bullet moves into a direction perpendicular to the direction of
the wind force.
It can be shown by a mathematical treatment that this condition is fulfilled,
if the gyroscopic stability factor s g exceeds unity. This demand
is called the gyroscopic stability condition. A bullet can
be made gyroscopically stable by sufficiently spinning it (by increasing w!
).
As the spin rate w decreases more slowly than the velocity v
w , the gyroscopic stability factor s g , at least close to
the muzzle, continuously increases.
Thus, if a bullet is gyroscopically stable at the muzzle, it will be gyroscopically
stable for the rest of its flight. The quantity s g also depends
on the air density r and this is the reason, why special attention has to
be paid to guarantee gyroscopic stability at extreme cold weather conditions.
Bullet and gun designers usually prefer s g > 1.2...1.5, but
it is also possible to introduce too much stabilization. This is called over-stabilization.
The gyroscopic (also called static) stability factor depends on only one aerodynamic
coefficient (the overturning moment coefficient derivative c M a )
and thus is much easier to determine than the dynamic stability factor. This
may be the reason, why some ballistic publications only consider static stability
if it comes to stability considerations.
However, the gyroscopic stability condition only is a necessary condition to
guarantee a stable flight, but is by no means sufficient. Two other conditions
- the conditions of dynamic stability and the tractability condition must be
fulfilled.
This figure shows the gyroscopic stability factor of
the 7.62 x 51 Nato bullet M80, fired at an angle of departure of 32°,
a muzzle velocity of 870 m/s and a rifling pitch at the muzzle of 12 inches.
The M80 bullet shows static stability over the whole flight path as the static
stability condition s g >1 is fulfilled everywhere. The value
of s
g adopts a minimum of 1.35 at the muzzle.
Generally it can be stated that if a bullet is statically stable at the
muzzle, it will be statically stable for the rest of its flight. This can
be easily understood from the fact, that the static stability factor is proportional
to the ratio of the bullet´s rotational and transversal velocity (see
formula ).
As the the rotational velocity is much less damped than the transversal velocity
(which is damped due to the action of the drag), the static stability factor
increases, at least for the major part of the trajectory.
Bullet and gun designers usually prefer s g > 1.2 ..1.5 at
the muzzle, however it has been observed that many handgun bullet show excessive static
stability.
The Dynamic Stability
Condition
- c D - Drag coefficient
- c L a - Lift coefficient derivative
- c Mp a - Magnus moment coefficient
derivative
- c mq +c m a - Pitch damping moment
derivative
- s g - Gyroscopic (static)
stability factor
- s d - Dynamic stability
factor
Explanation: A projectile is said to be dynamically
stable , if its yawing
motion of nutation and precession is damped out with time, which means that
an angle of yaw induced at the muzzle (the initial yaw) decreases.
A dynamic stability factor s d can be defined from the linearized
theory of gyroscopes (assuming only a small angle of yaw) and the above dynamic
stability condition can be formulated. An alternate formulation
of this condition leads
to the illustrative stability triangle .
s d however depends on five aerodynamic coefficients. Because
these coefficients are hard to determine, it can become very complicated
to calculate the dynamic stability factor, which varies as a function of
the momentary bullet velocity.
The Stability Triangle
- s g - Gyroscopic stability factor
- s d - Dynamic stability
factor
Explanations: The
dynamic stability condition can
be expressed in an alternate way. leading to a very illustrative interpretation
of bullet stability.
In using a quantity s , according to the above definition, the dynamic stability
condition takes a very simple form (see above formula). This means that for a
bullet to be gyroscopically and dynamically stable, a plot
of s vs. s d has to remain completely within the stability triangle
(green area in the figure below).
The Tractability Condition
- f - Tractability factor
- f l - Low limit tractability factor; fl » 5.7
- s g - Gyroscopic stability factor
- d p - Yaw of repose vector
Explanation: The tractability
factor f characterizes
the ability of the projectile's longitudinal axis to follow the bending trajectory.
The quantity f can simply be defined as the inverse of the yaw
of repose. It can be shown that the tractability factor f is
proportional to the inverse of the gyroscopic stability factor.
Over-stabilized bullet
This figure schematically shows
an over-stabilized bullet on a high-angle trajectory.
An over-stabilized bullet rotates
too fast and its axis tends to keep its orientation in space. The bullet´s longitudional axis becomes uncapable
to follow the bending path of the trajectory. Over-stabilization is said
to occur, if the angle enclosed between the bullet´s axis of form and
the tangent to the trajectory (the yaw of repose) exceeds a value of approximately
10°.
Over-stabilization of a bullet is most probable, if a bullet has excessive
static stability (a high value of s g and a low value for the
tractability factor )
and is fired at a high angle of departure, especially when fired vertically.
An over-stabilized bullet on a high-angle trajectory lands base first.
However, when firing bullets from handguns, over-stabilization is of minor
importance in normal shooting situation, but must be considered
when firing at high angles of elevation.
The Yaw of Repose
- c M a - Overturning moment
coefficient derivative coefficient
- d p - Yaw of repose vector
Explanation: The repose angle of yaw (or yaw of repose ,
also called equilibrium
yaw ) is the angle, by which the momentary axis of precession deviates
from the direction of flight.
As soon as the transient yaw induced at the muzzle has been damped out for
a stable bullet, the yaw angle d equals the yaw of repose.
The magnitude of the yaw of repose angle is typically only fractions of a
degree close to the muzzle, but may take considerable values close to the summit
especially for high-elevation angles.
The occurrence of the yaw of repose is responsible for the side drift of spin-stabilized
projectiles even in the absence of wind. The spin-dependent side drift is also
called derivation .
It can be shown that for right-hand twist, the yaw of repose lies to the right
of the trajectory. Thus the bullet nose rosettes with an average off-set to
the right, leading to a side drift to the right.
The above formula for the yaw of repose vector is an approximation
for stable bullet
flight.
If a bullet flies stable (gyroscopically and dynamically!)
and the transient yaw has been damped out, usually after a travelling distance
of a few thousands of calibres, the bullet´s axis of symmetry and the
tangent to the trajectory deviate by a small angle, which is said to be the
yaw of repose .
For bullets fired with right-handed twist, the longitudinal axis points
to the right and a little bit upward with respect to the direction of flight,
leading to a side drift to the right. The yaw of repose, although normally
measuring only fractions of a degree, is the reason for the side deviation
of spin-stabilized bullets.
