• Articles
• Articles about science and technology
Night of the living drag coefficients —
ballistics applied to zombie control
The first article in my series on mathematics applied to the zombie
apocalyze, night of the living differential equations,
dealt with modelling the population dynamics of the humanvszombie
struggle. This article deals with a more prosaic matter: how to ensure
that you use your limited supply of ammunition wisely, in order to
eradicate as many zombies as possible.
You and your small band of battleweary survivors of the zombie
apocalyse are trapped in a
burnedout building, and need to get to your vehicle which is parked
(say) 300 metres away. Unfortunately, said vehicle is surrounded by a
ravening zombie horde, attracted by the smell of the bratwuest
sausage that you left mouldering on the back seat (could happen.)
You have an old rifle and a small supply of
Hornady Zombie Max
ammunition, and
hope to shoot the zombies before trying to get away. The rifle's sights
are perfectly set for a range of 50 metres — you know this because
you've been using it to harvest the feral cats and dogs that now
form your staple diet. You know, however, that to dispose of a zombie
you must destroy its brain, which necessitates a shot right to the
middle of its mouldering forehead.
You're aware, of course, that your rifle does not shoot in a straight
line. If it's correctly sighted for 50m, the bullet will drop
considerably over the next 250m. But how far will it drop? You
can't afford to waste even a single shot, so should you aim two inches
high? Or a yard? With only a computer, the manufacturer's technical
data sheet for your ammunition, and a textbook on fluid dynamics, you
set to work...
The model
For the sake of simplicity, I will assume that you will be battling
zombies on a windless day with no precipitation. The problem is
thus twodimensional, as there is nothing to deflect the bullet in
the leftright direction. Moreover, you and
you targets are at the same height above ground (we will consider
situations where that is not the case later.) The diagram below
shows the important features of the model.
The problem: where to point your gun to achieve a perfect hit on the
zombie's brainbox?
The sight line is the straight line from your eye, through the
rifle sight, to the target. Since your
rifle is correctly sighted (zeroed to use the jargon) at some
distance s_{zero} from the gun muzzle, the bullet
trajectory crosses the sight line at that distance. You'll notice, perhaps,
that the bullet trajectory crosses the sight line at one other point.
This is the point between the muzzle and s_{zero} where
the bullet is still rising, owing to the upward angle of the rifle
barrel. I've drawn this angle hugely exaggerated in the diagram, just
to make the situation clear; in practice the angle between the gun sight
and the gun barrel will rarely be more than a few degrees.
In practice, the sight will be somewhat above the barrel (although
rarely as much as in the diagram) and, if we want to be very accurate,
we need to allow for this height s_{sight} of a few
cm in the calculations.
s_{range} is the distance from the gun muzzle to the
target, measured horizontally. drop is how far below the target
the bullet will fall if we do not make any allowance.
The physics of flying lumps
If you studied physics at school then most likely you were taught that
objects thrown into the air (bullets, baseballs, hats, pizza) follow
a parabolic path, that is, one described by a quadratic function.
The object has a certain constant horizontal momentum imparted by the original
throw, but its vertical momentum will eventually be overcome by
the force of gravity. The mathematics of this situation are pretty
straightfoward, and it's possible to write down a couple of formulae
that describe the horizontal and vertical positions of the flying
object. To be fair, at ranges of up to 100m or so, the flight of
a rifle bullet follows this simple model fairly well. Unfortunately,
this model neglects air resistance (drag) and so, at longer
distances, it is hopelessly inaccurate. Allowing for drag, we see
that the horizontal momentum is eventually lost, and the projectile
will enter what is effectively free fall. A modern rifle
projectile might cover several kilometres before it loses all its
forward momentum, but the trajectory will depart from the ideal model
a long time before that happens (see diagram below.)
Ideal parabolic path (blue line), compared with a more realistic
dragretarded trajectory (red line). Eventually, the bullet will
essentially be in free fall, when drag has overcome all its
horizontal momentum.
Drag can be modelled as a force along the path of the projectile, that is,
it acts against the direction of motion at any instant in time. Drag
depends on a number of relatively constant factors, most notably
the density of
the air surrounding the projectile, and the projectile's shape and size.
However, drag also depends on variable factors, in particular the
projectile's velocity. In simple situations, drag is
proportional to the square of the bullet's axial velocity (that is,
the velocity along its trajectory). What this means (all other things being
equal) is that, as drag tends to reduce the velocity, the drag itself
decreases, so drag does not create a constant retarding force on
motion. This means that it is difficult, perhaps impossible, to reduce
the physical system to a simple set of formulae into which values can
be plugged.
To make matters worse, all other things are not equal. Drag,
in fact, is not really proportional to velocity squared, for all
sorts of reasons. Ballistics physics tends to take the approach of
prentending that this proportionality still holds, and then introduce
velocityrelated compensating factors to allow for the fact that it
doesn't. But we'll get onto that later.
A simple drag model
We'll begin the analysis by pretending that the drag force is proportional
to velocity squared, as is the case for many fluid dynamics problems.
The bullet leaves the muzzle with a known velocity v_{muzzle}
(which will depend
on the ammunition and the barrel construction; the ammunition
manufacturer should be able
to give us a reasonably reliable average value). We can resolve
this velocity into horizontal
components v_{x} and v_{y} — the horizontal
component will be affected by drag only; the vertical by drag and
the effect of gravity. We need to decide which will be the positive and
which the negative direction of v; for no particular reason in
what follows I will assume that v is positive downwards and
to the right (i.e., in reading direction).
From simple mechanics, considering a small increment in time δt
δv_{x} = a_{x} δt,
δv_{y} = a_{y} δt
where a_{x} and a_{x} are the acceleration
in the horizontal and vertical directions, which we assume are constant
during δt. From the velocities we can determine
incremental changes of the positions
s_{x} and s_{y}:
δs_{x} = v_{x} δt,
δs_{y} = v_{y} δt
In this formulation I am measuring s_{y} so that it is
positive in a downwards direction, with s_{y}=0 being
the sight line (and not the line of the gun muzzle). This is an arbitrary
decision, but it necessary to be quite careful about the signs in
this sort of work.
The acceleration a will depend on the forces acting on the bullet.
Again, from basic considerations of mechanics we have:
a_{x} = F_{x} / m,
a_{y} = F_{y} / m + g
where F is the drag force operating on the bullet and g,
as ever, is the accelaration due to gravity. The negative signs
in these expressions reflect the fact that the force of drag will
be opposite to the direction of the bullet's velocity; g is
a positive term because it always acts downwards,
which is the positive direction of velocity in this formulation.
The textbook formulation of the drag force F is
F = ρ v^{2} c_{d} A / 2,
where c_{d} is a (sort of) constant, the coefficient
of drag, and ρ is the air density. A, which has units of
area, is typically called the 'reference area' or 'effective area'.
The idea is that the parameter c_{d} models the shape
of the moving object, and A its size. In practice A is
not usually the crosssectional area of the moving object, because
different objects can have identical areas but very different drag
forces. This means that, in reality, the meanings of
c_{d} and A vary according to the discipline of
study. How these terms are interpreted in ballistics will be discussed
below.
For now, however, we have an expression for F that is dependent,
for a particular projectile, on v^{2} and some constants.
It would be nice to be able to arrange the above expressions into a
pair of secondorder differential equations for s_{x}
and s_{y} in terms of t and solve them.
However these equations will be nonlinear because of the dependence
of F on v^{2} and, if there is an analytic
solution, I don't know what it is. It isn't really worth looking for
one, because when we realize that the drag force is not, in reality,
proportional to v^{2}, but varies in a way that has to
be determined experimentally, it is clear that a numerical solution
is required.
It's straightforward enough to solve the problem numerically, by
starting with initial values of s and
v, taking a small value of δt
(10 milliseconds seems to work quite nicely), and evaluating
F, and thus a, δv, and δs at
that time. Then we increment the s and v, and repeat.
Since we have a known distance that the projectile must cover, we
just iterate until s_{x} reaches that distance, and
read off the corresponding s_{y}, which is the drop
of the bullet at the target distance.
For completeness, we need to adjust the calculated drop
by s_{sight},
the distance that the muzzle is below the sight. In all the
calculations in this article I have assumed that this distance
is 1cm, although it doesn't make a huge difference to the final
results.
Drag forces in ballistics
In ballistics, the drag force is not proportional the square of the
bullet velocity. It's particularly troublesome at velocities
around mach1, the speed of sound. As the bullet slows from
just above supersonic to just below subsonic speeds, there is a
reduction by almost a half in the drag force. Since many bullets
will experience this transition at some point along their trajectories,
this effect cannot be ignored.
In ballistics it is common to formulate the drag equation like this:
F = ρ v^{2} c_{d} A i / 2.
Here the reference area A might just as well be taken to be the
crosssection area of the bullet, because at least we can easily measure it.
Of course, different formulations of A will lead to different
calculations of drag force unless we adjust the drag coefficient
C_{d} accordingly; but since the drag coefficient is — on the whole — something that has to be determined experimentally.
its actual value is irrelevant. In this formulation, C_{d}
is the drag coefficient for a "standard bullet," whose properties I will
describe later.
The term i — not to be confused with imaginary numbers — is
a form factor. This is simply a multiplier that adjusts the drag
that would be observed for the standard bullet by an amount that compensates
for the fact the projectile is not a standard bullet. It turns out that,
for a specific bullet shape, the drag force is broadly proportional to
the mass of the bullet, and inversely proportional to the crosssection
area. So, to a reasonable approximation,
i = (m/d^{2}) / (m_{std}/d_{std}^{2})
where m and m_{std} are the masses of the real projectile
and the standard bullet, and d and d_{std} are
their diameters. Presumably to save time when computations of this sort
had to be done by hand, the standard bullet was defined to be an
object of mass one pound, and diameter one inch. The term
(m_{std}/d_{std}^{2}) is therefore
numerically equal to 1.0, and is frequently omitted in the analysis.
Unfortunately, although this term has no numerical significance, it
has units of pounds/square inch, and this means that great care
has to be taken to insert the correct conversion factors when manipulating
expressions.
This formula for i was derived in the 19th century, when
it was originally
assumed that the specific shape of the bullet would be unimportant. All
that mattered, it was assumed, was the mass and the area exposed to the
air in the direction of flight. Later it was found that, in fact,
differently shaped bullets of the same mass and calibre (diamter)
could have very
different drag properties, and so an additional term,
the ballistic coefficient was introduced to allow for shaperelated drag variation. So
the form factor was rewritten:
i = (m/d^{2}) / (m_{std}/d_{std}^{2}) / BC.
Both form factor and BC are to some extent arbitrary; if they are defined
differently then the differences can, up to a point, be absorbed into
the values determined for C_{d}. However, BC is the figure
that ammunition vendors have largely chosen to measure and publicise for
their products. BC is probably more useful for end users, because it
describes an actual bullet with a specific size and weight. Form factor
is an abstract measurement dependent on projectile shape alone.
Incidentally, there is a certain amount of debate about whether the
term
(m_{std}/d_{std}^{2})
is part of the ballistic coefficient, part of the form factor, or
part of the drag equation itself. Because it has numerical value
1.0 in usual practice, it makes no difference to the results.
However, because of this uncertainty,
both
BC and form factor potentially have units (of pounds per square inch),
although the units are rarely written.
Now we can write the drag equation like this:
F = ρ v^{2} c_{d} A(m/d^{2}) / (m_{std}/d_{std}^{2}) / BC / 2.
Note that A here is in SI units if F is, but m and
d are in pounds and inches, because that is how the reference
bullet is specified. So, although
area = πd^{2}/4
generally, in this case
A = π(d/k_{i})^{2}/4
where k_{i} is the number of inches per metre.
In the expression for F the d^{2} terms
cancel (apart from the conversion constant just mentioned) and,
since there are no further unit conversions to trouble us, we
can omit the
(m_{std}/d_{std}^{2})
term from the drag equation and rewrite it like this:
F = ρ v^{2} (π / 4k_{i}) c_{d} m / BC / 2,
being careful to remember that the mass m is in pounds, even
when all other quantities are expressed in SI units.
It is interesting that, apart from the contribution it makes to mass, the
physical size of the bullet does not feature in this expression. This
leads to the somewhat surprising result that a bullet made of lead,
and a much larger bullet made of, say, marshmallow, would appear to
have identical
ballistic properties (so long as they have the same shape). However, appealing
as the prospect of marshmallow bullets is, it is unlikely that that
would have comparable C_{d} to metal ones.
The G tables
We don't yet have enough information to model the flight of a real bullet
because, although vendors publish values of BC, that figure along does not
capture the variation in drag with velocity. Remember that, for bullets,
drag is not proportional to v^{2}. We could capture
the relationship between drag and v^{2} by using a value
of BC that is velocitydependent, rather than constant, and some models
do exactly that. In practice, however, most ammunition vendors quote
a fixed BC relative to some standard aerodynamic model.
It turns out that, while the relationship between drag and
v^{2} does depend on the bullet, that dependence is very
similar for whole families of bullet shapes. In the mid20th century
various organizations did extensive research into the aerodynamics of
various bullet shapes, and this led to the socalled G tables or
G models. Each table presents the coefficient of drag with
varying velocity for a particular standard projectile. For modern
ammunition, the G1 and G7 tables are most relevant, with G1 being
used more commonly. For reference, the G1 and G7 standard projectiles
are shown below.
The G1 and G7 standard projectiles, along with their coefficients of
drag for varying velocity. Dimensions are expressed as multiples of the
diameter. Note the rapid change of drag coefficient around mach1 for
both bullet shapes.
Using a different ballistic model will lead to a different determination
of BC, and manufacturers should indicate the model they have used
when quoting BC figures.
Application: comparing ammunition
The computation method was described earlier; all we need do now
is to incorporate
the more sophisticated drag model into the math. The Download section of
this article provides a simple Java program that will perform the necessary
trajectory calculation, given the relevant physical and ballistic
properties.
To illustrate the importance of proper ballistic modelling, I will compare
two brands of rifle ammunition at moreorless the opposite ends of
the performance spectrum: Hornady Zombie Max in 0.223 calibre Winchester
format, and a standard 0.22 calibre 'long rifle' subsonic round of
the kind used all over the world for target shooting practice. The
Hornady round fires a 55grain (3.6gm) bullet with a BC of 0.225 at a muzzle
velocity of 3240 feet per second (fps). The standard 0.22 subsonic fires a
roundnosed, 38grain (2.5gm) bullet at just under the speed of sound,
about 1000fps. These subsonic bullets typically have a BC of around 0.125.
The higher BC of the Hornady bullet reflects its more aerodynamic shape.
We might expect that the subsonic bullet will take about three times
as long to reach the target as the Hornday ZMax, since it is ejected
from the barrel
at only one third the speed. If this is true, gravity will have
about three times as long to work on the bullet before impact.
Looking at the plots below, we can see that the subsonic
0.22 bullet has a drop of over 5 metres at 300m range.
That's nearly ten times as much as the ZMax bullet, which itself
drops a notinconsiderable 50cm. What this means is that even
with topquality ammunition, you would still have to aim 50cm
above the zombie's head to be sure of dispatching it. Fortunately
this is within the range of adjustment of most rifle sights, so
it should not be an impossible shot.
Drop of the bullet from the muzzle for two different kinds of
rifle ammunition: Hornady Zombie Max in 0.223 Winchester format,
and a roundnose 0.22 subsonic target round
Application: dealing with elevation
One common problem in ballistics lies in dealing with situations where
the target is above or below the firing point. Suppose for example that
the zombies, as well as being 300m away, are on top of a bridge 50m above
you. Or perhaps they are at the bottom a hill, shambling towards your
position.
The usual finding in situations like this, if you adjust your rifle
so that the point of impact would be correct for the actual distance
to the target, is that the impact is above the target,
whether the target is above or below the firing point.
The actual distance to the target is further than the horizontal
distance whether the target is higher or lower (this is just an
application of Pythagoras' theorem), and this might give
the impression that the bullet would fall further, and that the
point of impact would to too low in both cases. However, in practical
shooting situations the bullet strikes high because gravity has the
greatest effect on a projectile that is travelling perfectly
horizontally. Consider a bullet that is travelling upwards at an
angle of, say, 45 degrees. Gravity will make the bullet drop, of
course, but it has to overcome the initial upwards component of
motion of the bullet. The overall drop is therefore reduced.
If the target is below the firing point, gravity will tend
to speed the bullet on its way to the target. There is therefore
less time for gravity to act on the bullet and, again, it impacts
high. The effects of upwards elevation and downwards elevation are
not symmetrical — in most cases the tendency to impact high is
less pronounced for a low target than a high one.
A common way of compensating for a tendency to shoot high, which
has been known for at least a hundred years, is to adjust
the rifle sight for the correct horizontal distance to the target.
So if, for example, the zombies are on top of a tall building and
you are at the bottom, you should adjust the rifle sight
for the base of the building, not the zombies on top.
The effect of sighting this way is to lower the trajectory of the
bullet whether shooting upwards or downwards, but how effective is
it? The effect of sighting short will tend to turn an error in
shooting high into an error in shooting low but, with luck,
there will be a region of elevation where these two effects tend to
cancel.
We will investigate this elevation compensation approach
using the Hornady Zombie Max
projectile described in the previous section. We work out the
angle of elevation of the barrel (departure angle) to give
a zero point at 300m — the horizontal distance to the target — and then add or substract enough angle to
raise the sight point to the target elevation. If the target
is elevated by 50m, for example, we must raise the muzzle by
9.46 degrees [=atan(50/300)] to sight on the zombie on the
top of the building. Then we can use the
procedure described previously to compute the trajectory iteratively,
and work out the height of the bullet at 300m horizontal distance.
The graph below shows the effect of sighting this way. Note that
there is a region of relatively small impact error for
a target between about 20m above and 30m below the shooting position.
Error in height of point of impact with targets at various elevations
with respect to the shooter. Positive elevation means that the zombie
is above the shooter (aargh!), negative below. Given that a zombie brain
is about 10cm in diameter, the grey box indicates the range within which
we need not adjust for elevation — about 20m above to 35m below.
So the method works, to a limited extent. For angles of elevation more
than about 10 degrees from the vertical it will not really deliver the
accuracy required for zombie control; but, very broadly speaking,
it turns out that this method will reduce the error in point of impact
by about 50% over a relatively wide angle.
Further work
Readers interested in this kind of thing might like to consider how
windage — the effect of wind on the bullet — might be treated.
Winds close to the Earth's surface tend to blow horizontally so,
to a large extent, can be treated independently of
of gravity.
Wind will affect both the lateral motion of the projectile and the motion
towards the target, according to the wind direction. Wind blowing
from the shooter towards the target will have the effect of increasing
the projectile's velocity and ultimately reducing the amount of drop.
Wind directly towards the shooter from the target will have the opposite
effect. Neither of these winds, however, will blow the bullet off course
and lead to a miss in the leftright direction. Wind that is blowing
directly across the direction of flight will cause the largest
leftright error, although it will have only a small effect on drop.
Mathematically, windage can be modelled as a drag force in the
same way as aerodynamic drag. For windage, however, the coefficient
of drag will depend not only on the projectile shape, but the angle that the
wind makes with the direction of travel, which complicates matters
considerably.
Download
BDC.java is a Java program that implements the
iterative algorithm discussed in this article, for specified ballistic
and physical properties.
