How do bullets fly?

Author: Ruprecht Nennstiel, Wiesbaden, Germany

   

 Abstract

This document attempts to explain the basics of the complicated subject of bullet motion through the atmosphere and avoids formulas as well as mathematics, but expects familiarity with the way of physical thinking. It includes new experimental observations of bullets fired from small arms, both at short and at long ranges. Numerous illustrations are included and can be viewed via links to promote further understanding. This article is also thought as an introduction for all types of readers (hunters, sportsmen, ballisticians, forensic scientists), interested in the "mysteries" of the exterior ballistics of bullets, fired from small arms.

This document can also be downloaded and can then be read offline.

 Contents   

In reading this paper, it is recommended to follow the succession of chapters below.

Introduction Flying bullets on a microscopic scale

Forces and moments affecting flying bullets

Bullet stability

Bullets at short ranges

Bullets at long ranges

Anomalies of bullet flight

Summary, Acknowledgments, References

FAQs

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   Figures   

Viewing of figures is highly recommended!

Shadowgraphs

Shadowgraph photography - experimental set-up Shadowgraph of a .308 Win. bullet at supersonic flight

Shadowgraph of a 9mm Luger FMJ bullet at supersonic flight

Shadowgraph of a .32 ACP bullet at subsonic flight

Basics

The wind force Adding two forces to the wind force

The overturning moment

The gyroscopic effect

The Magnus effect

The Magnus force

The Magnus moment

Static stability factor (example)

Over-stabilized bullet on a high-angle trajectory

The yaw of repose

The yaw of repose for a 7.62 x 51 NATO FMJ bullet fired at 32°

Long range Doppler radar velocity measurement of a 7.62 x 51 NATO bullet

Coning motion of destabilized NATO bullet

Doppler radar velocity measurement of coning bullet

Yawing motion of handgun bullets

Two arms model of yawing motion Yawing motion in general

Experimental set-up to investigate yawing motion

Yawing motion of M193 bullet cal. 5.56 x 45

Yawing motion of an armor-piercing bullet, cal. 5.56 x 45

Yawing motion of M74 bullet, cal. 5.45 x 39

Yawing motion of an armor-piercing bullet, cal. 7.62 x 51 NATO

Yawing motion of M80 bullet, cal. 7.62 x 51 NATO

Yawing motion of KTW bullet, cal. .357 magnum

Yawing motion of 9mm Luger FMJ RN bullet

Yawing motion of a wadcutter bullet, cal. .38 special

Yawing motion of .32 ACP FMJ RN bullet

 Formulas   

Something for those readers who love physics!

The force of gravity The centrifugal force

The Coriolis force

The drag force

o The drag coefficient

o The ballistic coefficient (bc)

The lift force

The spin damping moment

The Magnus force

The overturning moment

The Magnus moment

The gyroscopic stability condition

The dynamic stability condition

The stability triangle

The tractability condition

The yaw of repose

  Software 

A recommended link for present and future ballisticians - download a test version!

EBV4 - Exterior ballistics software for the PC

This document has been continuously improved by the input of readers!  Many thanks to all of them! Further suggestions are welcome! Before posing a question, please read the FAQ chapter!

Click to send an eMail to the author.

Last Update of this document: March, 13th, 2005    

IntroductionThis document is an improvement of a presentation given by the author at the 1995 AFTE meeting in San Diego, CA (see reference [1]) and a publication in the AFTE Journal (see reference [2]). It is mainly intended for the following readership:

Hunters and sportsmen, having an interest in exterior ballistics, Students and teachers,

Forensic firearm examiners, especially "newcomers" in the field.

The title of this document "How do bullets fly?" seems to be an odd question, and appears to be almost foolish at first sight..

However, as in many fields of science and technology, studying an apparently simple matter more thoroughly, may bring to light complex and complicated facts. This indeed is the case with regard to the motion of spin-stabilized bullets fired from guns.

Most people expect that bullets fly nose-forward and remain stable from the muzzle to the target, but this is not necessarily so. For short ranges, most trajectories could be approximated by a straight line, whereas bending of the trajectory must be considered for longer ranges.

Most firearm experts accept that bullets may tumble when grazing an object or when leaving an intermediate target. However, as it will be outlined,  some physical conditions must be fulfilled to guarantee stable flight, and bullets are by no means automatically stable. Causing a bullet to spin endowes it with gyroscopic properties which are very important - but by no means exclusively - in maintaining bullet stability.

However, from a teacher's point of view, the motion of a spinning gyroscope is one of the most complicated motions with which a student of physics is confronted during lectures on classical mechanics. Although the general motion of gyroscopes can be explained and completely understood only by a thorough mathematical treatment, this introduction makes an attempt to explain the elements of the subject by means of numerous illustrations. The use of formulas is limited to those who wish to see them (note various links to view

formulas, indicated by the  icon).

For the explanation of some general physical terms used in this article, the interested reader is requested to refer to an elementary physics textbook.

How to use this document

I suggest reading this document according to the order of chapters given in the main page.

The main page can always be reached by a click on the  button.

The  button always returns you to the top of the currently selected page.

Within the text, you will find numerous links to figures and formulas. Viewing figures is highly recommended. Formulas can be viewed by those readers who are interested in a more mathematical description.  However no mathematical derivations should be expected.

You can link to a figure by a click on the  button.

You can link to a formula  by a click on the  button.

A click on the  button will return you back to the text from a page displaying a figure or a formula.

This document may be copied and used freely for non-commercial use, especially for educational purposes.

Downloading this document to your local hard drive and reading it offline is highly recommended.

Microscopic ScaleAll aerodynamic forces affecting a bullet´s flight through the air result from the interaction of the bullet with the surrounding airflow. To understand this interaction, it is worth viewing the subject on a microscopic scale.

A quite simple experimental photographic technique which enables the visualization of the flow of air in the vicinity of a moving body produces a picture called a "shadowgraph".

This technique requires a short duration flash of light, which must originate from a point.

As shown in the figure  , not even a photographic lens is required. The shadowgraph of the bullet, passing at very close distance in front of a film or photographic plate, visualizes the pressure differences of the flowfield particularly well.

The pictures shown in the three following figures were taken by applying this simple but effective technique.

The first photograph shows a .308 Winchester (7.62 x 51 Nato) FMJ bullet traveling at

approximately 2800 ft/s (approx. 850 m/s) (see picture  ; Be patient! Depending on your connection, loading of this high-resolution grey scale picture may take some time!).

One may distinguish at least three different shock waves. The first and most intensive one emerges from the bullet's nose and is called the Mach cone. A second shock wave originates from the location of the cannelure, and the third shock wave forms behind the bullet's base. Additionally one can see highly turbulent flow behind the base, which is called the wake.

The flow type at the bullet's surface changes from a laminar boundary layer at the forward region of the bullet, which is characterized by parallel stream lines, into a turbulent flow showing vortexes, beginning at the cannelure.

For a 9 mm Luger FMJ pistol bullet, moving slightly faster than the speed of sound (see

picture  Be patient! Depending on your connection, loading of this high-resolution grey scale picture may take some time!) one finds the following significant differences: the Mach cone is still present but no longer attached to the bullet' s tip, and the opening angle of this cone has increased. The wake is still visible, but the boundary layer appears to be laminar from the tip to the base, all along the bullet' s surface.

Finally, for a cal. .32 ACP pistol bullet, moving at a speed considerably below the speed of

sound (see picture  Be patient! Depending on your connection loading of this high-resolution grey scale picture may take some time!), all shock waves are absent, and what remains is the turbulences behind the bullet's base.

Forces and momentsHint: If you are not familiar with the physical concept of forces and moments, reference is made to any elementary physics textbook (chapter classical mechanics, forces, moments).

The shadowgraphs have shown that the flowfield in the vicinity of a bullet most generally consists of laminar and turbulent regions. The flowfield depends in particular on the velocity at which the bullet moves, the shape of the bullet and the roughness of its surface, just to mention the most important factors. The flowfield obviously changes tremendously,

as the velocity drops below the speed of sound, which is about 1115 ft/s (340 m/s) at standard atmospheric conditions.

The mathematical equations, by means of which the flowfield parameters (for example pressure and flowfield velocity at each location) might be determined are well known to the physicist as the Navier-Stokes equations. However, having equations and having useful solutions for these equations are entirely different matters. With the help of powerful computers, numeric and practically useful solutions to these equations have been found up to now only for very specific configurations.

Because of these computational restrictions, ballisticians all over the world consider bullet motion in the atmosphere by disregarding the specific characteristics of the flowfield and apply a simplified viewpoint: the flowfield is characterized by the forces and moments affecting the body. Generally those forces and moments must be determined experimentally, which is done by shooting experiments and through wind tunnel tests.

Generally, a body moving through the atmosphere is affected by a variety of forces. Some of those forces are mass forces, which apply at the CG (center of gravity) of the body and depend on the body mass and the mass distribution. A second group of forces is called aerodynamic forces. These forces result from the interaction of the flowfield with the bullet and depend on the shape and surface roughness of the body. Some aerodynamic forces depend on either yaw or spin or both. A summary of the most important forces affecting a bullet's motion through the atmosphere is shown in the table below. As an example, another table gives the magnitude of forces for a typical military bullet.

Table: Forces, affecting a bullet's movement through the air

Forces Requires

Remarks

  Yaw Spin  

Mass Forces      

Gravity N N responsible for bending of trajectory

Coriolis Force N N usually very small

Centrifugal Force N N small; usually included in gravity

Aerodynamic Forces

     

Drag Y N major aerodynamic force

Lift (Cross-wind Force)

Y N responsible for side drift

Magnus Y Y very important for stability

Pitch Damping Y Y usually very small, important for stability

Transversal Magnus

Y Y usually very small

Mass forces

The most simple "ballistic model", considering only the force of gravity , was discovered by Galileo Galilei (1590). A discussion of this important force can be found in any elementary physics book.

As we intend to study the movement of bullets on earth, we have to consider its rotation. However, Newton's equations of motion are only valid in an inertial reference system - which either rests or moves with constant speed. As soon as we consider bullet motion in a reference frame bound to the rotating earth, we have to deal with an accelerated reference frame. But we can compensate for that - and still use Newton's equations of motion - by

adding two additional forces: The centrifugal force and the Coriolis force

.

Wind force and overturning moment

Let us consider the most general case of a bullet having a yaw angle . By saying so, the ballistician means that the direction of motion of the bullet' s CG deviates from the direction into which the bullet's axis of symmetry points. Innumerable experimental observations have shown that an initial yaw angle at the muzzle of a gun is essentially unavoidable and is caused by perturbations such as barrel vibrations and muzzle blast disturbances.

For such a bullet, the pressure differences at the bullet's surface result in a force, which is called the wind force. The wind force seems to apply at the center of pressure of the wind force (CPW), which, for spin-stabilized bullets, is located in front of the CG. The location

of the CPW is by no means stationary and shifts as the flowfield changes. The figure  schematically shows the wind force F1, which applies at its center of pressure CPW.

As shown in another figure  , it is possible to add two forces to the wind force, having the same magnitude as the wind force but opposite directions. If one let those two forces attack at the CG, these two forces obviously do not have any effect on the bullet as they mutually neutralize.

Now let us consider the two forces F1 and F2. It can be shown that this couple is a free vector, which is called the aerodynamic moment of the wind force or, for short, the overturning momentMW. The overturning moment tries to rotate the bullet around an axis, which passes through the CG and is perpendicular to the bullet's axis of form, just as

indicated in the figure  .

Summary: The wind force, which applies at the center of pressure, can be replaced by a force of the same magnitude and direction plus a moment. The force applies at the CG, the moment turns the bullet about an axis running through the CG.

This is a general rule of classical mechanics (see any elementary physics textbook) and applies for any force that operates at a point different from the CG of a rigid body.

You may proceed one step further and split the force, which applies at the CG, into a force which is antiparallel to the direction of movement of the CG plus a force, which is

perpendicular to this direction. The first force is said to be the drag force FD or

simply drag, the other force is the lift force FL or lift for short. The name lift suggests an upward directed force, which is true for a climbing airplane, but which is generally not true for a bullet. The direction of the lift force depends on the orientation of the yaw angle. Thus a better word for lift force could be cross-wind force, an expression which can be found in some ballistic textbooks.

Obviously, in the absence of yaw, the wind force reduces to the drag force.

So far, we have explained the forces, how the wind force and the overturning moment

are generated, but we haven' t yet dealt with their effects.

Drag and lift apply at the CG and simply affect the motion of the CG. Of course, the drag retards this motion. The effects of the lift force will be met later.

Obviously, the overturning moment tends to increase the yaw angle, and one could expect that the bullet starts tumbling and become unstable. This indeed can be observed when firing bullets from an unrifled barrel. However, at this point, as we consider spinning projectiles, the gyroscopic effect comes into the play, causing an unbelievable effect.

The gyroscopic effect can be explained and derived from general rules of physics and can be verified by applying mathematics. For the moment we simply have to accept what can be observed: due to the gyroscopic effect, the bullet' s longitudinal axis moves aside towards the direction of the overturning moment, as indicated by the arrow in the

figure  .

As the global outcome of the gyroscopic effect, the bullet's axis of symmetry thus would move on a cone's surface, with the velocity vector indicating the axis of the cone. This movement is often called precession. However, a more recent nomenclature defines this motion as the slow mode oscillation.

To complicate everything even more, the true motion of a spin-stabilized bullet is much more complex. An additional fast oscillation is superimposed on the slow oscillation. However, we will return to this point later.

Spin damping moment

Skin friction at the projectile's surface retards its spinning motion. However, the angular

velocity of the rotating bullet is much less damped by the spin damping moment   than the translational velocity, which is reduced due to the action of the drag force. As will be shown later, this is the reason why bullet's, which are gyroscopically stable at the muzzle will remain gyroscopically stable for the rest of their flight.

Magnus force and Magnus moment

Generally, the wind force is the dominant aerodynamic force. However, there are numerous other smaller forces but we want to consider only the Magnus force, which turns out to be very important for bullet stability.

With respect to the figure  , we are looking at a bullet from the rear. Suppose that the bullet has right-handed twist, as indicated by the two arrows. We additionally assume the presence of an angle of yaw . The bullet's longitudinal axis should be inclined to the left, just as indicated in the previous drawings.

Due to this inclination, the flowfield velocity has a component perpendicular to the bullet's axis of symmetry, which we call vn.

However, because of the bullet's spin, the flowfield turns out to become asymmetric. Molecules of the air stream adhere to the bullet's surface. Air stream velocity and the rotational velocity of the body add at point B and subtract at point A. Thus one can observe a lower flowfield velocity at A and a higher streaming velocity at B. However, according to

Bernoulli's rule (see elementary physics textbook), a higher streaming velocity corresponds with a lower pressure and a lower velocity with a higher pressure. Thus, there is a pressure difference, which results in a downward (only in this diagram!) directed force, which is

called theMagnus force FM (Heinrich Gustav Magnus, *1802, died 1870; German physicist).

This explains, why the Magnus force, as far as flying bullets are concerned, requires spin as well as an angle of yaw, otherwise this force vanishes.

If one considers the whole surface of a bullet, one finds a total Magnus force, which applies

at its instantaneous center of pressure CPM (see figure  ). The center of pressure of the Magnus force varies as a function of the flowfield structure and can be located behind, as well as in front of the CG. The magnitude of the Magnus force is considerably smaller than the magnitude of the wind force. However, the associated moment, the discussion of which follows, is of considerable importance for bullet stability.

You can repeat the steps that were followed after the discussion of the wind force. Again, you can substitute the Magnus force applying at its CP by an equivalent force, applying at

the CG, plus a moment, which is said to be the Magnus moment MM . This moment tends to turn the body about an axis perpendicular to its axis of symmetry, just as shown in

the figure  .

However, the gyroscopic effect also applies for the Magnus force. Remember that due to the gyroscopic effect, the bullet's nose moves into the direction of the associated moment.

With respect to the conditions shown in the figure  , the Magnus force thus would have a stabilizing effect, as it tends to decrease the yaw angle, because the bullet's axis will be moved opposite to the direction of the yaw angle.

A similar examination shows that the Magnus force has a destabilizing effect and increases the yaw angle, if its center of pressure is located in front of the CG. Later, this observation will become very important, as we will meet a dynamically unstable bullet, the instability of which is caused by this effect.

Two arms model of yawing motion

We have now finished discussing the most important forces and aerodynamic moments affecting a bullet's motion, but so far we haven't seen what the resulting movement looks like. For the moment we are not interested in the trajectory itself (the translational movement of the body), but we want to concentrate on the body's rotation about the CG.

The yawing motion of a spin-stabilized bullet, resulting from the sum of all aerodynamic moments can be modeled as a superposition of a fast and a slow mode oscillation and can most easily be explained and understood by means of a two arms model (see reference [4]).

Imagine looking at the bullet from the rear as shown in the figure  . Let the slow mode arm CG to A rotate about the CG with the slow mode frequency. Consequently point A moves on a circle around the center of gravity.

Let the fast mode arm A to T rotate about A with the fast mode frequency. Then T moves on a circle around point A. T is the bullet's tip and the connecting line of CG and T is the bullet's longitudinal axis.

This simple model adequately describes the yawing motion, if one additionally considers that the fast mode frequency exceeds the slow mode frequency, and the arm lengths of the slow mode and the fast mode are, for a stable bullet, continuously shortened.

With respect to the figure  imagine looking at a bullet approaching an observer's eyes. Then the bullet's tip moves on a spiral-like (also described as helical) path as indicated in the drawing, while the CG remains attached to the center of the circle. The bullet's tip periodically returns back to the tangent to the trajectory. If this occurs, the yaw angle becomes a minimum.

StabilityWe are now in a position to discuss the conditions a bullet has to fulfill to fly in a stable condition. By saying that a bullet flies in a stable state, we generally mean that the bullet's longitudinal axis tends to point into the general direction of movement.

It can be shown that a stable bullet has to fulfil three different conditions:

it must be statically stable, it must be dynamically stable,

it has to be tractable.

Static stability

If the gyroscopic effect takes place, so that a bullet responds to the wind force by moving its nose into the direction of the overturning moment, one says that the bullet is statically (or equivalently: gyroscopically) stable. If a bullet is not statically stable, for example, if it is fired from a smooth bore barrel, the overturning moment will cause the bullet to tumble. A bullet can be made statically stable by sufficiently spinning it.

Statically unstable handgun bullets will hardly be met in "real life", because such a projectile would be useless. However, when fired with insufficient spin, "well-designed" bullets may be statically unstable.

It is possible to define a static stability factor sg and derive a static (or gyroscopic)

stability condition , which simply demands that this factor must exceed unity.

As an example, the figure  displays the static stability factor for the 7.62 x 51 Nato M80 bullet, fired at 32° to the horizontal. The M80 bullet exits the muzzle with a static stability factor of 1.35. Obviously, the static stability factor continuously increases at least for the major part of the trajectory or more generally, always exceeds its value at the muzzle. Generally, it can be assumed that if a bullet is statically stable at the muzzle, it will be statically stable for the rest of its flight.

Dynamic stability

A bullet is said to be dynamically stable, if an angle of yaw, induced at the muzzle, is damped out with time, or in other words if the angle of yaw decreases as the bullet travels

on. It can be shown that this is true, if the dynamic stability condition is fulfilled.

If, on the contrary, a bullet is dynamically unstable, the angle of yaw increases.

The occurrence of an initial yaw close to the muzzle is by no means an indicator of bullet instability. In some recent publications, the statements "bullet is unstable" and "bullet shows a (big) yaw angle" are used synonymously which is incorrect. On the contrary, an initial yaw angle at the muzzle is inevitable and results from various perturbations.

Bullets fired from handguns are not automatically dynamically stable. Bullets can be dynamically unstable at the moment they leave the barrel. Other bullets are dynamically stable close to the muzzle and loose dynamic stability as they continue to travel on, as the flowfield changes.

Tractability

According to our general definition of stability, a bullet may become unstable by being over-stabilized. Over-stabilization means that the bullet rotates too fast and becomes incapable of following the bending trajectory, as its longitudinal axis tends to keep its direction in space. This effect is often observed for high-angle shooting, but is of minor interest in normal shooting situations.

The figure  schematically shows an over-stabilized bullet fired at a high angle of elevation, which lands base first.

Mathematically, a bullet is said to be tractable, if the tractability condition is fulfilled.

Short rangesIn the preceding chapter, the physics of bullet motion under the influence of mass forces and aerodynamic forces has been studied. The remaining part of this document will be dedicated to experimental observations for handgun bullets which will generally confirm these findings.

Experimental set-up

The experimental set-up which is used by ballistic research institutes to study the yawing motion

of bullets, is shown in the figure . The bullet coming into the field of view of a camera is illuminated by a light flash of short duration which must be concentrated in a point. By means of an optical system, consisting of two mirrors and two scotchlite reflecting foils, two shadowgraphs are taken: first, a direct side view picture from the standpoint of the camera, and second a picture taken from above.

From the two bullet views which are available on a single photographic plate, the spatial orientation of the bullet's longitudinal axis can be evaluated, especially the spatial yaw angle can be determined.

Numerous photographic stations can be set up, one behind the other, allowing to determine the yaw angle, the angle of precession, and the location of the CG as a function of the traveling distance.

Yawing motion of handgun projectiles

Using this sophisticated and labor-intensive photographic technique, numerous brands of handgun projectiles were investigated and the results for some selected bullets are presented hereafter.

Stable bullets

The figure shows the trace, the tip of a M193 bullet (caliber 5.56 x 45) would leave in space from the moment it exits the muzzle, up to a distance of 8000 calibers, which corresponds to approximately 150 feet (45 m). If one imagines that the bullet's CG moves on a straight line, which is located in the center of the box, the curved path displays the location of the bullet's tip in space as it travels through the air.

You may also read approximate values for the maximum yaw angle, which does not exceed two degrees in this example. Although the drawing does not display it very clearly, the yawing motion of this M193 bullet is undamped. However, other experiments have shown that the M193 bullet may show small damping as well.

The next example (see figure  ) shows the yawing motion of a hard core, armor piercing bullet of the same caliber (5.56 x 45). This time it is undoubted that the yawing motion is damped, or with other words, the projectile is dynamically stable. However, a maximum yaw angle of more than five degrees could be observed close to the muzzle.

The distance between two successive extremes in yaw is about seven meters.

The next figure  displays the yawing motion of the Russian M74 bullet. One can observe maximum yawing angles of up to three degrees close to the muzzle. Again the yawing motion is damped, but has become more complicated. It requires quite a bit of perseverance to follow the path of the bullet's tip.

The fast modal arm is damped to one half after a traveling distance of 30 meters, whereas the slow modal arms requires twice as much to be damped to one half.

Our consideration of military bullets will be finished by two investigations on 7.62 x 51 Nato bullets.

The next figure  presents the behavior of a hard core armor piercing bullet, showing a very symmetric and easy to follow path. However, yawing angles of more than ten degrees have been observed in this example.

The yawing period, the spatial distance between two successive extremes in yaw, is approximately eight meters.

The final example (see figure  ) refers to the standard M80 bullet (7.62 x 51 Nato). Obviously, the yawing motion has become very complicated and the path of the bullet's tip is not easy to follow, yet still shows a symmetric and repetitive structure.

So far we have only met good-natured bullets. All of them had sufficient static stability with a static stability factor at the muzzle ranging between 1.1 and 2. In almost all of the cases, the yawing motion was damped, saying that the bullets were also dynamically stable.

This conclusion is not very surprising. All of these bullets were designed for military use and warfare. Many engineers and scientists in ballistic research institutes were occupied in optimizing this ammunition. Each of those bullets probably has undergone multiple ballistic improvements and refinements, based on shooting experiments and wind tunnel tests. Therefore, it would be more than amazing, to find any bad exterior ballistic properties with these bullets.

Over-stabilized bulletsIt can be asked, whether bullets fired from pistols and revolvers show the same behavior as those well-designed military projectiles.

The next figure  shows an investigation for the .357 magnum KTW bullet, fired from a Colt revolver at a maximum shooting distance of approximately 240 feet (70 m). The path of the bullet's nose shown in this drawing is characteristic for an over-stabilized bullet. Too much spin is transferred to the bullet. The frequency of the fast mode oscillation, also called nutational frequency, is very high (more than 1000 revolutions per second) and the bullet responds in a very nervous way. Obviously, the yawing motion is damped, as the maximum yaw angle continuously decreases, with a half life for the fast mode oscillation of 22 meters.

A second example is presented in another figure  . A 9 mm Luger FMJ RN bullet displays a similar behavior. An evaluation shows that the bullet has a static stability factor at the muzzle of 22.5. This is much too high as compared with the necessary value of one. The bullet shows very good damping. After a traveling distance of approximately 6000 calibers (170 feet = 50 m) the maximum yaw has been damped to almost nothing.

It has been a general observation that many bullets fired from pistols and revolvers are over-stabilized. However, the question remains to be answered, whether excessive spin, as demonstrated for the last two examples, may express in any ballistic disadvantages.

If one considers only short ranges, let us say, up to a few thousand calibers, which is generally the distance, within which pistols and revolvers are used, excessive spin does not influence accuracy. However, if fired at high angles of elevation, the bullet's longitudinal

axis may not follow the curved trajectory path, tends to keep its orientation in space and, as a consequence, the bullet may impact base first.

Long rangesUp to this point, we have only considered bullets at close distances from the muzzle. We have met well-designed military projectiles and over-stabilized pistol and revolver bullets, but all of them showed dynamic stability. In other words, the maximum yaw angle, which occurs close to the muzzle, is damped out as the bullet moves on. After a traveling distance of a few thousand calibers, depending on the damping rate, the transient yaw angle practically approaches zero.

The yaw of repose

Now let us consider a stable bullet, which has traveled a considerably longer distance. If the transient yaw has been damped out for a dynamically stable spin-stabilized projectile, does that mean that the bullet's longitudinal axis exactly coincides with the direction of movement of the CG?

It can be found from a mathematical treatment that the bullet's longitudinal axis and the direction of the velocity of the CG deviate by a small angle, which is said to be the

equilibrium yaw or the yaw of repose   . For right-handed spin bullets, the bullet's axis of symmetry generally points to the right and a little bit upward with respect to the direction of the velocity vector- indicating the direction into which the CG moves - , just as

shown in the figure  .

As an effect of this small inclination, there is a continuous air stream, which tends to deflect the bullet to the right. Thus the occurrence of the yaw of repose is the reason for bullet drift to the right (for right-handed spin) or to the left (for left-handed spin).

Usually, the yaw of repose is a very small angle and measures only fractions of a degree.

The figure  shows the variation of the yaw of repose angle along the trajectory for a 7.62 x 51 NATO M80 bullet fired at 32°. Although, in this example, the yaw of repose never exceeds half a degree, the resulting side drift at impact almost amounts to 100 yards.

AnomaliesAll of the examples shown so far may lead to the conclusion that instabilities do not occur in reality and many of the theoretical aspects, which were considered in the beginning of this paper, are of no practical significance.

The remaining part of this article is dedicated to anomalies or - probably a better expression - unexpected bullet behavior, which a shooter may be confronted with.

Statically unstable bullet

As already said, static instability is normally nothing one has to be concerned with, when dealing with well-designed bullets fired from small arms. However, the results of an experimental investigation by Giles and Leeming (see reference [5]) for 7.62 x 51 NATO bullets, fired from a smooth bore barrel, can be extended, at least to some degree, to the post ricochet flight of unstable projectiles.

In their experiments, Giles and Leeming found "no evidence of purely end over end tumbling" and that the unspun NATO bullets flew "near to broadside-on" and "were simply slewing sideways following launch and then damping rapidly to an equilibrium position featuring a large angle of yaw". The authors were even able to estimate this stationary yaw angle on a theoretical base by applying the crossflow analogy and found that the statically unstable NATO bullet flies base forward with a stationary yaw angle of approximately 127°.

Dynamically unstable bullet

Close to the muzzle

The figure shows the yawing motion of a .38 Special Wadcutter bullet fired from a revolver. The bullet is statically stable, the static stability factor of more than four indicates that it even has too much spin. Obviously, the bullet is dynamically unstable. The slow modal arm oscillation continuously increases. The maximum yaw angle increases approximately by a factor of three, from a value of five degrees at the muzzle to 15° after a traveling distance of 8000 calibers (240 feet =73 m).

A similar conclusion, with the exception of the magnitude of the yaw angle, can be drawn for the .32 ACP  FMJ RN bullet (7.65 mm Browning), fired from a pistol. This is shown

in the figure  .

It is justifiable to ask, whether these instabilities have a significant practical effect. As far as short ranges within a few thousand calibers are considered, dynamic bullet instability may hardly be detected, except when applying highly sophisticated measuring techniques. If a bullet vastly exceeds that range, the yaw angle gets to a considerable magnitude, the drag increases, and accuracy suffers. Most probably, shot-to-shot variations will become enormous and the trajectories become unpredictable.

At a long distanceFinally, we will examine a bullet, which is stable close to the muzzle, but looses dynamic stability after having traveled a considerable distance.

The two drawings in the figure  display the velocity-vs.-time curve of a standard 7.62 x 51 NATO bullet, fired at almost 40°. The measurement has been taken by a long-range Doppler radar tracking system, which is capable of following the bending trajectory from the muzzle to the impact (see acknowledgements).

At first sight, everything seems to be normal. The bullet's velocity considerably decreases close to the muzzle, and after a total flight time of almost 30 seconds, the bullet impacts at a distance of more than 2.5 kilometers.

A closer examination of the velocity-vs.-time curve, starting at 14 seconds after launching, clearly displays an oscillating behavior. A zoomed sector of the velocity-vs.-time curve is

shown in the lower drawing of the figure  .

An evaluation shows that the frequency of this velocity oscillation increases from approximately one revolution per second at 20 seconds of flight time to almost two revolutions per second at 28 seconds.

It is beyond doubt that the Doppler radar measurements are not erroneous. On the other hand, we have not met an aerodynamic force which could be responsible for accelerating and decelerating a bullet to cause an oscillating velocity.

This experimental observation can be explained by the dynamic instability of the 7.62 x 51 NATO bullet at low velocities.

We have learned from a previous figure that the 7.62x51 NATO bullet is statically and dynamically stable close to the muzzle. Thus, the yawing motion will be damped, and after a certain traveling distance, the yaw, with the exception of the small yaw of repose, will practically be zero.

Once the bullet's velocity has been considerably retarded and it moves at a subsonic velocity, the flowfield has changed tremendously. It has been found by an experimental investigation of the BRL (see reference [6]) that one of the consequences of the flowfield

change is the displacement of the center of pressure of the Magnus force. For supersonic velocities, this point is located behind the CG, but moves in front of the CG for subsonic

velocities. As seen previously (see figure  ), the Magnus moment thus turns to be a strong destabilizing moment, and as a consequence, the bullet becomes dynamically unstable.

The slow mode oscillation, also called precession, will no longer be damped and slowly increases. However, the bullet still has excessive static stability and thus the gyroscopic effect continues to take place.

As a consequence, the bullet's longitudinal axis moves on the surface of a cone with the trajectory being the axis of the cone. As this oscillation is undamped, the opening angle of

the cone continuously increases. The figure  schematically shows the coning motion of the NATO bullet on the descending branch of the trajectory.

Keeping this in mind, the experimental findings of the Doppler radar velocity measurement can now easily be interpreted. It should be remembered that the Doppler radar technique is only capable to measure the radial velocity of an object in the radar beam. This means that the Doppler analyzer only detects velocity components, which either approach or withdraw from the antenna.

For a bullet on the descending path of the trajectory, describing a coning motion, the velocity of the body, the axis of which withdraws from the antenna adds to the radial velocity of the CG. On the other hand, the velocity of the body axis which approaches the antenna subtracts from the radial velocity of the CG. This explains the oscillating nature of the measured Doppler signal. It results from a superposition of the radial velocity of the CG and the slow mode oscillation of the bullet's longitudinal axis. This is schematically shown

in the figure  .

What is most amazing is the fact that the antenna detects all this for a tiny little bullet, flying far away at a distance of more than two kilometers.

Obviously, the dynamic instability of the NATO bullet has a tremendous effect on its trajectory. As the yaw increases, the drag increases, the bullet's velocity is far more retarded and the range decreases. It has been observed that for the studied brand of NATO bullets, instabilities were not reproducible and thus the ranges, even when firing with almost the same muzzle velocity and almost the same angle of departure vary enormously, simply by chance.

As a further consequence, exterior ballistic calculations (see reference [3]) based either on the point mass model or the modified point mass model will not be able to predict accurately the trajectory of such an unstable bullet.

SummaryBullets fired from handguns follow general rules of physics and behave like gyroscopes. The angular motion of these bullets can be understood as a superposition of two oscillations, most easily be demonstrated by a two arms model.

Practically all handgun bullets are statically stable, many pistol and revolver bullets even have excessive static stability.

However, dynamic stability is not automatically guaranteed. Some bullets are dynamically unstable at the moment they leave the muzzle, others may loose dynamic stability during flight after being decelerated.

At the moment no reliable method exists, except experimentation, to foresee dynamic bullet instability, especially at long ranges. Some highly sophisticated computerized procedures (numerical solutions to the Navier-Stokes equations) to attack these problems are just being developed by ballistic researchers.

AcknowledgementsI want to thank Prof. Dr. J.G. Hauck, Dresden for reviewing the manuscript and his most valuable suggestions to improve this document.

The experimental determination and evaluations of the yawing motion of various spin-stabilized handgun bullets were carried out at the German army proving ground WTD91 in Meppen. I have to thank this organization for leaving me the reports containing those unique and extremely valuable results.

The long-range Doppler tracking radar measurements of the 7.62 x 51 NATO bullet were made available to me by WEIBEL equipment, Gentofte, Denmark.

I also want to thank all readers of this article for their suggestions and support.

References[1] Nennstiel, R., AFTE Training Seminar, 5.6.-9.6.1995, San Diego, CA USA

[2] Nennstiel, R., "How do bullets fly?", AFTE Journal, Vol.28, No.2, April 1996, S.104-143

[3] Nennstiel, R., "EBV4 User's Manual", Exterior Ballistics for the PC, Wiesbaden, Germany, 1995.

[4] Farrar, C.L., Leeming, D.W., Military Ballistics - A Basic Manual, Brassey´s Publisher Limited, Headington Hill Hall, Oxford OX3 0BW, England, 1983.

[5] Giles M.J., Leeming D.W., "An Aerodynamic Model for Unstable Projectiles", Proc. of the 11th Int. Symp. on Ballistics, Brussels, May 9 - May 11, 1989.

[6] Piddington, M.J., "Aerodynamic Characteristics of the 7.62 mm Nato Ammunition", BRL MR 1833, Aberdeen Proving Ground, Maryland, USA, 1967.

[7] McCoy, R., Modern Exterior Ballistics, The Launch and Flight Dynamics of Symmetric Projectiles, USA

FAQ

Questions and remarks of readers

At which angle of departure does a bullet achieve its maximum range?

  Why is the rotation of the bullet, after leaving the muzzle, clockwise

and why not counterclockwise?  

Is it true that if a bullet and its shell are released simultaneously, they

will both hit the ground at the same time? Why?  

If a bullet is fired horizontally from a barrel that is perfectly level, will

the bullet, at some point, rise?  

If a bullet is fired horizontally from a barrel and another bullet is dropped from the same altitude at the same instant, will they both hit the ground at the same time?

  Centrifugal force is a ficticious force, it does not exist! There is only a

force radially inward which is the centripetal force.

  Can a conventional gun fire an ordinary bullet in the vacuum of space?

  Will a bullet stabilize in space (absense of atmosphere), or tumble?

 

How fast does a bullet lose its spin velocity?

How fast do bullets travel through the air?

What is the unit of the drag coefficient c D and what is the connection between cD and the projectile caliber?

Where might I find more information about estimating C M (overturning moment coefficient) for various shaped bullets, primarily boat tail and flat base match bullets. 

If a bullet is fired vertically from a rifle, what will its terminal velocity be if it strikes the top of someones head on its way back down ?

How can bullet drift can be calculated from spin?

I would like to photograph a bullet in flight with the shadowgraph technique. What equipment do I need?

Answers of the author

Q: At which angle of departure does a  bullet achieve its maximum range? A: 1. If one neglects the atmosphere and considers bullet motion in vacuum, the maximum range will be reached for a 45° departure angle.

2. For bullets fired from handguns through the atmosphere, the maximum range (typically in the range of a few kilometres or less) is usually reached for a 30°- 35° departure angle. This is a consequence of bullet retardation by the drag force.

3. Artillery shells may reach maximum ranges of a few dozens of kilometres. When fired at higher departure angles, theseprojectiles are capable to reach much higher altitudes than handgun bullets.However in those altitudes the air density is considerably smaller than the ground air density. Lower air density goes along with lower drag and this is the reason why artillery shells reach their maximum range for higher angles of departure (typically at 45°).

Q: Why is the rotation of the bullet, after leaving the muzzle, clockwise and why not counterclockwise?

A: Of course - in the real world - there exists clockwise (right handed) as well as counterclockwise (left handed) twist. For unknown reasons, atleast for handguns, the majority of barrels have clockwise twist. Just for the purpose of simplification this article restricts only to clockwise twist. In fact, for counterclockwise twist, some forces (e.g. Magnus) change their orientation.

Q: Is it true that if a bullet and its shell are released simultaneously, they will both hit the ground at the same time? why?

A: It would be exactly true if there was no drag - or with other words if those objects were dropped in vacuum. In vacuum the only active force is gravity

FG= mg; m=mass g= gravitational constant.As a consequence the fall time t is independent of the mass (s = falling distance):

If both objects are dropped in air, their fall times to reach the ground will not be exactly the same, because they experience different retardations, depending on their shape, the way they tumble while they fall, and other factors. However, while falling, the drag of both objects will be very small, and they will "approximately" hit the ground at the same time.

Q:  If a bullet is fired horizontally from a barrel that is perfectly level, will the bullet, at some point, rise?

A: There is no way a bullet can rise over the axis of bore of the gun, because there is no "upward" directed force: 1. The drag acts opposite to the direction of movement and simply retards the bullet 2. the force of gravity is directed downward. Some kind of confusion may arise, because the bullet  normally rises over the line of sight of a gun. The line of sight however is usually inclined with respect to the axis of bore.

Q: If a bullet is fired horizontally from a barrel and another bullet is dropped from the same altitude at the same instant, will they both hit the ground at the same time? A: This is an interesting question and the answer is not trivial. It is true that the horizontally fired bullet and the dropped bullet would hit the surface at the same moment, if the experiment happens in vacuum. In vacuum there is only the force of gravity which affects both objects in the same way. However if the shooting occurs in air there is the additional force of drag. Both objects - let us assume spheres - experience drag. The difference however is, that the horizontally fired bullet has a much higher velocity. Only the "downward" velocity components vy at t=0 are the same (vy=0) for both bullets. The force of drag is (roughly) proportional to the square of the velocity v (v = sqrt(vx

2 + vy2)) and not only to the vy component!  Thus, the drag experienced by

the fired bullet is much higher than the drag experienced by the dropped bullet. As a consequence the fired bullet will reach the surface later. Example: Sphere of 10 mm diameter, 10 g mass, fired at 500 m/s from 10 m height

1. Horizontally fired: flight time 1.649 s; terminal velocity 160.2 m/s; point of impact at approx. X =  400 m (range) 2. Dropped: fall time 1.432 s, terminal velocity 13.9 m/s

Q: Centrifugal force is a ficticious force, it does not exist! There is only a force radially inward which is the centripetal force.

A: For a discussion of that subject please see this source .

Q:Can a conventional gun fire an ordinary bullet in the vacuum of space? A: There is no reason why a conventional gun shouldn´t fire in vacuum. First, the primer in the cartridge (which contains explosive material) is mechanically ignited. Hot particles are produced which ignite the powder charge. The powder however already contains the oxygen which is needed for "burning". If this wouldn´t be the case (if e.g. the powder had to take oxygen from the surrounding atmosphere), the burning process would be too slow.

It would be interesting to ask NASA for a verification!

Please also read that Internet source! It seems that the Soviets already thought about using guns in outer space!

Q: Will a bullet stabilize in space (absense of atmosphere), or tumble? A: No, the bullet will not tumble. Except the (weak) gravitational forces of the planets, which apply at the bullet´s CG and do not cause any moment, there will be no other forces or moments, especially no destabilizing moments. However, any bullet precession induced at muzzle exit will continue.

This would be a second experiment for NASA!

Q: How fast does a bullet lose its spin velocity? A:This question cannot be answered in general. As a rule, spin is much less reduced than velocity: An estimate for the M80 bullet (7.62 x 51 Nato) fired vertically up gives the following values:

o all of the velocity has been lost at the summito only approx. 36% of the angular velocity has been lost at the summit.

Q: How fast do bullets travel through the air? A: The answer depends on the type of gun. Typically bullets fired from pistols and revolvers travel at 300 - 500 m/s.

Hunting or miltary bullets are faster (approx. 800 - 1000 m/s). Air rifles are in the 100 - 200 m/s range.

Q: What is the unit of the drag coefficient cD and what is the connection between cD and the projectile caliber?

A:The drag coefficient is a dimensionless number (in the area of 0.1 - 1) and depends on Mach number, Reynolds Number, Froude number,... The drag coefficient is usually measured by Doppler Radar or other velocity loss measurements. There is no simple relationship between bullet geometry (length, diameter, shape) and cD.

Q: Where might I find more information about estimating CM (overturning moment coefficient) for various shaped bullets, primarily boat tail and flat base match bullets. A: All the aerodynamic coefficients are usually hard to obtain. Military research institutes measure them but only for military bullets. Almost no data is available for bullets from the civilian market. There is some (expensive) software available which estimates the aerodynamic coefficients from bullet geometry.

Q: If a bullet is fired vertically from a rifle, what will its terminal velocity be if it strikes the top of someones head on its way back down?   A: This question is hard to answer in general. The best I can give is a "worst-case" estimation. When a gun is fired vertically, the bullet after some time reaches a summit where the velocity  is zero, and then falls back. The bullet will fall back base first which is hard to calculate. I can estimate the velocity if it would fall nose first, that is the normal flying position for which drag is well known - so the real terminal velocity will actually be smaller than the following prediction.  

For a .22 lr bullet (m=40 grain, v0 = 1150 ft/s) the summit will be at 1164 ft, the total flight time 30 seconds and the terminal velocity 270 ft/s

For a SS109 military bullet (m= 55 grain, v0=3200 ft/s)the summit will be at 2650 ft, the total flight time 44 seconds and the

terminal velocity 404 ft/s.For this bullet are indications that it will become unstable. This will further reduce summit height and terminal velocity considerably.

Q: How can bullet drift can be calculated from spin? A: This is not an easy task and can only be done with some accuracy by applying exterior ballistics software. There is a simple formula for estimation purposes.   z = k1*T2

T is the total flight time in seconds k1 is a factor in the area of 0,1 ... 0,12 m/s2 which depends on spin, muzzle velocity and bullet parameters z is the side deviation in meters

Generally bullet drift at short distances (100 - 300 yd) is by far smaller than the normal scatter. Drift is only of some practical importance for artillery shells, at ranges of several miles.

Q; I would like to photograph a bullet in flight with the shadowgraph technique. What equipment do I need? A: The most important (and expensive part) of the equipment will be a light source - a spark flash of very small flash time (in the area of 1 millionth of a second). This is something you do not get in the supermarket.

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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, bcs 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  bctest 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 cDo

G1(Ma) . Using

cDotest(B,Ma) = iD

test(B) * cDoG1(Ma)

one finds for the bc (assuming "standard" atmosphere conditions)

bctest=1 / iDtest(B) * mtest / d2

test

This formula also shows that the bc and the form factor iD 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 (see also here).

Abbreviations

cDotest  Zero-yaw drag coefficient of test bullet

cDoG1  Zero-yaw G1 standard drag coefficient

iDtest  Form factor of test bullet

bctest  Ballistic coefficient of test bullet

mtest  Mass of test bullet in lb

dtest  Diameter of test bullet in inches

More Abbreviations

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, bcs 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  bctest 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 cDo

G1(Ma) . Using

cDotest(B,Ma) = iD

test(B) * cDoG1(Ma)

one finds for the bc (assuming "standard" atmosphere conditions)

bctest=1 / iDtest(B) * mtest / d2

test

This formula also shows that the bc and the form factor iD 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 (see also here).

Abbreviations

cDotest  Zero-yaw drag coefficient of test bullet

cDoG1  Zero-yaw G1 standard drag coefficient

iDtest  Form factor of test bullet

bctest  Ballistic coefficient of test bullet

mtest  Mass of test bullet in lb

dtest  Diameter of test bullet in inches

More Abbreviations

The drag coefficientThe drag coefficient cD 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

The following assumptions 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 cD can be neglected.

2. dependency

Depending on the physical ballistic model applied, an angle of yaw is either completely neglected (=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) cD(B,Ma,) = cDo(B,Ma) + cD(B,Ma) * 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) cD(B,Ma,) = cDo(B,Ma) + F(B,Ma,Re,)

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" cDo

standard which depends on the Mach number alone and a form factor iD which depends on the bullet geometry alone according to:

cDo(B,Ma) = iD(B) * cDo 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.

Abbreviations

cD  Drag coefficient; cD(B,Ma,Re,)

cDostandard  Zero-yaw standard drag function

iD  Form factor

More Abbreviations

Abbreviations Press the Back button of your browser to return to the previous page!

A Bullet cross section area; A = d2/4

a Velocity of sound in air, a = a(p,T,h)

B Symbolic variable, indicating bullet geometry

d Bullet diameter

ec Unit vector into the direction of the bullet' s longitudinal axis

et Unit vector into the direction of the tangent to the trajectory

g Acceleration of gravity; g = g( y)

h Relative humidity of air

Ix Axial (or polar) moment of inertia of the bullet

Iy Transverse (or equatorial) moment of inertia of the bullet

l Bullet length

m Bullet mass

Ma Mach number

p Air pressure

Re Reynolds number

rE Mean radius of the earth; rE = 6 356 766 m

T Absolute air temperature

vw Bullet velocity with respect to wind system

y Altitude of bullet above sea level

Azimuth angle

Yaw angle

Angle of inclination of the trajectory

Air density = (p,T,h)

Absolute viscosity of air; = (T)

Degree of latitude

Spin rate of bullet (angular velocity)

Angular velocity of the earth´s rotation; E = 7.29.10-5 rad/s

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Forces and momentsHint: If you are not familiar with the physical concept of forces and moments, reference is made to any elementary physics textbook (chapter classical mechanics, forces, moments).

The shadowgraphs have shown that the flowfield in the vicinity of a bullet most generally consists of laminar and turbulent regions. The flowfield depends in particular on the velocity at which the bullet moves, the shape of the bullet and the roughness of its surface, just to mention the most important factors. The flowfield obviously changes tremendously, as the velocity drops below the speed of sound, which is about 1115 ft/s (340 m/s) at standard atmospheric conditions.

The mathematical equations, by means of which the flowfield parameters (for example pressure and flowfield velocity at each location) might be determined are well known to the physicist as the Navier-Stokes equations. However, having equations and having useful solutions for these equations are entirely different matters. With the help of powerful computers, numeric and practically useful solutions to these equations have been found up to now only for very specific configurations.

Because of these computational restrictions, ballisticians all over the world consider bullet motion in the atmosphere by disregarding the specific characteristics of the flowfield and apply a simplified viewpoint: the flowfield is characterized by the forces and moments affecting the body. Generally those forces and moments must be determined experimentally, which is done by shooting experiments and through wind tunnel tests.

Generally, a body moving through the atmosphere is affected by a variety of forces. Some of those forces are mass forces, which apply at the CG (center of gravity) of the body and depend on the body mass and the mass distribution. A second group of forces is called aerodynamic forces. These forces result from the interaction of the flowfield with the bullet and depend on the shape and surface roughness of the body. Some aerodynamic forces depend on either yaw or spin or both. A summary of the most important forces affecting a bullet's motion through the atmosphere is shown in the table below. As an example, another table gives the magnitude of forces for a typical military bullet.

Table: Forces, affecting a bullet's movement through the air  

Forces Requires

Remarks

Yaw Spin

Mass Forces

Gravity N N responsible for bending of trajectory

Coriolis Force N N usually very small

Centrifugal Force N N small; usually included in gravity

Aerodynamic Forces

Drag Y N major aerodynamic force

Lift (Cross-wind Force)

Y N responsible for side drift

Magnus Y Y very important for stability

Pitch Damping Y Y usually very small, important for stability

Transversal Magnus

Y Y usually very small

Mass forces

The most simple "ballistic model", considering only the force of gravity , was discovered by Galileo Galilei (1590). A discussion of this important force can be found in any elementary physics book.

The force of gravity

 

Abbreviations

ej  Unit vector, opposite to the direction of the acceleration of gravity 

FG  The force of gravity

More Abbreviations

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.

As we intend to study the movement of bullets on earth, we have to consider its rotation. However, Newton's equations of motion are only valid in an inertial reference system - which either rests or moves with constant speed. As soon as we consider bullet motion in a reference frame bound to the rotating earth, we have to deal with an accelerated reference frame. But we can compensate for that - and still use Newton's equations of motion - by

adding two additional forces: The centrifugal force and the Coriolis force

.

The centrifugal force

Abbreviations

FZ  Centrifugal force

More abbreviations

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

Abbreviations

Fc Coriolis force

v Velocity vector with respect to xyz - coordinate system

Vector of the angular velocity of the earth´s rotation with respect to xyz - coordinate system.

More abbreviations

Explanation

The magnitude of the fictitious Coriolis force is so small that it is usually completely neglected and - as a rule of thumb - only has to be considered in ballistics for ranges of 20 km or more (artillery shells).

Wind force and overturning moment

Let us consider the most general case of a bullet having a yaw angle . By saying so, the ballistician means that the direction of motion of the bullet' s CG deviates from the direction into which the bullet's axis of symmetry points. Innumerable experimental observations have shown that an initial yaw angle at the muzzle of a gun is essentially unavoidable and is caused by perturbations such as barrel vibrations and muzzle blast disturbances.

For such a bullet, the pressure differences at the bullet's surface result in a force, which is called the wind force. The wind force seems to apply at the center of pressure of the wind force (CPW), which, for spin-stabilized bullets, is located in front of the CG. The location

of the CPW is by no means stationary and shifts as the flowfield changes. The figure  schematically shows the wind force F1, which applies at its center of pressure CPW.

The wind force

 

For a bullet, moving at velocity v, having a yaw angle , the flowfield pressure differences result in a net force which is called the wind force F1. This force seems to apply at the centre of pressure CPW of the wind force, which, for spin-stabilized bullets, is located in front of the centre of gravity CG. The location of the CPW depends on the flowfield conditions and varies as the velocity decreases. Two forces FW and F2, applying at the CG,

which mutually neutralize, can be added to the wind force (see next figure  ).

As shown in another figure  , it is possible to add two forces to the wind force, having the same magnitude as the wind force but opposite directions. If one let those two forces attack at the CG, these two forces obviously do not have any effect on the bullet as they mutually neutralize.

Adding two forces to the wind force

For a bullet, moving at velocity v, having a yaw angle , the flowfield pressure differences

result in a net force which is called the wind force F1 (see also previous figure  ). This force seems to apply at the centre of pressure CPW of the wind force, which, for spin-stabilized bullets, is located in front of the centre of gravity CG. For fin-stabilized bullets the CPW is located behind the CG. The location of the CPW depends on the flowfield conditions and varies as the velocity decreases. Two forces FW and F2, applying at the CG, which mutually neutralize, can be added to the wind force. The forces F1 and F2 form the

overturning moment MW (see next figure  ).

The overturning moment

 

Abbreviations

cM  Overturning moment coefficient, cM(B, Ma, Re, )

eW  Unit vector

MW  Overturning moment

 

More Abbreviations

Explanation

The point of the longitudinal axis, at which the resulting wind force F1 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 MW is called overturning moment

or yawing moment (see figure  ).

The gyroscopic effect

  The overturning moment MW tends to rotate the bullet about an axis, which goes through the CG and which is perpendicular to the plane of drag, the plane, formed by the velocity vector v and the longitudinal axis of the bullet. In the absence of spin, the yaw angle would grow and the bullet would tumble.

If the bullet has sufficient spin, saying if it rotates fast enough about its axis of form, the gyroscopic effect takes place: the bullet´s longitudinal axis moves into the direction of the overturning moment, perpendicular to the plane of drag. This axis shift however alters the plane of drag, which then rotates about the velocity vector. This movement is called precession or slow mode oscillation.

For spin-stabilized projectiles MW tends to increase the yaw angle and destabilizes the bullet. In the absence of spin, the moment would cause the bullet to tumble.

Adding two forces to the wind force

For a bullet, moving at velocity v, having a yaw angle , the flowfield pressure differences

result in a net force which is called the wind force F1 (see also previous figure  ). This force seems to apply at the centre of pressure CPW of the wind force, which, for spin-stabilized bullets, is located in front of the centre of gravity CG. For fin-stabilized bullets the CPW is located behind the CG. The location of the CPW depends on the flowfield conditions and varies as the velocity decreases. Two forces FW and F2, applying at the CG, which mutually neutralize, can be added to the wind force. The forces F1 and F2 form the

overturning moment MW (see next figure  ).

Adding two forces to the wind force

For a bullet, moving at velocity v, having a yaw angle , the flowfield pressure differences

result in a net force which is called the wind force F1 (see also previous figure  ). This force seems to apply at the centre of pressure CPW of the wind force, which, for spin-stabilized bullets, is located in front of the centre of gravity CG. For fin-stabilized bullets the CPW is located behind the CG. The location of the CPW depends on the flowfield conditions and varies as the velocity decreases. Two forces FW and F2, applying at the CG, which mutually neutralize, can be added to the wind force. The forces F1 and F2 form the

overturning moment MW (see next figure  ).

Now let us consider the two forces F1 and F2. It can be shown that this couple is a free vector, which is called the aerodynamic moment of the wind force or, for short, the overturning momentMW. The overturning moment tries to rotate the bullet around an axis,

which passes through the CG and is perpendicular to the bullet's axis of form, just as

indicated in the figure  .

Summary: The wind force, which applies at the center of pressure, can be replaced by a force of the same magnitude and direction plus a moment. The force applies at the CG, the moment turns the bullet about an axis running through the CG.

This is a general rule of classical mechanics (see any elementary physics textbook) and applies for any force that operates at a point different from the CG of a rigid body.

You may proceed one step further and split the force, which applies at the CG, into a force which is antiparallel to the direction of movement of the CG plus a force, which is

perpendicular to this direction. The first force is said to be the drag force FD or

simply drag, the other force is the lift force FL or lift for short. The name lift suggests an upward directed force, which is true for a climbing airplane, but which is generally not true for a bullet. The direction of the lift force depends on the orientation of the yaw angle. Thus a better word for lift force could be cross-wind force, an expression which can be found in some ballistic textbooks.

The drag force

Abbreviations

cD  Drag coefficient

FD  Drag force

More Abbreviations

Explanation

The drag force FD is the component of the force FW in the direction opposite to that of the

motion of the centre of gravity (see figure  ). The force FW 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 FD is the only component of the force FW . The drag force is the most important aerodynamic force. Given the atmosphere conditions p,T,h, the reference area A and the momentary velocity vw, the drag force is completely determined by the the drag coefficient cD .

Obviously, in the absence of yaw, the wind force reduces to the drag force.

So far, we have explained the forces, how the wind force and the overturning moment

are generated, but we haven' t yet dealt with their effects.

Drag and lift apply at the CG and simply affect the motion of the CG. Of course, the drag retards this motion. The effects of the lift force will be met later.

Obviously, the overturning moment tends to increase the yaw angle, and one could expect that the bullet starts tumbling and become unstable. This indeed can be observed when firing bullets from an unrifled barrel. However, at this point, as we consider spinning projectiles, the gyroscopic effect comes into the play, causing an unbelievable effect.

The gyroscopic effect can be explained and derived from general rules of physics and can be verified by applying mathematics. For the moment we simply have to accept what can be observed: due to the gyroscopic effect, the bullet' s longitudinal axis moves aside towards the direction of the overturning moment, as indicated by the arrow in the

figure  .

The spin damping moment

 

Abbreviations

cspin  Spin damping moment coefficient; cspin(B,Ma.Re)

MS  Spin damping moment

More Abbreviations

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).

As the global outcome of the gyroscopic effect, the bullet's axis of symmetry thus would move on a cone's surface, with the velocity vector indicating the axis of the cone. This movement is often called precession. However, a more recent nomenclature defines this motion as the slow mode oscillation.

To complicate everything even more, the true motion of a spin-stabilized bullet is much more complex. An additional fast oscillation is superimposed on the slow oscillation. However, we will return to this point later.

Spin damping moment

Skin friction at the projectile's surface retards its spinning motion. However, the angular velocity of

the rotating bullet is much less damped by the spin damping moment than the translational velocity, which is reduced due to the action of the drag force. As will be shown later, this is the reason why bullet's, which are gyroscopically stable at the muzzle will remain gyroscopically stable for the rest of their flight.

Magnus force and Magnus moment

Generally, the wind force is the dominant aerodynamic force. However, there are numerous other smaller forces but we want to consider only the Magnus force, which turns out to be very important for bullet stability.

With respect to the figure 

The Magnus effect

In the figure above, a bullet is viewed from behind and is assumed to have right-handed twist. Additionally, the bullet should have an angle of yaw , its longitudinal axis should be inclined to the left. Then there is a component of the flowfield velocity vn, perpendicular to the bullet´s axis of symmetry.

Due to bullet spin and air molecules adhering to the bullet´s surface, the flowfield in the vicinity of the bullet becomes asymmetric. Air stream velocity and the rotational velocity of the body subtract at point A and add at point B (see above figure). However, according to Bernoulli´s rule (see any elementary physics book), this coincides with a pressure difference. A higher pressure at A and a lower pressure at B give rise to a downward directed force, which is called the Magnus force FM .

The Magnus force

 

Abbreviations

cMag  Magnus force coefficient; cMag(B,Ma,Re,,)

eM  Unit vector

FM  Magnus force

 

More Abbreviations

Explanation

The Magnus force FM 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

(see figure  ). 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.

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 FM   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.

, we are looking at a bullet from the rear. Suppose that the bullet has right-handed twist, as indicated by the two arrows. We additionally assume the presence of an angle of yaw . The bullet's longitudinal axis should be inclined to the left, just as indicated in the previous drawings.

Due to this inclination, the flowfield velocity has a component perpendicular to the bullet's axis of symmetry, which we call vn.

However, because of the bullet's spin, the flowfield turns out to become asymmetric. Molecules of the air stream adhere to the bullet's surface. Air stream velocity and the rotational velocity of the body add at point B and subtract at point A. Thus one can observe a lower flowfield velocity at A and a higher streaming velocity at B. However, according to Bernoulli's rule (see elementary physics textbook), a higher streaming velocity corresponds with a lower pressure and a lower velocity with a higher pressure. Thus, there is a pressure difference, which results in a downward (only in this diagram!) directed force, which is

called theMagnus force FM (Heinrich Gustav Magnus, *1802, died 1870; German physicist).

This explains, why the Magnus force, as far as flying bullets are concerned, requires spin as well as an angle of yaw, otherwise this force vanishes.

If one considers the whole surface of a bullet, one finds a total Magnus force, which applies

at its instantaneous center of pressure CPM (see figure  ). The center of pressure of the Magnus force varies as a function of the flowfield structure and can be located behind, as well as in front of the CG. The magnitude of the Magnus force is considerably smaller than the magnitude of the wind force. However, the associated moment, the discussion of which follows, is of considerable importance for bullet stability.

You can repeat the steps that were followed after the discussion of the wind force. Again, you can substitute the Magnus force applying at its CP by an equivalent force, applying at

the CG, plus a moment, which is said to be the Magnus moment MM . This moment tends to turn the body about an axis perpendicular to its axis of symmetry, just as shown in the figure 

The Magnus moment 

 

Abbreviations

cMp  Magnus moment coefficient; cMp(B,Ma,Re,,)

eMM  Unit vector

MM  Magnus moment

   

More Abbreviations

Explanation

As the Magnus force applies at the CPM, which does not necessarily coincide with the CG,

a Magnus moment MM (see figure  ) 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 moment

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 MM  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 .

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.

.

However, the gyroscopic effect also applies for the Magnus force. Remember that due to the gyroscopic effect, the bullet's nose moves into the direction of the associated moment. With respect to the conditions shown in the figure 

The Magnus moment

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 MM  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 .

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 Magnus force thus would have a stabilizing effect, as it tends to decrease the yaw angle, because the bullet's axis will be moved opposite to the direction of the yaw angle.

A similar examination shows that the Magnus force has a destabilizing effect and increases the yaw angle, if its center of pressure is located in front of the CG. Later, this observation will become very important, as we will meet a dynamically unstable bullet, the instability of which is caused by this effect.

Two arms model of yawing motion

We have now finished discussing the most important forces and aerodynamic moments affecting a bullet's motion, but so far we haven't seen what the resulting movement looks like. For the moment we are not interested in the trajectory itself (the translational movement of the body), but we want to concentrate on the body's rotation about the CG.

The yawing motion of a spin-stabilized bullet, resulting from the sum of all aerodynamic moments can be modeled as a superposition of a fast and a slow mode oscillation and can most easily be explained and understood by means of a two arms model (see reference [4]).

Imagine looking at the bullet from the rear as shown in the figure

Two arms model

The simple two arms model adequatly describes the yawing motion of spin-stabilized bullets, resulting from the action of all aerodynamic moments.

The yawing motion can be understood as a superposition of a fast and a slow oscillation, often called nutation and precession.

Imagine to look at the bullet from the rear. The slow mode arm CG to A should be hinged at the CG and rotates with the slow mode frequency. Consequently A moves on a circle around the CG (the red circle). The fast mode arm A to T, where T is the bullet´s tip, should be hinged at A and rotates with the fast mode frequency. Thus, T rotates on a circle about A. CG to T is the projection of the bullet´s longitudinal axis.

An animation of the two arms model is shown in the figure below.  

One additionally has to consider that the fast mode frequency exceeds the slow mode frequency (which is true in the animation) and the arm lengths CG to A and A to T are continuously shortened (which is not true in the animation) if the bullet is dynamically stable.

  . Let the slow mode arm CG to A rotate about the CG with the slow mode frequency. Consequently point A moves on a circle around the center of gravity.

Let the fast mode arm A to T rotate about A with the fast mode frequency. Then T moves on a circle around point A. T is the bullet's tip and the connecting line of CG and T is the bullet's longitudinal axis.

This simple model adequately describes the yawing motion, if one additionally considers that the fast mode frequency exceeds the slow mode frequency, and the arm lengths of the slow mode and the fast mode are, for a stable bullet, continuously shortened.

With respect to the figure

Yawing motion

This figure schematically visualizes the general angular motion of a spin-stabilized bullet close to the muzzle.

Imagine that the CG of the bullet is fixed at the centre of a co-ordinate system and that the bullet approaches an observer´s eye. Then its tip moves on a helical path (as indicated by the curved line) into the direction of the arrows. At muzzle exit (t=0) the yaw angle can be small, but increases to a maximum of approximately 1°, then decreases again to almost zero.

Note that the magnitude of successive maximum yaw angles is less than its predecessors, as the bullet in the drawing is assumed to be dynamically stable (the maximum yaw angle decreases as the bullet continues to move on).

  imagine looking at a bullet approaching an observer's eyes. Then the bullet's tip moves on a spiral-like (also described as helical) path as indicated in the drawing, while the CG remains attached to the center of the circle. The bullet's tip periodically returns back to the tangent to the trajectory. If this occurs, the yaw angle becomes a minimum.

Example: Magnitude of forcesThe table below shows the magnitude of some forces - as multiples of the force of gravity FG - for the M80 bullet (7.62 x 51 NATO) at standard air density ( = 1.25 kg/m3).

M80 bullet 7.62 x 51 Nato

Mass [g] 9.45

Calibre [mm] 7.82

Force Ma = 2.5 Ma = 0.6

Coriolis FC/FG 0.013 0.003

Centrifugal FZ/FG 0.003 0.003

Drag FD/FG 70.2 1.7

Lift FL/FG (=1°) 9.4 0.4

In the table above, the relative magnitude of the lift force appears to be relatively large. This is due to the fact that an angle of yaw of 1° has been assumed. If a bullet flies stable and the transient yaw has been damped out, the maximum yaw however is much smaller than 1° and consequently the lift force  is also much smaller.

SummaryBullets fired from handguns follow general rules of physics and behave like gyroscopes. The angular motion of these bullets can be understood as a superposition of two oscillations, most easily be demonstrated by a two arms model.

Practically all handgun bullets are statically stable, many pistol and revolver bullets even have excessive static stability.

However, dynamic stability is not automatically guaranteed. Some bullets are dynamically unstable at the moment they leave the muzzle, others may loose dynamic stability during flight after being decelerated.

At the moment no reliable method exists, except experimentation, to foresee dynamic bullet instability, especially at long ranges. Some highly sophisticated computerized procedures (numerical solutions to the Navier-Stokes equations) to attack these problems are just being developed by ballistic researchers.

AcknowledgementsI want to thank Prof. Dr. J.G. Hauck, Dresden for reviewing the manuscript and his most valuable suggestions to improve this document.

The experimental determination and evaluations of the yawing motion of various spin-stabilized handgun bullets were carried out at the German army proving ground WTD91 in Meppen. I have to thank this organization for leaving me the reports containing those unique and extremely valuable results.

The long-range Doppler tracking radar measurements of the 7.62 x 51 NATO bullet were made available to me by WEIBEL equipment, Gentofte, Denmark.

I also want to thank all readers of this article for their suggestions and support.

References[1] Nennstiel, R., AFTE Training Seminar, 5.6.-9.6.1995, San Diego, CA USA

[2] Nennstiel, R., "How do bullets fly?", AFTE Journal, Vol.28, No.2, April 1996, S.104-143

[3] Nennstiel, R., "EBV4 User's Manual", Exterior Ballistics for the PC, Wiesbaden, Germany, 1995.

[4] Farrar, C.L., Leeming, D.W., Military Ballistics - A Basic Manual, Brassey´s Publisher Limited, Headington Hill Hall, Oxford OX3 0BW, England, 1983.

[5] Giles M.J., Leeming D.W., "An Aerodynamic Model for Unstable Projectiles", Proc. of the 11th Int. Symp. on Ballistics, Brussels, May 9 - May 11, 1989.

[6] Piddington, M.J., "Aerodynamic Characteristics of the 7.62 mm Nato Ammunition", BRL MR 1833, Aberdeen Proving Ground, Maryland, USA, 1967.

[7] McCoy, R., Modern Exterior Ballistics, The Launch and Flight Dynamics of Symmetric Projectiles, USA