Internal ballistics of the sling Free

Shakespeare's “slings and arrows of outrageous fortune”¹ point to the importance of slings alongside bows as ancient weapons of long standing, yet the bow is much better known and studied. Projectile weapons have been part of human culture since the Stone Ages (literally—flint arrowheads have been found that date to the Upper Paleolithic, 50 000–12 000 years ago). They have been essential tools used against animals (hunting, protection of flocks against predators) and against other humans (warfare). Based on archaeological remains, the oldest of our projectile weapons is the bow (those arrowheads) but it is speculated that the sling is at least as old—slings would leave no traces in the archaeological records, as they are constructed from perishable material such as hemp or braided grass² and the earliest sling projectiles were unmodified stones. (For accounts about the evolution of ancient projectile weapons, see Refs. 3–8).

We investigate the physics of a sling.^* Ballistic science is conventionally divided into three phases: internal ballistics is concerned with a projectile's motion prior to its release, external ballistics deals with projectile motion through the air after release, and terminal ballistics is concerned with the effects of a projectile upon the target. Here, will be concerned mostly with internal ballistics.

The sling is an ancient weapon that has generated a large corpus of archaeological literature but has received insufficient attention by physicists, probably because the projectile trajectory prior to release is quite complicated—it moves in three dimensions while acquiring a significant spin. Our analysis carefully picks key aspects of this internal ballistic motion to shed light on the dynamics. The movement of both sling and slinger prior to launch is very complex, and there are many throwing styles as we will see, and so well-chosen approximations are important (for example, here we will in two cases replace a complicated process with its simpler time average). Indeed, our main goal in this paper is to emphasize the importance of approximations in physics analysis.

A secondary goal, also for the benefit of classroom teaching, is to suggest an interesting line of applied physics problems based on the analysis presented here. A third goal of the paper is to describe sling dynamics for the benefit of specialists (sling historians and archaeologists, and hobbyists); this description is concisely encapsulated in Eq. (1) below. (For analyses of the internal ballistics of other ancient projectile weapons, see Refs. 9–16).^†

Complications arise because of the various actions that are required by slingers to generate high projectile speeds; there are many parts to a successful sling bullet launch. (Henceforth, we adopt the common practice of calling the projectile, simply, a bullet.) Thus, they must time the release of their bullets exquisitely and must adjust their paces and thrust forward their torsos at just the right instant. The plane(s) of rotation of the sling must be chosen carefully and may be different for different bullet weights, sling lengths, and intended projectile ranges. For these reasons, the reader might suppose, correctly, that slinging requires finesse and experience; historically, slinging has always been regarded as an activity requiring great skill, with years of practice required to obtain that skill.

Due to the complexity of sling internal dynamics, it is more instructive to watch the action of a slinger launching a bullet^‡ than it is to read about it.²¹ Here, we will only briefly describe a sling and outline the action of the slinger, while recommending that the reader views some of the footnote videos. A sling consists of two braided cords of equal length connected to a pouch that holds the projectile. One cord (the retention cord) has a loop, which is to be attached to a finger (or sometimes the wrist) of the slinger's throwing arm. The other (release) cord has a knot or tassel at the end (see Fig. 1(a)).

The simplest sling action for the novice—and the most rapid-fire for the experienced slinger—is called the Greek style by modern slingers, though historically it appears to have been more widespread in classical antiquity than the name suggests. The slinger holds the loaded pouch in one hand above the head, and the two cords in the other, taut and pointed toward the target. A swift action then swings the sling toward the target. At the right moment, the sling having traveled through an arc of roughly 270°, the slinger lets go of the release cord, and the bullet is directed toward the target. More commonly, the loaded sling is whirled over the head like a cowboy's lariat in a circular path, accelerating the bullet, and then the hand is turned down so that the sling's plane of rotation is turned toward the vertical. After traveling nearly 360° in this plane, the bullet is released. For distant targets, the sling is initially rotated in a vertical plane and then released in that plane so that the bullet's velocity has an upward component.

It is likely that, in classical times, the range of a sling was greater than that of an arrow fired from a bow (the modern record for a sling bullet throw is 437 m²²). Accuracy at long distance is poor, but in ancient warfare the slingers would be aiming at large bodies of enemy soldiers and so accuracy was not so important. At short range, the sling can be very accurate when used by experts (the footnoted Youtube videos show that it is sometimes possible to hit a small moving target). For several thousand years, both the sling and bow were used widely in warfare. We refer the interested reader to the  Appendix for more historical details.

The reason why a projectile can be slung several times further than it can be thrown is, appropriately, a more effective use of the slingshot or double-pendulum effect due to the sling having negligible mass compared with a throwing arm and a greater length. In Secs. II and III, we will unpack aspects of sling internal ballistics that show this to be the case. Given the complexity of sling dynamics, it is not possible to provide a single overarching analysis that covers all features of bullet internal trajectory;^§ instead, we aim (so to speak) to provide insight into two key parameters: bullet initial linear velocity and angular (spin) velocity—how are these generated?

In this style of launching a sling bullet, the slinger's motions resemble those of an Olympic hammer thrower. The sling cords are long, and suitable for heavier (typically $∼$ 1 kg mass) bullets. Like the hammer thrower, the slinger spins (or pirouettes) around a vertical axis and the bullet, at the end of the cords, picks up speed. In this style, the bullet describes a circular arc in a plane at a fixed angle. It is easier than other slinging styles for the investigating physicist to analyze because of the relative simplicity of the sling motion—confined to one plane.

So, we begin our analysis with this style, and assume that the bullet moves around a circle n times prior to release (typically $n=3$⁠). Let $θ$ represent the polar angle of the bullet in its arc. Our first assumption is the following:

Assumption A1 is not essential to the development that follows—but it streamlines the calculations and presentation. It is straightforward to show that constant angular acceleration implies that the power—the rate at which energy is being transferred to the bullet—increases linearly with time. We might have chosen a model in which power is constant, but there is some slight indication from observing slingers in action that, during the internal phase of sling motion, power increases with time: a slinger begins by rotating the bullet with their arm and ends by thrusting forward and twisting their torso, so that just prior to release the muscles of their trunk and legs (as well as their arm) are involved in accelerating the bullet. Given that more muscles become involved in the later stages of the swing, it is reasonable to assume that the power is increasing.

So, the slinger accelerates the bullet at a constant rate $θ̈=α$⁠, which means that, at time t after pirouetting begins, the bullet has angular speed $θ̇=αt$⁠. It is released after traversing an angle $θn=2nπ=αtn2/2$⁠. Thus, the launch phase lasts for a time $tn=2nπ/α$⁠. The speed at release is $v=θṅr$⁠, where r is the radius of the circle (equal to arm length plus sling length). Thus, $v=2rnπα$⁠.

The average power that is transferred to the bullet by the slinger during this wind-up phase is given by $Pn¯=E/tn$⁠, where E is the bullet energy. It is straightforward to show that for the arc of Fig. 2(a) the gravitational energy is two orders of magnitude less than the rotational energy, and so here (and throughout this paper) we can neglect the influence of gravity on sling internal ballistics.^**

How much power is delivered by the slinger while accelerating the bullet? We can obtain an upper bound by comparing our slinger with modern athletes for whom there is an abundance of measured data. Consider a hammer thrower—the throwing action is the same as that of our pirouette-style slinger, but with a much heavier (7.26 kg) projectile. Hammer throw data for three world-class athletes yield a typical launch speed of 30 ms⁻¹ and thus a hammer kinetic energy of 3200 J.²³ This kinetic energy is more than seven times that of our sling bullet (⁠ $≈$ 430 J). The hammer throwers transfer energy to their projectiles at a rate exceeding 1 kW while accelerating their hammers.^†† Our slingers are not necessarily world-class athletes, so we assign a smaller value for the power they transfer to their projectile. We assign a nominal power of 200 W to the pirouette-style slinger—at this expenditure rate, they will be able to shoot many sling bullets before fatigue sets in.

Given A3, we are not considering the power needed to accelerate the sling cords nor the biochemical power needed to move the slinger. Let us assume a long sling, with $r=2.5$ m, a heavy bullet (⁠ $m=0.454$ kg, i.e., 1 lb), and $n=3$ rotations before the bullet is released. With these parameter values, we see from Eq. (1) that the bullet speed at release is about $v≈$ 40–50 ms⁻¹. The acceleration time is $t3=2.2$ s. This calculation ignores any effects of aerodynamic drag upon internal ballistics—this is reasonable for a phase of such short duration. It is unreasonable for external ballistics, however, and so we state our fourth assumption:

Given the bullet release velocity, we can determine the range over flat ground from the familiar equation $R=(v2/g) sin 2γ$⁠, in the notation of Fig. 2(a), with no aerodynamic drag. Assuming that the slinger's shoulders are at a height $h=1.6$ m above the ground (and so $γ=40°$⁠), we find that $R=190$ m. If quadratic drag acts to retard the projectile motion, then standard numerical calculations (for r = 2.5 m, m = 0.454 kg, n = 3, and for a spheroidal stone bullet with $b=0.026$ m, drag coefficient $cD$ = 0.4, and release speed in the middle of the range of speeds just estimated: v = 45 ms⁻¹) yield the curve shown in Fig. 2(b) and so a range of $R=170$ m. (⁠ $R(v)$ for a lead bullet with mass 50 gm is very similar.)²⁴ Henceforth, when we state a range expected for a given release speed, we will include quadratic drag effects.

There are a number of factors which render our results, for both release speed and range, approximate. These factors include

Thus, in practice a slinger's torso is often pushed forward and rotated just prior to releasing the sling, adding to launch speed. A sling bullet internal trajectory might well be three-dimensional (especially for the side-arm sling action that we will consider later) rather than confined to a plane as assumed here. Our slinger might be moving forward while pirouetting, as with hammer throwers. This will also add a little to release speed. Against this, release speed will be reduced somewhat from our estimated values due to aerodynamic drag acting on the sling and bullet. These approximations will change our internal ballistics predictions only slightly. Note that the numbers we obtain are entirely compatible with measured sling bullet release speeds and ranges.^21,22

Sling bullet spin rates have been observed to be (perhaps surprisingly) high: references measure spin rates of 45–70 Hz for tennis ball and lacrosse ball sling bullets and report spin rates of 60–250 Hz for smaller stone and lead bullets.²⁵ Can we anticipate such rates from the simple sling model developed here?

First, we need to describe these sling bullets. Originally just smooth, rounded stones, bullets were then made from clay or, better yet, lead and cast as spheroids (Fig. 3(a)) or bicones. Such elongated shapes were common across many cultures that developed slings, which implies that these shapes yielded better results. In fact, we will see that a sling imparts high spin rates so that the elongated bullets are spin-stabilized. The pirouette style of launching a sling bullet requires the slinger to rotate with the sling and so, if the sling is held correctly, the bullet can be made to always face its direction of motion. (That is, the spheroidal bullet velocity vector is parallel to the long axis of the spheroid.) Thus, when the slinger releases the bullet it will fly forward presenting the minimum area to the atmosphere it passes through and so will be subjected to the minimum aerodynamic drag. Further, the spin-stabilized bullet will fly true, and will not tumble end-over-end, which would reduce both range and accuracy. (In Sec. III, we will see that other slinging styles require considerably more finesse on the part of the slinger to ensure that, at release, the bullet is pointed toward the target.)

We note that $Δt$ decreases rapidly with target range and that the above-calculated precise timing requirement applies only to azimuthal error—the slinger must also be very precise in elevation and release speed. Thus, we see why shooting with a sling requires great skill.

We now consider the much more common one-arm slinging style. The bullet internal trajectory for this side-arm motion is more difficult to analyze than that of the pirouette style, both because it is three-dimensional and because it has a second type of rotation. Indeed, the many degrees of freedom of this style make the physical analysis of sling motion more complex than that of any other ancient weapon that the author has analyzed (atlatl, boomerang, javelin, onager, sword, trebuchet) and we will need to make drastic simplifications to obtain tractable equations of motion. This paper emphasizes the lesson that such simplifications can be made in a meaningful way, that carefully-chosen approximations can capture the essential physics and that reducing complexity can be managed in such a way as to avoid throwing out the baby with the bathwater.

In contrast to the pirouette style, in which the slinger's arms and the sling were aligned and considered to be a single stiff rod, we now permit one joint, where the cord meets the arm, as shown in Fig. 4(a). The arm is always (approximately) straight—no hinged elbow. Here, we introduce another simplifying assumption:

If the pirouette style is likened to a pendulum, then the side-arm motion is that of a double pendulum. Consequently, we can expect significantly more complexity of analysis and calculation, due to the extra degree of freedom. (Permitting jointed elbows would have led to a triple pendulum, with another degree of freedom and more complexity.) It turns out that the double-pendulum model is sufficient to approximate sling internal dynamics, and the analysis is not onerous so long as we make judicious approximations.

Before analyzing the side-arm sling trajectory that is sketched in Fig. 4, we should address the additional rotation of this style in order to fully conclude our discussion on bullet spin.

We complete our discussion of bullet spin with a few supplementary comments. Human anatomy influences the side-arm style of Fig. 4. The slinger's wrist is not a pivot—it cannot rotate $2π$⁠. To understand the complication, pick up a pen by its end and hold it horizontal while your forearm is vertical. Then, “swing” the pen around in a circle like a lasso rope. You'll see that you can't keep the same side of the pen facing upward during the full circle. As they complete a full circle around your arm, both the pen and the lasso rotate by $2π$ around their long axes. This means that a non-spherical bullet at the end of the cord must also rotate; its long axis cannot always be parallel to its direction of motion. This additional rotation complicates the release of a non-spherical bullet because the side-arm slinger must now ensure that, at the instant of release, the bullet is aligned properly with the long axis pointing at the target. Slingers learn to make this alignment by choosing parameters that best work for them: sling weight, length and stiffness, bullet weight and length, and the distance between the two cords held in their throwing hand. The technique for timely alignment may vary with sling action: the side-arm style may be applied to an underarm or overarm action (for which the plane of rotation of the sling is vertical—generally not the case for a side-arm release).

This style is well-illustrated in a recent Youtube video.²⁷ There are many variants of this style, with underhand or overhand release, or with different arc angles. In Fig. 4, we have restricted the angular acceleration phase of sling motion to the x–y plane; this is a significant simplification for two reasons. First, it implies zero vertical component for bullet launch speed. We can immediately dismiss this concern, however, because as noted in Sec. II A gravity is negligible compared with other forces acting on the bullet, and so we can orientate the plane of acceleration any way we like—it will not influence launch speed. Second, observations of slingers reveal the unsurprising fact that sling motion is not confined to one plane. Usually the slinger initially rotates the sling at some slow rate (Fig. 5(a)) with their wrist barely moving so that the bullet trajectory is circular. Then, the arm moves rapidly so that the bullet accelerates in a different plane (Fig. 5(b)). We will see, however, that for the purposes of estimating launch speed we can ignore the initial circling motion and consider the bullet to accelerate from rest to launch speed while confined to one plane.

The acceleration is again taken to be constant: $θ̈=α$⁠. When the bullet is moving along the y-axis (so that $Ẋ=0$⁠), it is released. For realistic choices of L and l, this simple double-pendulum model reproduces pretty well the action of a slinger, despite the considerable simplifications it makes, explicitly and tacitly. No doubt the angular acceleration of a real sling is not constant. Also, the three-dimensional nature of a real side-arm action (sketched in Fig. 5) will involve forces that are ignored in our two-dimensional trajectory. We expect that these considerations (plus twisting the sling to align the bullet, plus any change in length of the arm by straightening the elbow throughout the acceleration, plus ignoring the Earth's gravitational field, plus…) will slightly modify our predictions, rather than render them worthless, because our simple model captures most of the dynamics. In Sec. IV, we will see how even more drastic simplifications can also be useful, so long as their limitations are borne in mind.

For lower launch speeds, shorter slings are more appropriate—we see now why the Balearic slingers (Fig. 1(b)) carried slings of different lengths.

Can the extreme ranges claimed by ancient writers, and by the current world record holder, be reproduced in this model? We find by integrating Eq. (4) that a release speed of 76 ms⁻¹ for a 50-g bullet can be achieved for a one-meter sling by a slinger transferring to the bullet an average 269 W of power over 0.539 s. (The mechanical energy transferred is 144 J; this is equivalent to lifting a 20 kg sack onto a table.) Such a launch speed could send the bullet 400 m (allowing for air drag). Thus, we conclude that, yes, such ranges are certainly possible for an expert slinger.²⁸ 

There is merit in providing a simplified approximation of the internal dynamics for which analytic solutions exist, albeit approximate solutions. This is because, unlike the numerical solutions obtained in Sec. III, analytic solutions provide further insight into the dependence of key variables upon parameters such as release speed.

We revisit the internal trajectory of a side-arm style slingshot. Instead of approximating the physical system of throwing-arm plus sling, we now approximate the trajectory itself. Thus, we are not really modeling the dynamics of sling motion here; rather we are determining how a similar-shaped trajectory depends upon sling parameters. This will prove quite instructive. Let us assume that the sling bullet internal trajectory passes through an angle $Δθ$ during which the distance of the sling bullet from a center of rotation increases from an initial value of l to a final value at release of $l+L$⁠, as shown in Fig. 7(a). Thus, we suppose that at first the slinger holds their wrist in one position while swinging the sling (hence the initial radius l) and then they accelerate the bullet while rotating their arm so that at release the rotation radius is $l+L$⁠. We do not need to specify the exact manner of how the transition is made, only that it takes place over an angular interval of $Δθ=2nπ$⁠. For the internal trajectory sketched in Fig. 7(a), we have $n=3/4$⁠. (It is perhaps worth emphasizing that it is the slinger's arm that rotates through $Δθ=2nπ$⁠; the sling itself may rotate a different amount. Thus, for the Greek style the sling rotates through 270°.)

If we again assume that sling angular acceleration is constant, then we obtain Eq. (1) for release speed in terms of $nP¯n$⁠, with $L+l$ replacing r. Sling launch data are available, provided by hobbyists, from which we can construct the plots of Fig. 7(b). For three sling lengths l, the launch speed was recorded for different bullet masses. For each length, we have chosen a value for $nP¯n$ that yields the best fit (minimum mean square error) to the data for our predicted $vn(m)$ of Eq. (1). Note that the curves fit the data points well. The fitted parameter values are $nP¯n=327$ W for the longest sling, 281 W for the middle-length, and 196 W for the shortest sling. It is reasonable to expect that the power exerted by a slinger will vary with sling length, because technique is likely to vary with sling length. The largest $nP¯n$ applies for the longest sling, probably because the arc $Δθ=2nπ$ is largest for this sling. More data are needed to confirm this relationship between sling power/stroke length and sling length. Even with the data of Fig. 7(b), however, we see that these results are useful for a historical researcher or hobbyist: given a sling of unknown performance, launch speed (and so range) can be estimated for different bullet masses.^***

It is possible from the video data to visually estimate n and so to provide a rough estimate of $P¯$⁠, given these best-fit values for $nP¯n$⁠. The error bars would be large, however. This is because in practice the slinger transitions smoothly from the initial whirling actions (Fig. 5(a)) to the final acceleration phase (Fig. 5(b)). To provide a useful estimate of transferred power, more video data are needed (during which the slingers indicate the instant at which they transition to the acceleration phase—perhaps by opening or closing their free hand).

Compared with the side-arm model of Sec. III, our variable-length model has no theoretical basis—it does not describe dynamics but merely approximates the bullet trajectory. (For an arc length $Δθ=π$⁠, it most nearly resembles the trajectory of Fig. 5(b) with $θ0=−180°$⁠; we superimpose the prediction of Eq. (1) and obtain fairly good agreement.)

The sling is underrated by most people with an interest in historical weapons. However “…in experienced hands, the sling was arguably the most effective personal projectile weapon until the 15th century, surpassing the accuracy and deadliness of the bow and even of early firearms.”²² There has been much discussion in the archaeological literature about sling bullet external and terminal ballistics (range and penetrating power) and consequently much practical experimentation to determine the maximum bullet release velocity that could be attained. Bullet speeds measured in this way by archaeologists and most hobbyists seemed to not match the claims made by ancient authors. We asked: is the shortfall in observed sling ranges due to exaggeration by classical authors, or due to the difficulty of mastering the sling effectively (several modern slingers admit to novice status, e.g., Ref. 20)? Our analysis of sling internal ballistics, and recent reports by modern hobbyists, show that slings really are capable of projecting a bullet 400 m. That is, sufficiently high bullet speeds can be generated by a proficient slinger who applies modest power for a short time.

The release speed is approximately proportional to the cube root of average slinger power and angular arc, and inversely proportional to the cube root of bullet mass (Eq. (1)). These predictions are robust in that they do not depend on any details of the bullet internal trajectory—only upon its end points. They depend only on average power, not upon $P(t)$ which would be much harder to measure. This equation may be of use to archaeologists, to estimate the capabilities of ancient slings. Bullets are launched with high spin speeds (tens of Hertz). Bullet release timing error must be no more than a few milliseconds to hit a human-sized target at close range, and must be sub-millisecond at medium range. The target angular width is a degree or two and so the precision of aim is about that of a skeet shooter, though with a weapon that is harder to use and with a significantly slower projectile. We see why becoming an expert slinger requires much training.

We have of necessity made a large number of approximations in order to attain these results, due to the complexity of sling motion prior to release of the bullet. The equations resulting from these approximations have been easily solved either numerically or algebraically. The errors introduced by our approximations are small, though we have calculated these explicitly only for A1 and A7. The interested student is invited to relax the other assumptions, note the burgeoning complexity of the resulting equation(s) of motion,^††† and observe the small difference in final results.

The author thanks David Derbes for a careful reading of an earlier version of the MS, and three anonymous referees whose comments have led to an improved paper. Also, the author is grateful to Archaic Arms for permission to use their sling data.

The author has no conflicts to disclose.

There is evidence for the use of arrows, and so presumably of bows, from 50 000 years ago in southern Africa, Asia, and Europe. The first Americans likely brought the bow with them across the land bridge from Asia. Other ancient projectile weapons include the crossbow (the earliest evidence suggests that they originated in China, 2700 years ago), the ballista, onager and other siege equipment from classical antiquity (eighth century BCE to fifth century CE), and the atlatl or spear-thrower (found on every continent except Africa, they are thought to have been developed about 30 000 years ago). The trebuchet is a product of the Middle Ages.⁶ 

Korfmann⁷ and Harrison²² discuss the prominent role played by slings in ancient and medieval warfare. Yet today the sling is largely forgotten, except by professional historians—the rest of us recall the David and Goliath story and relegate slings to biblical times and places. In fact, the sling was used from ancient times across the classical world by Assyrians, Baleares, Britons, Egyptians, Greeks (Achaeans, Athenians, Minoan-Mycenians, Rhodians), Irish, Israelites, Persians, Phoenicians, Romans, Sumerians and others.^30–33 It was also widely used in Asia,⁸ Mesoamerica, and Oceania.^34,35 In that classical period, slingers from the Balearic Islands fought as mercenaries for the Carthaginians and later for the Romans. The difficulty of mastering the sling meant that it took some years of practice to become expert, and so tended to become a specialty; Balearic slinger were famous for their skill in the Roman world. Archery required less training, but bows and arrows were more difficult and expensive to manufacture than slings and sling bullets.

Almost universally, bullets were spheroidal or biconical in shape, for aerodynamic reasons. This implies that they were spin-stabilized when shot correctly.^‡‡‡36 Romans manufactured lead bullets: their increased density relative to stone or clay meant reduced volume and hence reduced drag, and so longer range for a given weight.

With comparable ranges and roughly comparable accuracies (at least at short range) and with similar rates of fire, it is perhaps not so surprising that both slings and bows found a place in many human societies for about 10 000 years.^7,22 There is a suggestion in the statistics of archeological finds⁷ that slings and bows were to some extent mutually exclusive; societies with slings largely eschewed the bow, and vice versa. In warfare, the bow did eventually dominate, however. The reason is that archers could be used in close order—a line that is 400 meters long might contain 300 archers, but fewer than 100 slingers. Thus, the concentration of arrow fire would exceed that of sling bullets. The bows' superiority in organized warfare was thus largely due to battlefield geometry, rather than to an inherent superiority of the weapon.

Yet the sling survived as a weapon of war in the New World and the Near East until well into the 1700 s,²² with its last recorded martial applications being in 1936 during the Spanish Civil War and in 2018 by Palestinian protesters.³⁷