**3**

# Terminal Ballistics - A model to predict penetration depth

**Summary**

In this blog entry, I use data published by The Slingshot Channel to come up with a simplified formula to predict the depth of penetration of slingshot projectiles in ballistics gelatin. Gelatin has some initial resistance to penetration, but once you get over 20m/s (65fps), you can expect the the projectile to penetration its own diameter for every 2.5m/s (8.2fps) of velocity that it is travelling at impact. The data further shows that there is an optimum projectile size/mass for any given bandset.

**Experimental Data**

Jörg has kindly published data from his video Gelatin Block of Truth. In the video, he shoots solid steel ball bearings at a big block of 'certified ballistic gelatin'. In other words, it's a medium that's been scientifically engineered to behave like human flash when shot by a bullet.

The range of projectile sizes that he shoots is truly impressive, from a tiny 8mm ball to a mammoth 38mm ball that weighs over a hundred times as much. By using several bandsets of increasing strength, Jörg manages to do a good job shooting each projectile within meaningful velocity range.

Jörg's results are reproduced below. The first figure is the diameter of the steel ball, the second is the velocity and the third is the penetration.

- 8mm 94m/s 21cm
- 10mm 78m/s 27.5cm
- 12mm 72m/s 31cm
- 15mm 68m/s 35cm
- 20mm 59m/s 45cm
- 25mm 50m/s 33cm
- 30mm 40m/s 38cm
- 38mm 32m/s 6-7cm

**Searching for a Model to Fit the Data to**

What we all want at the end of the day is to use the experimental data to make predictions of penetration for any given projectile travelling at any given velocity. To do that, we need to find the pattern behind the data. At first glance, there's not much of a pattern to it. Medium sized balls seem to go in deeper. The deepest penetration is the 20mm ball. However, there's more to it than that. They're all travelling at different speeds, have a different amount of kinetic energy and make a hole of a different size. What would happen if you used a different bandset? What if the ball was made of stone or lead? To answer questions like these, we need to take a closer look at ballistic gelatin and how the material behaves and consider the physical mechanism of penetration and how all the variables would affect that mechanism.

Luckily, this question is relevant to the well funded US Army, which makes it their business to know the finer details about poking holes in people with bullets. The army has several research labs whose scientists looked into this and shared their knowledge with us. Seeing as it's not really desirable to do countless experiments shooting at cadavers or anaesthetised pigs, scientists discovered that concentrated gelatin behaves a lot like flesh when it comes to replicating gunshot cavities and have formulated a particular consistency that they consider a pretty good analogue for what happens when people get shot. They call this ballistic gelatin and this is exactly what Jörg used for his tests.

One of these scientists (Segletes) realised that the ease of which a bullet passes through gelatin is related to how much the gelatin has been deformed from its resting state. You see, gelatin is a bit elastic. When Jörg punches the block in his video, his fist bounces off the block and doesn't penetrate it. Why? Because the gelatin absorbs the blow; it doesn't get stretched enough to start separating to allow his fist in. When a bullet travelling faster than a punch hits the gelatin, it makes a big indentation as it stretches the gelatin, then when the gelatin reaches a breaking point, it gradually gives way and the bullet passes through the block losing momentum as it does the work of penetrating the block. Eventually, the bullet slows below the speed at which it can deform the gelatin enough to tear its way through and the gelatin again behaves elastically and the ball comes to a juddering, bouncing halt.

It turns out that once the gelatin gives way and starts to separate then the ease at which the gelatin penetrates is more or less linear. That is to say, at slingshot speeds at least, it takes a certain amount of force to push apart a given volume of gelatin. That's pretty much a constant until projectiles start to reach hundreds of meters per second and slingshots don't shoot that fast.

What this means is that we can start to make formulae to describe the process. Wetted surface area and momentum are linked, because the density and shape is fixed. To simplify things, we just look at two variables: (1) penetration measured in terms of the diameter of the ball; and (2) the velocity it was travelling at. Other variables such as density and diameter of the ball are proportional, you just scale the penetration accordingly.

Let's plot the data in a chart and see what it looks like. In case you're wondering what I meant when I described Selegetes' theoretical model, I have sketched on a curve to describe what I mean.

It appears that the data fits the conceptual model pretty well. Unfortunately the method is not scientifically valid and short of referring to the chart, it's not easy to calculate the expected penetration given measurements of ball diameter, density and velocity. For that, we need to do a mathematical procedure called a regression fit.

**Linear Regression Model**

A regression fit involves calculating a line that would go through all the points as closely as possible, minimising the total prediction error. Once we have the line, we can literally read off the slope of the line the values of the unknown constants in the equation and we will have a usable formula.

What we are supposed to do is come up with a the equation of the formula that accurately describes the underlying physical processes and fit the data to that. Segletes gives that equation, but it is exquisitely complex and honestly, I don't have a sufficient working knowledge of calculus to integrate his formula. Fortunately, I don't need to; there is a practical shortcut.

As all the data points lie on a straight line, that means a straight line adequately describes the mechanics of the process within the velocity, size and density range of the data sample. Beyond this range, I cannot say. For example, below 20 to 30m/s, a straight line would indicate a negative number for penetration and that obviously doesn't make sense.

Conveniently, most spreadsheets have a built-in subroutine to calculate the regression line and assess how closely it fits the data.

We assess how close the model comes to predicting actual measurements with something called the correlation coefficient, R

^{2}. You can take this to mean how much of the observed values can be explained by the model and the rest is down to experimental error or your model lacks the sophistication to fit the data. In this case, R

^{2}=0.82, so the data fits the trend line well, but could be better. Let's consider what fit we would achieve if we presumed that the outliers were the result of experimental error and should be excluded.

By dropping the slowest and fastest projectiles, we achieve a better fit of R

^{2}=0.92, but it is always best to avoid excluding outliers unless you either have good reason to suspect a datapoint was subject to experimental error, or the fit would be significantly improved without it (implying either experimental error or an imperfect model).

Indeed, if we exclude only the fastest projectile's data point,we get a better fit of R

^{2}=0.93.

**The Predictive Formula**

The slope of this line that the computer has drawn through the sample data gives us a formula which we can use to make predictions of penetration depth for projectiles of similar shape, size, density and velocity:

*Penetration{mm} = (0.5223*Velocity{m/s}-11.663)*Diameter{mm}*Density{g/cm ^{3}}/7.85{g/cm^{3}}*

To use it, all you do is enter figures for velocity, diameter and density into the formula and the answer is the penetration. With a bit of conversion, you can also use Imperial units.

**Does It Work?**

I have used this model to make to predict values for the data points gathered:

You can see the formula fits the data well. The red bars are the prediction and the green bars are the actual test results. Certainly, the formula fits the data better than the model I previously used (the blue bars). I derived the blue bars based on data presented in a study by another US Army scientist (Minisi) which were based on much smaller projectiles fired at much higher velocities.

Please note my earlier caveat about the model having an upward bias at low velocities and the exclusion of the fastest projectile gives me less confidence in the predictive value of projectiles with an initial velocity of above 78m/s.

**What's it Used For?**

So far may seem that all the model is good for is predicting experimental observations that we already know, but what it does it tell us that we don't already know?

For a start, it has already told us a lot about the nature of gelatin (and by extension muscle tissue). It tells us that it's not like a fluid and that we need to overcome a certain initial resistance before a bullet will penetrate and once it slows down below this threshold it comes to an abrupt stop.

The model tells us how changes in velocity, size and density will affect penetration. It even allows us to make predictions. Let's say I have a lead sinker that's pretty much a round ball. It weighs 9.3g and my bands shoot that at 57m/s. How much penetration would I expect? Well, the density of lead is 11.34g/cm3, so the diameter of a lead ball weighing 9.3g is about 11.6mm. We enter that data into our formula:

*Penetration{mm} = (0.5223*Velocity{m/s}-11.663)*Diameter{mm}*Density{g/cm ^{3}}/7.85{g/cm^{3}}*

*Penetration{mm} = (0.5223*57}-11.663)*11.6*11.34/7.85* = 303mm

I would estimate that the ball would penetrate 303mm (one foot) into ballistic gelatin.

If we do a series of these calculations, we can answer interesting questions, such as "Should you pick bigger or smaller ammunition for better penetration and/or wound cavity?" and "Is there an optimal for a given bandset?"

If you also have a model for slingshot band velocity (or are prepared to Chony lots of different sized projectiles) you can use the ballistic gelatin penetration model to avoid needing to shoot against ballistic gelatin. A time consuming and expensive experiment is reduced to a quick and costless desktop study.

We take a bandset and using another model that I made earlier, we can predict the velocity the projectile will travel at for any given mass.

This relationship reflects that even without a projectile, there is a maximum speed that the band will travel at. Loading up the projectile mass progressively slows the speed of contraction. We then feed the output of this model into models for kinetic energy and penetration, based on the density of lead (11.34/cm

^{3}).

This chart shows that light projectiles don't deliver a lot of kinetic energy (the red line), as most of the elastic potential energy is expended contracting the bands and moving the dead mass of the pouch and ties.

Look at the predicted penetration (green line). This is something I didn't know before and wasn't expecting, but now I see it, it makes sense. At very small diameters, the ball carries little momentum and while the penetration in terms of the ratio of diameters of the ball is high, the depth of penetration is low. However, penetration rises quickly and peaks early at 3.7g/8.5mm diameter and travelling at 74m/s it achieves a maximum penetration of 334mm. Penetration then decreases gradually as the projectile gets larger and heavier. By 83g/24mm (22m/s), the projectile does not have enough velocity to penetrate the gelatin and bounces off.

What if we used steel instead of lead? Substituting 7.85g/cm

^{3}as the density, we recalculate and replot the curve.

The shape of the curve, including the velocities of the peak penetration and penetration failure point is the same, but the penetration depth is less. At very small diameters, the ball carries little momentum and while the penetration in terms of the ratio of diameters of the ball is high, the depth of penetration is low. However, penetration rises quickly and peaks early at 3.7g/9.7mm diameter and travelling at 74m/s it achieves a maximum penetration of 260mm. Penetration then decreases gradually as the projectile gets larger and heavier. By 83g/27mm (22m/s), the projectile does not have enough velocity to penetrate the gelatin and bounces off.

Let's see what would have happened if we had changed to a heavier bandset instead. I changed the parameters doubling the maximum energy of the bandset to 43J and scaling up the bandset inertia proportionally.

The new bandset achieves the same unladen velocity, but being harder to draw, is less peaky and is less slowed by additional inertia from the projectile. Let's see what it does to penetration.

We can see that not only has the penetration increased at all projectile masses, the optimum mass has increased and the maximum mass for penetration has increased too:

- Optimum: 7.4g, 11mm, 420mm
- Maximum: 166g, 30mm, bounce

**What Should A Hunter Do With This Information?**

Should we select a projectile that gives the optimum penetration? That's for your consideration. You don't necessarily want to stab a target with a needle-like wound canal with a projectile that carries little mass and doesn't deliver it all on the target. Maybe we should be picking a less dense material; whatever does damage but doesn't exit still carrying most of its kinetic energy. I would probably choose lighter bandset and pick a projectile that lies a little on the heavy side of the optimum. Some would choose a lighter projectile and lighter bands yielding flatter trajectory. That's up to you, your shooting style, your comfort zone and the quarry that you were shooting at.

**Next Steps - Out of Sample Backtesting**

What we need to do is to back test this model against out-of-sample data, i.e. to shoot projectiles of different velocities, sizes and densities and see how well the model predicts their ultimate penetration depth. If the model does this well, then we can rely on its predictive ability with confidence. If anyone cares to provide this data, I would be keen to see it.

**Yes, It's Pseudoscience**

Please remember that what I am doing is making a model to predict the ultimate penetration depth of slingshot projectiles

*in ballistic gelatin*. Hunters please note that I cannot say for certain that this projectiles will penetrate flesh, human or otherwise, in the same way.

Why?

(1) Knowing what I know now, it's quite likely that the formulation of ballistic gelatin was calibrated at impact velocities and energies far removed from those that a slingshot produces.

(2) Furthermore, based on penetration anecdotes in vivo (living subjects such as Blood, Pain, and Slingshots) and in vitro (blocks of meat such as Slingshot vs. Pig Cheek - Schweinebacke), penetration of skin, flesh and bone is far less than the ballistic gelatin data would suggest. As I wrote previously,

*animals are not made out of gelatin and they do not sit on a pedestal waiting to be shot.*

That said, the formula and constants represents the best model that I can currently offer to the slingshot community.

**Ballistic Gelatin vs Computer Modelling**

If we are able to model the penetration of slingshot bullets in ballistic gelatin, then why do firearms engineers and law enforcement technicians continue to use ballistic gelatin in their tests? Whereas slingshot ammunition travelling at 30m/s to 100m/s remains intact and stable, penetrates in a straight line and decelerates at a predictable rate, firearms ammunition and high velocity rifle bullets in particular do anything but that. In order to deliver all their energy within their target, firearms bullets, which travel at hundreds to over a thousand m/s must expand, fragment, bend and/or tumble, and will create one or more large temporary cavities and a bent or corkscrew wound track. This is a highly complex mechanical and hydrodynamic process that to this day cannot be modelled entirely accurately.

That’s tough luck for them, but good news for us. Once our models are calibrated and confirmed, we do not have to continue testing in ballistic gelatin to test bandset and bullet combinations, as long as we are not using exotically shaped ammunition or bandsets that shoot outside the calibrated velocity range.

If I may propose another series of experiments, what we need to do is to calibrate our model against muscle and other tissues instead of ballistic gelatin. After all, we have established that ballistic gelatine may not be an exact analogue for body tissue.

Scientists have established that the physical properties of muscle differ along and across the grain of the fibres, but as we might presume that muscles are penetrated side on then this should be the direction of incidence. In other words, we need to shoot against the side of a block of muscle, rather than down a stack of chops.

I propose that, having established the penetration characteristics of muscle tissue, that we also ascertain the characteristics of skin that is attached to underlying muscle by its natural connective tissue. I acknowledge that our friend Jörg has already tested a sample of slingshot ammunition against Schweinebacke (pig’s cheek) however, the block of meat used was not properly secured and he did not offer velocity measurements. In addition, a domestic pig is a poor analogue for the kind of animal that people would hunt with a slingshot. Instead, we should consider using some rabbit meat rabbit, slaughtered for the food trade, hide on and against a block of gelatin. As we know the penetration characteristics of the gelatine, we can deduce the characteristics of the rabbit.

Of course, this is no substitute for actual hunting experience. In my previous blog entry on this subject, I presented several anecdotes of hunters describing the penetration and knock-down power of their band and projectile combinations.

Perhaps all this testing, whether on whole slaughtered animals, ballistic gelatin, or computer models, should be used for is to build a platform of understanding of how slingshot bullets do their work. We now know that small, dense ammunition penetrates best, that there is an optimum configuration for both the penetration depth and displacement volume of the wound canal and we have an indication of what region of the mass-velocity curve that these optima lie. We also have a predictive formula that will allow anyone who knows the size and density of a projectile and the velocity at which his slingshot throws that projectile to calculate the approximate depth of penetration that we would expect to achieve in ballistic gelatin.

**Internal References:**

**External References:**

- Sturdivan, L. M. A Mathematical Model of Penetration of Chunky Projectiles in a Gelatin Tissue Simulant; ARCSL-TR-78055; U.S. Army Chemical Systems Laboratory: Aberdeen Proving Ground, MD, December 1978.
- Segletes, S. B. Modeling the Penetration Behavior of Rigid Spheres Into Ballistic Gelatin; ARL-TR-4393; Weapons and Materials Research Directorate, Army Research Laboratory, March 2008.
- Minisi, M. LS-Dyna Simulations of Ballistic Gelatin; U.S. Army ARDEC: Picatinny Arsenal, NJ, report in progress, 31 October 2006.
- Nicholas NC, Ballistics Gelatin, INLDT, 2004 http://www.firearmsi...atin Report.pdf