# A Drag formula used to calculate bullet hole depth?

1. Apr 3, 2017

### darkdave3000

I'm writing a real time 3D physics simulator that simulate bullet ballistics.

Can I use the air drag formula that factors in drag coeficient and outputs drag force to calculate how deep a bullet will penetrate a target material?

Say I substitute air pressure with target material density. And then I can use the drag force to calculate when the bullet will come to a full stop and the distance travelled. (The force of the bullet would have to be over the tensile compression strength of the material to travel in the material medium or else I will have to assume the bullet bounced off it.)

That would work right?

Furthermore I could also use the drag force to work out energy due to air friction and increase the heat of the material and the bullet to make the computer simulation decide if the bullet or the target material is hot enough to melt and therefore change properties in density. But how do I decide what percentage of air friction goes into the bullet vs the target medium? In fact for a bullet travelling in air, how does the heat dissipate? 50:50 between bullet and air?

2. Apr 3, 2017

### PhysicsPhan

Well firstly, you can use drag to calculate the trajectory and speed at impact. The heat of friction as the projectile is moving through the air should be negligible: the heat of the bullet is due to the firing mechanism. You cannot accurately use drag force to calculate how deep the projectile will penetrate for obvious reasons. One being that flesh does not act as a fluid. Also, to find out the dissipation of heat from the bullet into the air, you have to consider the specific heat of the bullet for one, and the problem is very complicated.

3. Apr 4, 2017

### jbriggs444

Your project faces enough daunting challenges without worrying about how a bullet deforms within the target.

Even if a soft lead projectile does not melt, it can deform very significantly. I once fired a soft-point 0.45 bullet at an oblique angle (about 10 degrees from tangent) onto the surface of a pond which was frozen some 4-6 inches deep. The bullet punched out a hole in the ice -- a crater.

Did it penetrate? No. It bounced. We found it afterward on the surface of the ice where it had rebounded toward the firing position. Did it melt? No. But it deformed. It was squashed badly. Did the ice melt? No.

The same test performed with a shotgun load fired under the same conditions did not penetrate. Instead, a shallow divot was formed in the surface of the ice. [A second shot into the face of the divot penetrated handily and blew out a hole in the ice]

Would your simulation model any of these three events accurately?

4. Apr 4, 2017

### A.T.

I doubt such simplistic ideas will work. Even the most advanced currently available computer modelling technology would have problems to predict such impacts. That's why it's still mostly tackled experimentally:

5. Apr 4, 2017

### darkdave3000

I think so.

The simulator would factor in the tensile strength of ice. I do not have that data but I would have to look it up or find some way to calculate it. The bullet will richochette off the ice if the force does not exceed the tensile strength of the ice. I would assume also that a head on 0 degree angle collission on the ice would transfer 100% of the force to the ice and a 45 degree would only transfer 50% to the ice for test if tensile strength was exceeded.

The heat of the bullet in this case would not be important. But my simulator also will simulate re-entry capsules from rockets and MIRV from ballistic missiles and in those situations drag heat would be relevant.

The simulator like most simulators wont be perfect but should be able to give as a minimum above 90% accuracy of flight performance from firing/launching of a projectile/rocket to impact.

6. Apr 4, 2017

### jbriggs444

It would be nothing anywhere near so binary. In any case, there is no such thing as "the force of a collision".

7. Apr 4, 2017

### darkdave3000

Im not a very pedantic person and I speak lazilly sometimes and just assume people will "get it". It's an asian trait. My apologies. When I say force of collission what I mean is tensile strength (compression type) is meassured by pressure devided by surface area. The bullet will have a surface area and if the force of the bullet over the surface area the bullet has exceeds the tensile strength of the material then penetration will occur and the bullet will deflect slightly into the material depending on the angle, if not then the bullet and most of its kinetic energy will be deflected away from the material.

8. Apr 4, 2017

### jbriggs444

Can you explain that more clearly? What is compressive tensile strength? What is pressure divided by surface area?

Furthermore, how will you compute the "force of the bullet" without knowing whether (and how far) it will penetrate?

9. Apr 5, 2017

### darkdave3000

This is "Compressive - Tensile Strength"
https://en.wikipedia.org/wiki/Compressive_strength
Basicly it means that if the bullet cannot overcome this strength it will bounce back at an angle depending on angle of shot.

The metric unit for pressure Pascal is defined as force(newtons)/meter square.
The bullet will have kinetic energy and force. It will also have a cross sectional surface area. If the force/cross sectional area (bullet pressure excerted on contact area of target surface) exceeds that of the target material's tensil strength then there will be penetration since this is the minimum amount of force required to break the bonds of the atoms.

How far it penetrates the material will depend on deceleration rate of the bullet and remaining velocity from this point onward. The deceleration will depend on the force acting against the bullet's forward velocity which will be the drag force with the density of the target material substituted for air density. So the material now becomes the "fluid" in the drag formula instead of oxygen, nitrogen and carbon dioxide(our atmosphere).

Also while inside the material: once the bullet slows down below the force required to overcome tensil strength then the bullet will attempt to bounce back either by back tracking or deflecting at the deflection angle while under the material's surface. If there is material in the way then the bullet will attempt to deflect through the material. If there is little material then the bullet might be able to do this since then we will instead be overcoming the weaker tensil shearing strength instead of compression. So it should be able to simulate your bullet in the ice scenario, the bullet penetrates the ice at a shallow angle and then deflects and sprays ice upward as it does so.

Is that clear enough?

Above is the theory anyways, I hope it pans out in real life beyond 90% accuracy.

Last edited: Apr 5, 2017
10. Apr 5, 2017

### jbriggs444

That reference fails to define "compressive tensile strength". What do you mean when you use the term?

That fails to provide a meaning for pressure divided by area. Instead it speaks to pressure multiplied by area.

Kinetic energy yes. Force, no. Force is not an attribute that a bullet possesses.

In the process of a collision, the force between bullet and target will vary both over time and across position within the contact area. Even that is an over-simplification since neither bullet nor target will be rigid during the collision. Both will be subject to internal forces and deformations as well.

11. Apr 5, 2017

### darkdave3000

When the bullet strikes something there will be force according to the drag formula. In the drag formula you get drag force in newtons when you enter the material density and the drag coefficient of the bullet. The formula works out the force due to the deceleration of the bullet as it flies through the fluid that is the material. Remember you have force when you have acceleration, deceleration is negative acceleration. The formula accounts for this negative acceleration caused by the bullet slowing down in the fluid which causes friction.

Force is an attribute a bullet possess while it transitions any material with density. When a bullet strikes you force is being transferred as momentum (but not all momentum at once, it happens gradually). Imagine your body as a fluid. In the air the bullet actually is already decelerating as it pushes air out of it's way, when it hits the target it decelerates even more (more force) as it pushes tissue aside for a human or synthetic material aside for a synthetic target.

Force per unit area = pressure.

Try googling "Compressive Strength" alone without "tensile".
https://en.wikipedia.org/wiki/Compressive_strength

Last edited: Apr 5, 2017
12. Apr 5, 2017

### Staff: Mentor

@jbriggs444 example of the ice on a pond is REALLY important.

The ballistic coefficient (what you seem to want to work with) does affect the terminal or impact velocity of a projectile, at distance. What determines penetration is a collection of factors that are paramount out in in the field. Not just velocity. I would start with something simple - old versions the Speer reloading manuals have calculation examples for ballistic coefficients. And tables of ballistic coefficients. These values can be determined using a chronograph by firing projectiles under what is termed 'standard meteorological atmosphere'. Note the effect of ogives, boat tail, and sabots, for example:

http://www.appliedballisticsllc.com/Articles/ABDOC2.1 - OGIVECOMPARISON.pdf

You need to learn some interesting ideas about penetration. It is very definitely not all about projectile terminal velocity.

https://en.wikipedia.org/wiki/Ballistic_gelatin
And an example on the receiving end is "chobham" or ceramic armor:
https://en.wikipedia.org/wiki/Chobham_armour

I think PF has someone who knows a lot about exterior ballistics and may have some good advice - @Dr. Courtney

13. Apr 5, 2017

### darkdave3000

Was any of that advice for me? Or was that just for JBrigs?
What are your thoughts on my idea of using drag formula to calculate depth of penetration by treating the target as a fluid with different density?

Also what formula should I use if the bullet is supersonic?

14. Apr 5, 2017

### Dr. Courtney

Predicting bullet penetration from first principles is an extremely challenging problem, and is not yet solved in the general case. People who really want to know the answer for a given projectile and target have no recourse at the present time than to perform experiments.

Some simplified cases are beginning to be understood: nondeforming bullet in a homogeneous isotropic medium like water or calibrated ballistic gelatin.

One challenge continues to be that material properties are not known at the high strain rates of ballistic impact.

15. Apr 5, 2017

### ZapperZ

Staff Emeritus
16. Apr 5, 2017

### Dr. Courtney

http://www.investigativesciencesjournal.org/article/view/17572/11372

The link shows a recent publication illustrating our experimental approach of using high speed video to quantify the interaction forces between a bullet and ballistic gelatin. The attached figure from the paper suggests the difficulty of predicting these force curves from first principles. The ripples in the force curve for distances beyond 5 cm may be attributable to experimental noise, but the second peak near 4.5 cm is real (it is repeatable). Most simple computational approaches predict a single peak, and there are none that we know of that can accurately reproduce the peak interaction force, the depth at which it occurs, or provide quantitative agreement with the tail.

Theorizing with drag equations is easy. Obtaining quantitative agreement between theoretical predictions and measured force curves is hard, even though the area under the curve is simply the initial kinetic energy of the projectile.

17. Apr 9, 2017

### darkdave3000

So youre saying publicly available college level physics offer no significant insight into this? Remember the solution doesnt have to be 100%

18. Apr 9, 2017

### darkdave3000

I like this illustrated reply , what I wonder is why couldn't they(experts in general) use the drag formula to account for various densities of tissue/material(target). They could have also use laws of refraction to calculate the bullet changing course slightly as it transitions from say the liver to the heart (change in densities).And then they can compare the theorized results to the experimental.

References:
https://en.wikipedia.org/wiki/Drag_equation
https://en.wikipedia.org/wiki/Snell's_law

19. Apr 9, 2017

### Dr. Courtney

Going through a fluid, the drag coefficient is not constant, but varies with velocity. Going through a visco-elastic medium (tissue or ballistic gelatin), there is the added complexity that the medium is being damaged to allow the bullet to pass, so quantifying the energy needed to rupture the tissue is also challenging, as it depends on velocity and bullet shape and diameter. Finally, in many cases, an additional complicating factor is the deformation of the bullet. Even in a homogeneous medium, it is a very complicated problem.

20. Apr 10, 2017

### darkdave3000

Hi thanks for this insightful reply. You say the Drag Coefficient changes according to speed of the relative fluid. Is there a formula to predict this or is this all experimental testing only?

Regarding tearing the tissue, I believe I mentioned a solution to this not too while back in this thread about using the tensile strength of the tissue? So if you know the force of impact on the new medium you can calculate the pressure and compare this pressure to the compressive strength of the material being impacted upon.

Here is a reference: https://en.wikipedia.org/wiki/Compressive_strength

If it fails to overcome this compressive strength then the bullet should deflect, if it does succeed then it should refract.
Regarding the bullet deforming and melting etc I have decided I dont need this level of accuracy for now in my simulator. That just the big picture newtonian mechanics and fluid dynamics should be suficient to give a reasonably accurate simulator.