## Introduction

I have always been interested in the shooting sports, but I have not pursued any of them while I was raising my kids. I suddenly find myself with my kids gone and my interest in shooting has reappeared. As part of my interest in shooting, I have been reading the book “Modern Practical Ballistics” by Pejsa. While reading this book, I quickly learned that not all bullet shapes are created equal. So I started to look at how bullet shapes are defined. To give you an idea of the diversity of bullet shapes, I have included Figure 1, which shows a small number of bullet shape examples.

 Figure 1: Example of a Few Bullet Shapes. Source

To make this learning exercise more concrete, I decided to focus on developing a general algorithm for computing the mass of a spitzer bullet. I made this choice after reading an interesting article that computed bullet mass using a BASIC program. I used to do a lot of BASIC programming myself, but in recent years I have found computer algebra systems to be more convenient for the working engineer to use on a daily basis. I thought the mass calculation would be a good example to use to demonstrate the power of computer algebra system to solve real problems. I will develop an algorithm for computing the mass of bullet and will test this algorithm on a real bullet design.

The overall exercise was a good exercise in applied mathematics. It also demonstrates the power of a computer algebra system. In this case, I am using Mathcad.

## Descriptive Geometry of a Spitzer Bullet

The Pejsa book is focused on the spitzer (German for pointed) bullet shape, and I will concentrate on the spitzer shape as well. Today, it is the most commonly used bullet shape for hunting and target shooting applications. Figure 2 shows how this shape is defined.

Figure 2: Descriptive Geometry of a Spitzer Bullet (Tangent Ogive with Radius of 6 Diameters).

As shown in Figure 2, the spitzer bullet can be modeled geometrically in three pieces:

• Ogive (pronounced “Ojive”)
The ogive shape forms the front of the bullet. The ogive shape is formed from the arcs of two circles. The ogive may or may not be tangent at the point of intersection to the cylindrical portion of the bullet. When the circles are tangent to the cylinder portion, we call say this is a tangent ogive. When the circles are not tangent to the cylinder portion, we say we have a secant ogive. The rationale behind the use of the term “secant” can be seen in Figure 6, where there are two horizontal reference lines (brown color) that are both secant lines.
• Cylinder
The cylindrical portion of the bullet is what engages the rifling of the barrel.
• Frustum of a Cone
The back of the bullet (aka “boattail”) geometrically is in the shape of the frustum of the cone Tapering the back of the bullet reduces drag, particularly at speeds less than supersonic.

I have included three examples of ogive-shaped projectiles. Figure 3 shows a bullet with a tangent ogive nose. Figure 4 shows a bullet with a secant ogive nose. Figure 5 shows one of the most famous secant ogive noses, which is on the Honest John missile. I cannot imagine the Honest John ogive being used for a bullet, but it does give a feeling for the range of shapes possible using an ogive model.

 Figure 3: Tangent Ogive Example. Figure 4: Secant Ogive Example. Figure 5: Secant Ogive Example. Source Source Source

## Bullet Mass Calculation

### Approach

We will compute the volume of the spitzer bullet as follows.

• Compute the volume of the frustum portion (VFrustum)
• Compute the volume of the cylinder portion (VCylinder)
• Compute the volume of the ogive portion (VOgive)
• Compute the total volume by summing all the volumes of the pieces
(${{V}_{Total}}={{V}_{Frustum}}+{{V}_{Cylinder}}+{{V}_{Ogive}}$)
• Compute the mass using the density of lead ($M=\rho_{LeadAlloy} \cdot {{V}_{Total}}$)

### Frustum Volume

The volume of the backside of a bullet (known as the boattail) has the shape of the frustum of a cone. The formula for the frustum is well known and is given in Equation 1.

 Eq. 1 ${{V}_{Frustum}}=\frac{\pi \cdot L_F}{3}\cdot \left( {{R}^{2}}+R\cdot r+{{r}^{2}} \right)$

### Cylinder Volume

Equation 2 is used to calculate the volume of the cylindrical portion of the bullet.

 Eq. 2 ${{V}_{Cylinder}}=\pi \cdot {{R}^{2}}\cdot L_C$

### Ogive Volume

#### Types of Ogives

The real work is in computing the volume of the ogive. Let’s begin with the variables defined in Figure 6.

Figure 6: Basic Ogive Variable Definitions.

Using Figure 6, we can see that the different types of ogives are defined by the angle γ. There are three cases (for each case, two criteria are listed– their equivalence is shown at the bottom of this post) :

• γ < π/2 $\left( \rho >\frac{{{R}^{2}}+{{L_O}^{2}}}{2\cdot R} \right)$
The ogive’s circular arc is not tangent to the cylinder at the point of intersection. This case results in a rather pointy bullet.
• γ = π/2 $\left( \rho =\frac{{{R}^{2}}+{{L_O}^{2}}}{2\cdot R} \right)$
The ogive’s circular arc is tangent to the cylinder at the point of intersection. This results in a rather curved bullet.
• γ > π/2 $\left( \rho <\frac{{{R}^{2}}+{{L_O}^{2}}}{2\cdot R} \right)$
The ogive’s circular arc is not tangent to the cylinder at the point of intersection. This case results in a bulbous shape like that of the Honest John missile (see Figure 5).

#### Analysis

The most important cases for bullet design are when γ π/2 and the following drawings will focus on these cases. However, the equations are general and apply to all cases. We begin our ogive analysis with Figure 7, which contains the definitions of the critical angles and lengths.

Figure 7: Definitions of Ogive Angles.

We need to calculate angles α and γ. To accomplish this task, we will be working with two triangles from Figure 7 (see Figures 8 and 9).

 Figure 8: Angles Alpha and Gamma Derivation. Figure 9: Triangle for Deriving Angle Beta Equation.

Using Figures 7, 8, and 9, we can derive the key equations (Equations 3-6). I have used basic trigonometry and will the let the figures stand for themselves. These equations will be used in part 2 to compute the mass of two projectile examples.

 Eq. 3 $\alpha ={{\cos }^{-1}}\left( \frac{\sqrt{{{R}^{2}}+{{L_O}^{2}}}}{2\cdot \rho } \right)-{{\tan }^{-1}}\left( \frac{R}{L_O} \right)$
 Eq. 4 $\gamma =\pi -{{\tan }^{-1}}\left( \frac{R}{L_O} \right)-{{\cos }^{-1}}\left( \frac{\sqrt{{{R}^{2}}+{{L_O}^{2}}}}{2\cdot \rho } \right)$
 Eq. 5 $y(x)=\sqrt{{{\rho }^{2}}-{{\left( \rho \cdot \cos \left( \alpha \right)-x \right)}^{2}}}-\rho \cdot \sin \left( \alpha \right)$
 Eq. 6 $V_{Ogive}=\int_{0}^{L_O}{\pi \cdot y{{(x)}^{2}}\cdot dx}$

Note that two equivalent criteria were listed for identifying the type of ogive. Equation 2 can be used to demonstrate this equivalence. Included below is a short derivation showing that $\gamma < \frac{\pi }{2}$ => $\rho > \frac{{{R}^{2}}+{{L_O}^{2}}}{2\cdot R}$ . The equivalence between the other criteria can be demonstrated similarly.

Derivation Demonstration Using Equation 2

Continued on Part 2

I am an engineer who encounters interesting math and science problems almost every day. I am not talking about BIG math here. These are everyday problems where a little bit of math really goes a long way. I thought I would write some of them down and see if others also found them interesting.
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### 10 Responses to Ballistics, Ogives, and Bullet Shapes (Part 1)

1. Dan Marotta says:

I’ve always wondered about bullet performance. You provide some great analysis on the Spitzer design. Looking forward to when you tackle the other examples you mentioned above. Some other users of Mathcad increased my interest because they talk about muzzle velocity, projectile motion and firemarm design.

• mathscinotes says:

Thanks for the comments. Once I finish going through my ogive examples, I will start discussing basic ballistics. I usually blog about books I am reading. My general approach is to work through problems while reading the books so that I know I am learning the material. I want to put together material that encourages people to use computer algebra systems in their daily work.

2. Hello. This is fantastic material and has helped me greatly in an essay about drag coefficients of nosecone shapes. However, could you please explain the derivation of equation 3. How do you get it?

Thank you.

3. Joel says:

Now there are ventilated bullets to change the dynamics in the barrel.
http://gizmodo.com/5865769/the-compbullet-reduces-recoil-with-rocket-vents

4. sgeorge says:

It may be useful to read this material: http://www.if.sc.usp.br/~projetosulfos/artigos/NoseCone_EQN2.PDF. I also made some subroutines in Matlab from this material.

• mathscinotes says:

Thanks! I will go through it.

mathscinotes

5. STEVE KLOSSING says:

I COULD USE SOME HELP WITH THE FOLLOWING
Spherically blunted tangent ogive
Elliptical,Haack series
AS TO BULLET NOSE SHAPES
THANKS
STEVE

6. Nice website. Do you have an email address? Regards, Alastair

• mathscinotes says:

I will send you a private note.

Mathscinotes