Quote of the Day

Criticism may not be agreeable, but it is necessary. It fulfills the same function as pain in the human body. it calls attention to an unhealthy state of things.

— Winston Churchill

## Introduction

I received the following question on one of my earlier blog posts:

Naval guns claim a range of up to 16 miles, and apparently do so with an initial velocity of only approx. 3000 fps [, which is similar to rifle bullet]. how is this possible[?]

You can view my answer on that blog post, but I started to think about alternative, but equivalent, ways to answer this question. From a numerical standpoint, one way to answer his question is by comparing the ballistic coefficient of a 16-inch projectile (Figure 1) to that of a standard firearm projectile. A projectile with a large ballistic coefficient is less affected by drag than a projectile with a smaller ballistic coefficient. We can use the the ballistic coefficient to compare the effect of drag on different projectiles. A 16-inch projectile goes so much farther than a rifle bullet because the drag on the 16-inch projectile is relatively small compared to its momentum. Ultimately, this is because mass increases by the cube of the projectile dimensions and drag increases by the square of the projectile dimensions. This means that larger projectiles tend to have higher ballistic coefficients and drag has less effect.

Let's put some numbers together ...

## Background

The Wikipedia defines the ballistic coefficient as follows:

- Ballistic Coefficient
- The Ballistic Coefficient (BC) of a body is a measure of its ability to overcome air resistance in flight. It is inversely proportional to the negative acceleration — a high number indicates a low negative acceleration.

A projectile with a small deceleration due to atmospheric drag has a large BC. Projectiles with a large BC are less affected by drag and have performance closer to their performance in a vacuum.

While the Wikipedia definition is accurate as far as it goes, it does not allow you to compute the BC of a projectile. Equation 1 shows you how to compute the BC of any projectile.

Eq. 1 |

where

*a*is the acceleration of a reference projectile (eg. G7)._{ReferenceProjectile}*a*is the acceleration of the projectile we are interested in._{Projectile}

## Analysis

I will compute the BC of the US Navy's 16-in, high-capacity (1900 pound) shell that was fired by the Iowa-class battleships using a couple of different methods.

- Estimate the deceleration of the battleship projectile and compute Equation 1.
- Apply a geometry-based formula.

Both methods give similar answers.

### Method Based on Deceleration Estimate

To estimate the 16-inch projectile's deceleration. There is data available for this. Figure 2 shows a snippet from a US Navy range table for the 16-inch projectile (Source).

For my BC estimate calculation (Figure 3), I will use only the forward velocity component because that component is experiencing the bulk of the drag.

### Ballistic Coefficient Calculation Using Formula

Figure 4 shows the ballistic coefficient calculation using a geometric formula that compares the 16-inch projectile to a 30-06 projectile (Ball, M2, 152 grain, 0.308 in diameter).

## Conclusion

Here is a quote that supports my analysis from "Modern Practical Ballistics" on page 115 (ISBN 0-96212776-3-7).

Modern long-range guns like the US Navy's 16-inch guns are able to attain ranges in excess of 25 miles using an initial elevation angle near 45 degrees; their shells reach an altitude of approximately seven miles. Although their muzzle velocities are generally only about 2600 to 2800 fps, their enormous size and weight -- in excess of 2000 pounds -- gives them ballistic coefficients of around 15, which are 30 to 50 times as great as small arms projectiles!

In Figure 4, you scale up the 30-06 data to the larger reference projectile using the factor 0.473 . What is the origin/derivation of this scaling factor?

Hi Lawrence,

The scaling factor appears in the formula for the ballistic coefficient, which I show below. I also show how to compute the ballistic coefficient of the M2 0.308 bullet. This calculation comes from Pejsa.

The shape factor is usually determined through testing that generates a velocity versus range table, which can be used to determine the ballistic coefficient. I show a calculation for determining the drag coefficient (related to the ballistic coefficient) in this post.

In the old days, people would estimate the shape factor using a template like the one I show below from Hatcher's Notebook.First, you would take your bullet, which for the M2 is 30 caliber (0.30), and find the best ogive fit using the following figure.

According to Hatcher, the best ogive fit is ~7.

You then use the following table, interpolate between the ogive 6 and 8 values to get about ~0.575, which I show below.

The value is significantly off from Pejsa shape value 0.473. However, this is how people used to do it when designing bullets.

I hope this helps. If it does not, ask another question.

mathscinotes

Thanks. So, I am still confused. You calculate the BC of the 30-06 bullet to be 0.473 using d=0.308in, w=152 grains, and shape factor = 0.484 . I checked this, to be sure I was following, and the BC = 0.473 (rounded). However, in Figure 4, shouldn't you have used the shape factor, i =0.484, instead of the BC? They are so close, it really doesn't matter, but which is correct?

Anyways, I really appreciate you taking the time to explain these concepts. I am trying to recreate thru analysis the Table of Fire for my group's 10-pdr Parrott Rifle, and this has renewed my interest in ballistics. The 10-pdr Parrott shell is big and relatively slow (MV=1230 fps), and I don't know how much firearm ballistics applies to it, but it is fun learning.

You found a bug. I should have used the shape factor i = 0.484. The BC and shape factor are close, so it did not make a difference with 2 significant digits. I will fix it. Thanks for helping me correct the site.

mathscinotes

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