## Introduction

I was received an email this weekend from a dad struggling to help his son with a project involving aerodynamic drag and and BB gun. I did some quick calculations which I document here. I will try to look at pellets tomorrow. I was able to use basic principles to duplicate the empirical results quoted by manufacturers.

I will be computing three numbers associated with an air rifle shooting a BB:

- The force of drag on the BB as it leaves the muzzle
- The deceleration of the BB as it leaves the muzzle
- The ballistic coefficient of the BB

## Background

### Definition of Ballistic Coefficient

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}

### Coefficient of Drag Given Reynolds Number

Figure 1 shows the coefficient of drag graph that I digitized using Dagra for this example.

### Density and Viscosity Data for Interpolation

Figure 2 shows the table from this web site that I interpolated so that I could get the density and viscosity of air at various temperatures.

## Analysis

### Interpolation of the Air Data

Figure 3 shows how I used Mathcad to interpolate all this data.

### Calculations

Figure 4 shows how I calculated the

- force of drag on the BB
- acceleration experienced by the BB
- ballistic coefficient of the BB

## Conclusion

I computed three numbers associated with an air rifle shooting a BB:

- The force of drag on the BB as it leaves the muzzle: 0.17 N = 0.038 pound = 0.613 ounce
I cannot find corroborating information on the web. However, I use this number to compute the ballistic coefficient, for which I do find corroborating evidence.

- The deceleration of the BB as it leaves the muzzle: 1691 ft/sec
^{2}Straightforward application of Newton's second law.

- The ballistic coefficient of the BB: 0.014

Next, I will look at a 22 caliber pellet.

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