Air Rifle Math

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 $\displaystyle B{{C}_{\text{Projectile}}}\triangleq \frac{{{a}_{\text{ReferenceProjectile}}}}{{{a}_{\text{Projectile}}}}$

where

• aReferenceProjectile is the acceleration of a reference projectile (eg. G7).
• aProjectile is the acceleration of the projectile we are interested in.

Coefficient of Drag Given Reynolds Number

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

Figure 1: Coefficient of Drag Versus Reynolds Number.

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.

Figure 2: Air Density and Viscosity Data.

Analysis

Interpolation of the Air Data

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

Figure 3: Digitization Code for Graphical 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

Figure 4: Calculations for the Drag and Ballistic Coefficient of a 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/sec2

Straightforward application of Newton's second law.

• The ballistic coefficient of the BB: 0.014

This agrees with data I have seen on forums here and here.

Next, I will look at a 22 caliber pellet.

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