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.

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.

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.

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.

Figure 4: Calculations for the Drag and Ballistic Coefficient of a BB.

Table of Viscosities Table of Densities Diameter of a BB Graph of Drag Coefficients Vs Reynolds Number Wikipedia article on drag coefficients Wikipedia article on drag coefficients Reference for Ballistic Coefficient of a BB Reference for mass of a BB Reference on types of ballistic coefficient

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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