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
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.
- 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.
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.
Interpolation of the Air Data
Figure 3 shows how I used Mathcad to interpolate all this data.
Figure 4 shows how I calculated the
- force of drag on the BB
- acceleration experienced by the BB
- ballistic coefficient of the BB
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
Next, I will look at a 22 caliber pellet.