Archery Ballistics Study | Drag Approach

Overview

Projectile motion through air is strongly influenced by aerodynamic drag. For typical archery or small arms projectiles, drag is modeled as proportional to the square of the velocity:

• F_D is the drag force
• C_d is the drag coefficient
• A is the reference area
• ρ is the air density
• v is the velocity

In this analysis, the product C_d · A is treated as a single drag constant K, since the effective area A can be difficult to define for irregular or deformable projectiles (e.g., arrows with fletching). This simplifies the model:

This approach assumes C_d remains approximately constant over the velocity range of interest, which holds for most subsonic projectiles but may not apply at transonic or supersonic speeds.

The model simplifies the projectile to a point mass acted upon by gravity and aerodynamic drag, without explicitly accounting for phenomena like spin, yaw, or flexing. Any aerodynamic consequences of these effects are subsumed into the fitted drag constant K for each configuration. Variability was further reduced by using standardized projectiles and ensuring consistent bow tuning throughout all tests.

Physical Model

• Gravity, modeled as a constant downward acceleration
• Aerodynamic drag, opposing motion and quadratic in speed

The resulting acceleration is:

• v: projectile velocity vector
• g = (0, -g, 0): gravitational acceleration vector
• m: mass of the projectile
• ρ: air density (measured using a Kestrel ballistics meter)

The coordinate system used is:

• x: downrange (horizontal)
• y: vertical
• z: crossrange (not used here, but supported in simulation)

Experimental Procedure

• Testing Facility: All drag and lift measurements were conducted indoors at the Easton Salt Lake Archery Center (Easton Foundation), Salt Lake City, UT, to eliminate wind and environmental variability.
• Projectile Launch: Arrows were fired from a Hoyt AX-2 29, shot with Easton Archery's custom shooting machine to ensure repeatable launch conditions and precise alignment past each velocity chronograph.
• Velocity Sampling: For each arrow build, velocities were measured at 0.5 yards (near-bow), 30 yards, and 60 yards using Garmin Xero C1 Pro Doppler chronographs. At each distance, 12 shots were measured. A few of the broadheads broke or bent on impact, and some radar readings were missed. As such, the number of samples varied by arrow configuration and are reflected in the confidence intervals.
• Atmospheric Conditions: Density altitude was measured with a Kestrel 5700 Elite; during testing, density altitude averaged 5697 ft.
• Arrow Mass: Arrow mass was measured with an OMP Digital Accu-Arrow scale. Every arrow was measured and averaged for each build.
• Vane Helical: Arrows were fletched with a straight Bitzenburger jig, and right-helical Bitzenburger jig, and a Arizona E-Z Fletch. Helical amount was measured and quantified with the DCA Custom Arrows Vane Angle Tool.
• Data Reduction: Velocities at each range were averaged and 95% confidence intervals calculated for each arrow configuration at each distance. 95% confidence intervals were computed based on the t-distribution, with degrees of freedom equal to the sample size minus one, to appropriately reflect the increased uncertainty associated with small sample sizes.

This multi-point approach enables precise modeling of drag effects, using robust indoor environmental controls and consistent measurement geometry.

Numerical Integration Method

The projectile trajectory is computed using a 4th-order Runge-Kutta (RK4) numerical method, ideal for systems with non-linear forces (like drag ∝ v²).

Each simulation step computes intermediate velocity and acceleration estimates. The solver uses an adaptive time step based on instantaneous speed to maintain stability and performance.

Determining the Drag Constant from Measured Velocities

Downrange velocity measurements are collected at known distances using Doppler radar. These measured velocities are compared to simulated velocities from the RK4 solver.

The drag constant K = C_d · A is inferred by adjusting simulation inputs until modeled velocity loss between two points matches observed data. No prior knowledge of the projectile's shape or exact surface area is needed.

With known values for:

• Initial mass
• Measured air density
• Downrange velocities at two or more distances

The drag constant K can be estimated accurately under controlled indoor conditions.

Practical Considerations

• Wind effects are not modeled; the study was conducted in a controlled indoor environment.
• Air density is measured using a Kestrel ballistics meter for accurate environmental parameters.
• All data points are reported with 95% confidence intervals to quantify measurement and statistical uncertainty.