Components | Drag Calculations

Test Equipment

Drag was fit only for the standard-speed component test in 2026. Downrange velocity was not collected for the high-speed-vane builds, so no drag constant is reported for that subset. The equipment below covers the standard-speed protocol.

• Bow: Hoyt AX-3 33, 28″ draw length, 70 lb draw weight.
• Indoor facility: Easton Salt Lake Archery Center (Easton Foundation), Salt Lake City, UT.
• Shooter: Easton Archery's custom shooting machine, used to ensure precise, repeatable shots and eliminate shooter variability.
• Mass measurement: Last Chance Archery Pro Grain Scale 2.0. Every arrow was weighed and per-build averages were used in downstream calculations.
• Fletching jigs: a left-helical Bitzenburger jig, a right-helical Bitzenburger jig, and an Arizona E-Z Fletch were used to fletch arrows across three deliberate helical conditions per vane.
• Downrange velocity: Garmin Xero C1 Pro Doppler chronographs at 0.5 yd, 30 yd, and 60 yd.
• Environmental measurement: Kestrel 5700 Elite. Density altitude averaged 5670 ft across the test sessions.

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 with the Kestrel 5700 Elite ballistics meter)

The coordinate system used is:

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

Experimental Procedure

For each arrow build, velocities were measured at three downrange distances (0.5 yd, 30 yd, and 60 yd) using the Garmin Xero C1 Pro Doppler chronographs listed in the Test Equipment block above. At each distance, 12 shots were measured per build. A small number of broadheads broke or bent on impact, and some radar readings were missed; the number of valid samples therefore varied slightly by configuration and is reflected in the per-shot fits and the resulting confidence intervals.

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.

Per-Shot vs. Averaged-Velocity Fits (New in 2026)

In 2025, per-build velocity averages at each radar distance were the inputs to the drag fit, producing one K per build with a single CI propagated from the averaging step. In 2026, the drag fit is run per shot instead of on the per-build averages: each individual triple of (0.5 yd, 30 yd, 60 yd) velocities is treated as its own observation, an independent K is fitted for that shot, and the per-build K and 95% confidence interval are computed from the resulting distribution of per-shot fits. This better reflects the underlying shot-to-shot variability in drag for a given build and avoids losing information to early averaging.

Practical Considerations

• Wind effects are not modeled; the study was conducted in a controlled indoor environment.
• Air density is measured using the Kestrel 5700 Elite ballistics meter for accurate environmental parameters.
• Per-build Kand 95% confidence intervals are computed from the distribution of per-shot fits using a t-distribution (df = n − 1).