The three forces acting on a shaft
From the moment the nock leaves the string, an arrow is governed by three accelerations. Gravity pulls it down at a constant 9.81 m/s². Aerodynamic drag opposes its motion through the air and bleeds speed over the whole flight. Wind coupling — really just drag acting on the air-relative velocity — pushes the shaft sideways when the air itself is moving. The Teamstoviewticker engine integrates all three every 2 milliseconds of simulated time, which is why no two arrows in a gusty match fly exactly alike.
Gravitational drop
In a vacuum, the vertical position of an arrow launched at speed v₀ and angle θ follows the classic projectile equation. Horizontal speed is constant, so flight time to a target at distance D is simply the distance divided by horizontal velocity.
y(t) = v₀ · sin θ · t − ½ · g · t²t ≈ D / (v₀ · cos θ) Δy_drop = g · D² / (2 · v₀² · cos² θ)Plug in a full-draw arrow at 62 m/s and a 70 m target, and the vacuum flight time is about 1.13 seconds. Over that time gravity pulls the shaft down roughly 6.3 m relative to the line it was launched on. That is why an archer never points the arrow at the gold: the launch line sits several metres above the target and the arrow falls onto it. The launch angle that exactly cancels the drop satisfies sin 2θ = g·D / v₀², which gives about 5.15° at 70 m in still, frictionless air.
Aerodynamic drag
The textbook drag force is F = ½ · ρ · Cd · A · v², where ρ is air density, Cd the drag coefficient, A the reference area and v the air-relative speed. For an arrow, most of the resistance comes from skin friction along the shaft and the fletching rather than the tiny frontal area, so the engine lumps every aerodynamic term into a single constant k and applies it to the air-relative velocity vector:
a_drag = −k · |v − w| · (v − w) with k = 0.0012 m⁻¹Treated along the line of flight, this gives dv/dz = −k·v, so speed decays exponentially with distance: v(D) = v₀ · e−kD. A 62 m/s arrow arrives at 70 m moving about 57 m/s, an 8% loss. Because the arrow is slower in the back half of its flight, the real flight time grows to roughly 1.18 seconds and the drop grows with it — the drag-corrected sight angle at 70 m is closer to 5.6° than 5.15°.
Wind drift and the lag-time rule
Wind does not push an arrow the way a hand pushes a box. Instead, the moving air changes the relative velocity the drag term sees. With a crosswind wx, drag slowly accelerates the arrow sideways toward the air speed. Solving that linearised equation produces an elegant result known in ballistics as the lag-time rule: drift equals crosswind speed multiplied by the difference between the real flight time and the vacuum flight time.
drift ≈ w_x · (t_actual − D / v₀) t_actual = (e^(kD) − 1) / (k · v₀)The rule explains why drift grows much faster than distance. The lag term is tiny at short range and balloons at long range, because the arrow spends more and more of its flight slower than its launch speed. Fletching adds a weathervaning effect — the vanes turn the nose slightly into the relative wind — so the engine multiplies the lateral coupling by 1.6, producing about 60% more drift than the baseline rule.
| Distance | t actual | t vacuum | Lag | Baseline drift | Engine (× ≈1.6) |
|---|---|---|---|---|---|
| 18 m | 0.293 s | 0.290 s | 0.003 s | 0.9 cm | ≈ 1.5 cm |
| 30 m | 0.493 s | 0.484 s | 0.009 s | 2.6 cm | ≈ 4.2 cm |
| 50 m | 0.831 s | 0.806 s | 0.025 s | 7.4 cm | ≈ 11.9 cm |
| 70 m | 1.178 s | 1.129 s | 0.049 s | 14.6 cm | ≈ 23.4 cm |
| 90 m | 1.533 s | 1.452 s | 0.081 s | 24.4 cm | ≈ 39.0 cm |
Headwind and tailwind effects
The same relative-velocity treatment handles winds blowing along the range. A headwind increases |v − w|, so drag rises, the arrow slows faster, flight time lengthens and the shot lands low. A tailwind does the reverse and lifts impacts slightly. At 70 m a 3 m/s head component typically costs a few centimetres of elevation — less dramatic than crosswind drift, but enough to push a 9 into an 8 on a tight group.
Range takeaway: at 70 m, every 1 m/s of full-value crosswind moves the simulated arrow roughly 7.8 cm — a little more than one 6.1 cm ring on the 122 cm face. Read the flags, estimate the cross component, and hold off by that amount into the wind.