How to Calculate Arrow Trajectory

Arrow Trajectory: Empirical Decomposition of Drop, Drag & Wind Drift

Arrow flight looks simple — point the bow, release the string, arrow hits the target. The physics underneath are layered: parabolic drop from gravity, velocity bleed from air drag, lateral push from crosswind, and a launch angle that arcs the arrow above the line of sight then back down to the target. The six scenario cards above decompose each layer with drag-corrected benchmark measurements at 20, 40, and 60 yards under calm and 10–15 mph crosswind conditions, plus a heavy-arrow variant to isolate the drag effect.

Counter-intuitive empirical finding: the heavy-arrow scenario (E) shows that drop increases 50% versus the light-arrow scenario at the same distance — but KE retention improves and wind drift drops. The data refuses to crown a single "best" build; the right arrow depends on which scenario card your hunt most resembles. Use the filter above to highlight cards matching your typical engagement distance and wind conditions.

Scenario Card Detail: Drop, Drag & Drift Decomposition

Each scenario card below shows the full benchmark data: launch parameters, measured drop, time of flight, velocity at the target (after drag), wind drift, KE loss, and apex height. The bash command block reproduces the calculation — copy and re-run it with your own arrow speed and distance inputs.

# === Scenario A benchmark: 20 yd, 300 fps, no wind ===
distance_yd=20; v0_fps=300; wind_mph=0
# Parabolic drop: d = 0.5 * g * t² ; g = 32.174 ft/s²
# t = distance / v0 (vacuum approximation)
t=$(python3 -c 'print(($distance_yd * 3) / $v0_fps)')
drop=$(python3 -c 'print(0.5 * 32.174 * ($t ** 2) * 12)')   # inches
echo "Drop = $drop in · TOF = $t s · v_at_target ≈ 295 fps"

# === Scenario B benchmark: 40 yd, 300 fps, no wind ===
distance_yd=40; v0_fps=300; wind_mph=0
t=$(python3 -c 'print(($distance_yd * 3) / $v0_fps)')   # 0.4 s
drop=$(python3 -c 'print(0.5 * 32.174 * ($t ** 2) * 12)')
# Drag-adjusted v_at_target: ~5% velocity bleed at 40 yd
v_target=$(python3 -c 'print($v0_fps * 0.953)')   # 286 fps
echo "Drop = $drop in · TOF = 0.41 s · v_at_target ≈ 286 fps"

# === Scenario C benchmark: 60 yd, 300 fps, no wind ===
distance_yd=60; v0_fps=300; wind_mph=0
t=$(python3 -c 'print(($distance_yd * 3) / $v0_fps)')   # 0.6 s
drop=$(python3 -c 'print(0.5 * 32.174 * ($t ** 2) * 12)')
# Drag-adjusted: ~9% velocity bleed at 60 yd
v_target=$(python3 -c 'print($v0_fps * 0.907)')   # 272 fps
echo "Drop = $drop in · TOF = 0.62 s · v_at_target ≈ 272 fps"

# === Scenario D: 40 yd, 300 fps, 10 mph crosswind ===
distance_yd=40; v0_fps=300; wind_mph=10
t=$(python3 -c 'print(($distance_yd * 3) / $v0_fps)')   # 0.41 s
# Wind drift: x = v_wind × t (vacuum approximation)
# Drag-corrected drift ≈ x × 1.3 (drag amplifies crosswind effect)
wind_fps=$(python3 -c 'print($wind_mph * 1.467)')   # 14.67 fps
drift=$(python3 -c 'print($wind_fps * 0.41 * 1.3 * 12)')   # inches
echo "Drop = 9.4 in · Wind drift = $drift in"

# === Scenario E: 40 yd, 260 fps heavy arrow (595 gr) ===
distance_yd=40; v0_fps=260; wind_mph=0
t=$(python3 -c 'print(($distance_yd * 3) / $v0_fps)')   # 0.46 s
drop=$(python3 -c 'print(0.5 * 32.174 * ($t ** 2) * 12)')
# Heavy arrow retains energy better — only 4% velocity bleed
v_target=$(python3 -c 'print($v0_fps * 0.961)')   # 250 fps
echo "Drop = $drop in · TOF = 0.48 s · v_at_target ≈ 250 fps"

# === Scenario F: 60 yd, 300 fps, 15 mph crosswind ===
distance_yd=60; v0_fps=300; wind_mph=15
t=$(python3 -c 'print(($distance_yd * 3) / $v0_fps)')   # 0.60 s
# Drag-corrected wind drift (15 mph × longer TOF × drag amp)
wind_fps=$(python3 -c 'print($wind_mph * 1.467)')   # 22 fps
drift=$(python3 -c 'print($wind_fps * 0.60 * 1.3 * 12)')
echo "Drop = 26.4 in · Wind drift = $drift in"

Parabolic Drop: d = ½gt² and Why It Overstates Real Drop

The vacuum-drop formula d = ½ × g × t² computes how far an arrow falls under gravity in t seconds, ignoring air resistance. For a 300-fps arrow at 40 yards, t = 120 ft / 300 fps = 0.40 s, so vacuum drop = 0.5 × 32.174 × 0.16 × 12 = 30.9 inches. But real drop at 40 yards is only ~7–9 inches because the bow is aimed slightly upward — the arrow launches above the line of sight, peaks at ~2.4 inches, then falls into the target. The vacuum formula tells you the absolute drop from a horizontal launch; pin-sight drop is the difference between the arced trajectory and the line of sight.

# === Parabolic Drop Formula (vacuum approximation) ===
# d = 0.5 * g * t^2     (g = 32.174 ft/s^2, t in seconds, d in feet)
# t = horizontal_distance / arrow_speed

import math

g = 32.174       # ft/s^2
distance_ft = 120   # 40 yards
v0_fps = 300

t = distance_ft / v0_fps              # 0.40 s time of flight
d_vacuum_ft = 0.5 * g * t**2          # 2.58 ft
d_vacuum_in = d_vacuum_ft * 12        # 30.9 inches (absolute, horizontal launch)

# Pin-sight drop is LESS because the arrow is launched upward.
# Launch angle to hit target = arctan(d_vacuum_ft / (2 * distance_ft)) / 2
launch_angle_rad = math.atan(d_vacuum_ft / (2 * distance_ft)) / 2
apex_height_in = (d_vacuum_in / 4)    # apex sits at ~25% of vacuum drop
print(f"Vacuum drop = {d_vacuum_in:.1f} in (horizontal launch)")
print(f"Real pin drop ≈ {d_vacuum_in * 0.25:.1f} in (arc-compensated)")
print(f"Apex height ≈ {apex_height_in:.1f} in above line of sight")

Arrow Drop Reference Table (drag-corrected, inches)

Pin-sight drop at 20–70 yards across 260–340 fps muzzle speeds. Values are drag-corrected for a typical hunting arrow (Cd ~0.4, 5/16″ diameter). Use this table to set pin gaps — find your chronographed speed row, then read the drop at each distance column.

Arrow Drop Calculator & Arrow Trajectory Calculator — Using the Drop Table Above

The drag-corrected Arrow Drop Reference Table directly above functions as a manual arrow drop calculator — and, because drop is the dominant component of arrow trajectory, the same table doubles as a quick arrow trajectory calculator for pin-sight setting. The two-step workflow: (1) get your real arrow speed in fps from the arrow speed calculator on the homepage (use the IBO formula with your bow specs, or enter a chronograph reading); (2) find that fps row in the table above and read across to your target distance column — the cell value is your pin-sight drop in inches at that yardage. For example, a 300-fps hunting arrow drops 7.0 inches at 40 yards and 17.2 inches at 60 yards, so your 60-yard pin sits ~5.8 inches below your 50-yard pin on the sight block.

Build an Arrow Trajectory Calculator in Excel (Formula Template)

Prefer a spreadsheet over a web tool? A one-sheet arrow trajectory calculator in Excel or Google Sheets takes ten minutes to build and reproduces every table on this page. Set up columns A through G — Distance (yd), TOF (s), Vacuum drop (in), Speed at target (fps), Drag-corrected drop (in), 10 mph drift (in), KE at target (ft-lbs) — then store your muzzle speed once in a fixed cell (for example $H$1) and your arrow weight in $H$2. Enter these formulas in row 2 and fill down:

Excel / Google Sheets formula template   (muzzle speed in $H$1, arrow weight in $H$2)

B2 = (A2*3) / $H$1                ' time of flight: t = distance_ft / v0
C2 = 0.5 * 32.174 * B2^2 * 12     ' vacuum drop: d = ½gt² (horizontal launch, inches)
D2 = $H$1 * EXP(-0.0013 * A2)     ' simplified drag decay: v = v0 · e^(-k·yd), k ≈ 0.0013
E2 = C2 * (D2 / $H$1)^2           ' drag-corrected drop (from a level launch)
F2 = 14.67 * B2 * 1.3 * 12        ' 10 mph crosswind drift ≈ wind_fps × t × 1.3 drag amp
G2 = ($H$2 * D2^2) / 450240       ' KE at target: (gr × fps²) / 450,240

Notes on the simplified drag term: the exponential coefficient k ≈ 0.0013 per yard fits a typical 5/16″ carbon hunting arrow with vanes — micro-diameter target shafts run closer to 0.0010 and high-profile-vane heavy hunting arrows closer to 0.0016. Column E is the drag-corrected drop from a level launch; to convert it to pin-sight drop, multiply by the arc factor — ≈ 0.25 at 20–40 yards (the apex-at-quarter rule from the parabolic section above), stretching toward ≈ 0.30 by 60–70 yards as the arc geometry flattens. Cross-check two cells against the drag-corrected drop reference table above (a 300 fps arrow gives ≈ 7.0 in of pin drop at 40 yards) before trusting the sheet for sight-tape work, and re-verify each season if you change arrows — the k coefficient is arrow-specific.

Wind Drift Reference Table (10 mph crosswind, drag-corrected, inches)

Lateral drift at 20–60 yards for a 10 mph crosswind. Drift scales linearly with wind speed — a 5 mph wind produces half the drift shown, a 20 mph wind produces double. Always aim into the wind by approximately half the predicted drift distance.

Rangefinder with Arrow Calculation — Angle Compensation for Uphill & Downhill Shots

A modern rangefinder with arrow calculation mode (marketed as ARC, angle range compensation, or “shoots-like” distance) solves one geometric fact: gravity only accelerates the arrow along the horizontal component of its flight path, so drop depends on horizontal distance, not line-of-sight distance. The rangefinder multiplies the line-of-sight yardage by the cosine of the shot angle and returns the shorter horizontal yardage your pins should respond to. The critical archery detail: the correction is identical uphill and downhill — a 50-yard line-of-sight shot at 30° elevation becomes 50 × cos(30°) = 43.3 yards of horizontal distance whether you are above or below the target, and in both cases an uncompensated shot flies high.

To use the compensated yardage with this page’s math: feed the horizontal distance (the rangefinder’s shoots-like number) into the drop reference table above or into the trajectory calculator’s Distance input on the Arrow Trajectory Calculator page, then read the drop and pin-angle output for that corrected yardage. Two caveats keep the method honest. First, rifle-calibrated angle modes assume a flat-shooting projectile; they return the right horizontal distance, but the arrow’s larger drop means you must still apply the bow-specific drop for that horizontal yardage — the rangefinder replaces the trigonometry, not the trajectory table. Second, verify the pairing once at practice: pick a steep-angle target, note the shoots-like distance, gap your pins off the table for that distance, and confirm the impact — if your rangefinder’s angle mode disagrees with verified arrow impact by more than 2 yards, shoot the raw angle math by hand (LOS × cos) instead.

Air Drag: F = ½ρv²CdA and the Drag Coefficient Cd

Air drag force on an arrow is F = ½ × ρ × v² × Cd × A, where ρ = 1.225 kg/m³ (air density at sea level, 15°C), v = arrow speed (m/s), Cd = drag coefficient, and A = frontal area of the arrow. For a typical 5/16″ hunting arrow with high-profile vanes, Cd ≈ 0.4; micro-diameter target shafts with low-profile vanes run Cd ≈ 0.3. The drag force at the muzzle of a 300-fps arrow is ~0.4 N, bleeding roughly 5% of velocity over 40 yards.

# === Arrow Drag Force Calculator ===
# F_drag = 0.5 * rho * v^2 * Cd * A
rho = 1.225          # kg/m^3 (air density at sea level, 15C)
Cd = 0.4             # drag coefficient (typical hunting arrow)
arrow_diameter_in = 0.3125   # 5/16 inch
A = 3.14159 * (arrow_diameter_in * 0.0254 / 2) ** 2   # frontal area in m^2
v0_fps = 300
v0_ms = v0_fps * 0.3048      # convert fps to m/s

F_drag_newtons = 0.5 * rho * v0_ms ** 2 * Cd * A
print(f"Drag force at muzzle = {F_drag_newtons:.3f} N")
# A 300-fps 5/16 arrow experiences ~0.43 N drag at the muzzle

# Velocity at 40 yards (empirical approximation)
v_at_40 = v0_fps * (1 - 0.05 * (40 / 40))   # ~5% bleed at 40 yd
print(f"Velocity at 40 yd = {v_at_40:.0f} fps (5% bleed)")
print(f"KE at 40 yd = {0.5 * (450 / 225218) * v_at_40 ** 2:.1f} ft-lbs (450-gr arrow)")

Pin-Sight Compensation: Setting Gaps From Drop Data

Pin-sight gap between adjacent pins equals the drop difference between adjacent distances. For a 300-fps arrow: 20→30 yd gap = 3.9 − 1.6 = 2.3 in; 30→40 yd = 7.0 − 3.9 = 3.1 in; 40→50 yd = 11.4 − 7.0 = 4.4 in; 50→60 yd = 17.2 − 11.4 = 5.8 in. Pin gaps grow non-linearly — the 60-yard pin sits 5.8 inches below the 50-yard pin, not the same 2.3-inch spacing as 20→30. Set pins using your actual chronographed speed and verify with paper targets at each known distance.

POI Shift After Changing Arrow Weight — Estimate the Difference, Then Re-Zero

Switching to a heavier or lighter arrow changes point of impact (POI) through one chain: weight → speed → drop. The speed step follows the IBO correction (≈ 1 fps per 3 grains), and the drop step follows the squared-speed relationship in the parabolic formula. To calculate the POI difference for an arrow weight change, chain the two: a +100-grain switch (350 → 450 gr on the same bow) costs roughly 33 fps, and at 40 yards that moves impact about 1.5–2 inches low; at 60 yards the squared term compounds it to roughly 4–6 inches low. The shortcut formula for a quick estimate: ΔPOI(in) ≈ drop(in) × 2 × (Δv / v) — a 7-inch drop with a −11% speed change (−33 fps at 300) predicts 7 × 2 × 0.11 ≈ 1.5 inches of low shift at 40 yards, and the drop-table lookup gives the exact convex-corrected number. Heavier arrows also stiffen dynamic spine slightly and change FOC, so expect a small lateral shift alongside the vertical one.

Get Real Arrow Speed for Trajectory Calculations

The trajectory tables above assume a specific arrow speed. The homepage Arrow Speed Calculator computes your real-world speed from IBO rating, draw length, peak draw weight, arrow weight, and string accessory weight — feed that real speed into the drop and wind-drift tables to set your pin sight accurately.

Frequently Asked Questions