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Bow and arrow
Aim, range and wind
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The science
Stored energy. A bow is a spring. The pull force rises with draw length d roughly as F(d) = k d (1 + β (d/D)²), F(D) = W where W is the draw weight at full draw D and β is a stacking term (0 = ideal linear spring, a straight English longbow is close to 0.1; recurves stack more). Integrating the force over the draw gives the energy you put in:
E = ∫F dd = k D² (1/2 + β/4)For a linear bow this is the familiar W D / 2: a 100 lb (445 N) bow drawn 28 in (0.71 m) stores about 150 J.
Arrow speed. Only a fraction η of the stored energy reaches the arrow, the rest is limb and string motion, hysteresis and sound. The limbs and string also have to be accelerated, which we lump into a virtual mass mL that moves with the arrow:
η E = ½ (marrow + mL) v²Heavier arrows leave slower but carry more momentum and kinetic energy; this is why a war arrow of 70 to 100 g is slower than a light club arrow of 30 g.
Flight. The arrow is a point mass with gravity and quadratic drag relative to the moving air, integrated with a fourth-order Runge-Kutta scheme (step 2 ms):
dv/dt = g − (½ ρ CdA / m) |v − w| (v − w)Here w is the wind vector, ρ the air density and CdA the drag area of shaft, fletching and head together. The wind pushes the arrow sideways for as long as it is in the air, so the drift grows faster than linearly with distance and is largest for light arrows. Release height is 1.55 m, the target gold is 1.30 m high and the standard 122 cm face has ten zones 6.1 cm wide.
Checks run in the page. (1) The numerical integral of F(d) is compared with the closed-form energy. (2) With drag and wind switched off, the integrated ground range is compared with the closed form R = v cosθ (v sinθ + √(v² sin²θ + 2 g h)) / g. (3) Halving the time step must not move the arrow at the target. The elevation solver and the wind-drift solver are bisection searches over the same integrator.
Honest simplifications. The arrow is a point mass: no archer's paradox flex, no spin, no angle of attack or fishtailing, no lift. A fixed efficiency stands in for the real bow dynamics, which depend on limb shape and string mass; efficiencies of 0.6 to 0.8 are typical for a self bow, so treat the speeds as plausible rather than measured. The wind is uniform and steady, with no gusts or height gradient. The drag area is a representative figure for a field-pointed shaft with three feathers, not a measurement of any particular arrow. Historical draw weights (the Mary Rose bows are usually quoted at roughly 110 to 180 lb) are context only; the simulator does not model armour or any target other than a straw butt.