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root(bow / arrow) * (material +- sd) = velocity

bow = the draw weight of the bow in kg (or a measuring unit the same as the arrow).

arrow = the mass of the arrow in kg (or a measuring unit the same as the bow).

material = the material multiplier, shown below for Compound, Yew, Composite Buffalo Horn, Maple, Steel.

sd = the standard deviation of the material multiplier.

velocity = the starting velocity of the arrow, in meters per second. For feet per second, add “/ 0.3048” to the end of the first half of the equation, e.g. “root(bow / arrow) * (material +- sd) / 0.3048 = velocity in fps”.

Compound Bow Equation:

root(bow / arrow) * (2.50 +- sd 0.05) = velocity

Yew Bow Equation:

root(bow / arrow) * (2.00 +- sd 0.05) = velocity

Composite Buffalo Horn Crossbow Equation:

root(bow / arrow) * (1.264 +- 0.037) = velocity

Maple Crossbow Equation:

root(bow / arrow) * (0.965 +- 0.014) = velocity

Steel Crossbow Equation:

root(bow / arrow) * (0.708 +- 0.024) = velocity

Empirical measurements of bow starting velocities, for various bows and arrows, and the equation’s given results, in comparison:

COMPOUND BOWS: root(bow / arrow) * (2.50 +- 0.05) = velocity

25 lb Carbon Fiber Bow = 11.3398 kg draw weight. 650 grain arrow = 0.04209986 kg arrow. 42 m/s velocity at the start.

root(11.3398) / 0.04209986 * (0.510 + 0.015) = 41.99 m/s.

45 lb Carbon Fiber Bow = 20.4117 kg draw weight. 650 grain arrow = 0.04209986 kg arrow. 54 m/s velocity at the start.

root(20.4117) / 0.04209986 * (0.510 – 0.007) = 53.98 m/s.

60 lb Carbon Fiber Bow = 27.2155 kg draw weight. 650 grain arrow = 0.04209986 kg arrow. 62 m/s velocity at the start.

root(27.2155) / 0.04209986 * (0.510 – 0.010) = 61.96 m/s.

60 lb Carbon Fiber Bow = 27.2155 kg draw weight. 725 grain arrow = 0.046959782 kg arrow. 55 m/s velocity at the start.

root(27.2155) / 0.046959782 * (0.510 – 0.015) = 54.99 m/s.

YEW BOWS: root(bow / arrow) * 2.00 +- 0.05 = velocity

48 lb Yew Bow = 21.77244 kg draw weight. 500 grain arrow = 0.0323995 kg arrow. 52.1208 m/s velocity at the start.

root(21.77244) / root(0.0323995) * (2.005 + 0.006) = 52.13 m/s

110 lb Yew Bow = 49.89512 kg draw weight. 972 grain arrow = 0.063 kg arrow. 56.63184 m/s velocity at the start.

root(49.89512) / root(0.0323995) * (2.005 + 0.007) = 56.62 m/s

145 lb Yew Bow = 65.77084 kg draw weight. 972 grain arrow = 0.063 kg arrow. 64.43472 m/s velocity at start.

root(65.77084) / root(0.063) * (2.005 – 0.011) = 64.43 m/s

110 lb Yew Bow = 49.89512 kg draw weight. 1157 grain arrow = 0.075 kg arrow. 53.40096 m/s velocity at the start.

root(49.89512) / root(0.075) * (2.005 + 0.065) = 53.39 m/s

145 lb Yew Bow = 65.77084 kg draw weight. 1157 grain arrow = 0.075 kg arrow. 59.31408 m/s velocity at start.

root(65.77084) / root(0.075) * (2.005 – 0.002) = 59.32 m/s

160 lb Yew Bow = 72.57472 kg draw weight. 1157 grain arrow = 0.075 kg arrow. 60.8076 m/s velocity at start.

root(72.57472) / root(0.075) * (2.005 – 0.050) = 60.81 m/s

160 lb Yew Bow = 72.57472 kg draw weight. 1312 grain arrow = 0.085 kg arrow. 56.7 m/s velocity at start.

root(72.57472) / root(0.085) * (2.005 – 0.065) = 56.69 m/s

Composite Buffalo Horn Crossbow: root(bow) / root(arrow) * 1.264 +- 0.037 = velocity

1270 lb Buffalo Horn Crossbow = 576.06184 kg draw weight. 4012 grain bolt = 0.26 kg bolt. 57.74 m/s velocity at start.

root(576.06184) / root(0.26) * (1.264 – 0.037) = 57.76 m/s

1270 lb Buffalo Horn Crossbow = 576.06184 kg draw weight. 5370 grain bolt = 0.348 kg bolt. 52.92 m/s velocity at start.

root(576.06184) / root(0.348) * (1.264 + 0.037) = 52.93 m/s

Maple Crossbow Equation: root(bow) / root(arrow) * 0.965 +- 0.014 = velocity

276 lb Maple Crossbow = 125.191392 kg draw weight, at x draw length. 673 grain bolt = 0,04363 kg bolt. 50.96 m/s velocity at the start.

root(125.191392) / root(0.04363) * (0.965 – 0.014) = 50.94 m/s

276 lb Maple Crossbow = 125.191392 kg draw weight, at x draw length. 845 grain bolt = 0,05476 kg bolt. 46.84 m/s velocity at the start.

root(125.191392) / root(0.04363) * (0.965 + 0.014) = 46.81 m/s

Steel Bow Equation: root(bow) / root(arrow) * 0.708 +- 0.024 = velocity

960 lb Steel Crossbow = 435.44832 kg draw weight. 1034 grain bolt = 0.067 kg bolt. 55.1688 m/s velocity at start.

root(435.44832) / root(0.067) * (0.708 – 0.024) = 55.14 m/s

960 lb Steel Crossbow = 435.44832 kg draw weight. 1389 grain bolt = 0.09 kg bolt. 50.9016 m/s velocity at start.

root(435.44832) / root(0.09) * (0.708 + 0.024) = 50.92 m/s

976 lb Steel Crossbow = 442.705792 kg draw weight. 1482 grain bolt = 0.096 kg bolt. 47.8536 m/s velocity at start.

root(442.705792) / root(0.096) * (0.708 – 0.003) = 47.88 m/s

THOUGHTS:

The multiplier in the equation seems to change by 3 different things: (1) a difficult pull and shorter draw length release from very heavy draw weight bows, reduces multiplier (2) the projectile being underweight compared to the bow draw weight, reduces multiplier, (3) the projectile being overweight compared to the bow draw weight, reduces multiplier.

The equation gets more accurate, the more accurate the measurements are. The measured Yew Bows are 30″ draw length, the Carbon Fiber Bows are 28″ draw length. The rest are of unknown draw length, the crossbows being shorter. The multiplier should change, depending upon the draw length upon release. Perhaps the original draw weights, as expressed in full pounds, should be expressed more accurately. More test data of measurements are needed for more conclusive results.

Test data should preferably include: many bows of different draw weights yet same material, and many arrows of different masses yet same draw length, all bows shooting all arrows. There should also be a bow rest period between shootings, because it seems the yew bows shoot less fast if they’re shot repeatedly in a short amount of time.

One of the most interesting empirical tests would be to shoot various draw weight bows of the same material(s), with the lightest possible arrows or bolts, so as to reach a measurement of “the maximum bounce velocity of the bow material at given draw weight i.e. of a given volume”. This measurement would allow a mathematician to calculate the optimal projectile mass for any given bow, for maximizing the projectile energy and/or momentum, by setting the “maximum bounce velocity” as equation velocity, and putting x to the projectile mass. Another test that could yield possible useful results contains an opposite: to shoot the heaviest projectiles possible, such as metal shafted bolts. These tests would be useful, because the data seems to support the fact that, at least very heavy draw weight crossbows, should and did historically shoot very heavy projectiles, as the bow starting velocity wouldn’t increase past the “maximal bounce velocity parameter(s)”, but also wouldn’t slow down linearly with higher weight projectiles. The test projectiles should optimally go above and under the “bow’s speed limit optimal”, which requires many projectiles of enough variation in mass, the heavier projectiles above the bow’s “optimal”, the lighter projectiles under the bow’s optimal. Maximally light projectiles of the same draw length should be most useful for finding the bow material’s “bounce velocity maximal”, shot with different draw weight bows of the same material, and combined with the “bow’s optimal” i.e. underweight and overweight projectiles.

Your own bow(s) can be measured and put into the equation with their own specific multiplier(s), by replacing the multiplier with x and solving the equation with measured parameters (draw weight and arrow mass), presuming the bow material follows the same simple equation pattern of “root(bow) / root(arrow) * (materialMultiplier +- theThreeParametersMentionedAbove) = velocity”, or same equation form but arrow mass without root, for carbon fiber bows.

With more data, new mathematical patterns may emerge from the correspondences between the equations’ results, as aiming to the empirical measurement results, by searching akin to game-tree navigation, by combining the available parameters with the different available mathematical operations in turn, to form the equation results, the same operations upon the same parameters. This is how I found this equation, and how I am going to find better accuracy, with more test results.

The above equations can, of course, be converted to different measurement unit systems, such as the imperial system.