Every archer knows the effort of the draw: on each arrow you bring the string back to the anchor against a few dozen pounds, hold it on aim and let it go. It is natural to wonder how much of that effort ends up in the arrow and how much is lost on the way. In physics the question has a precise form, because the work done to draw the bow and the energy the arrow leaves with are both measured in joules, and the ratio between the two is called the efficiency of the bow.
The account here covers the work that goes into the bow, not what the muscles spend. The short answer is that on paper a target bow passes about three quarters of the drawing work to the arrow. That figure, though, comes mostly from mathematical models, and the measurements concern bows other than an Olympic recurve in an archer's hands: a kyudo bow, a modern recurve mounted in a rack, a compound. On where the rest goes there is a measured budget, on a compound: the quarter that does not reach the arrow stays almost entirely in the parts of the bow that move with the string. Machine tests show the share rising with arrow weight; on the string the tests disagree, and on limb shape there are only models.
From drawing work to arrow energy
Drawing a bow is work, and its value is the area under the curve that links force to draw length: more pounds and more centimetres of draw mean more joules. The limbs store it and give it back at release, but they do not hand all of it to the arrow. A small part is lost even before that, to internal friction in the limbs and, on a compound, in the cams. Another part stays in the bow, because when the arrow leaves the string the limbs and string are still moving and keep their own energy.
An item-by-item budget exists for one bow only. A physicist measured a production compound in the laboratory with a 20.6 g arrow, rebuilding the draw curve photograph by photograph and timing the arrow with a chronograph. Drawing the bow took 73.7 joules; at release the bow returned 71.0, because 3 to 4% was lost to friction; the arrow carried off 57.1. Against the drawing work that is a little over three quarters, against the energy returned it is 80%, and the second figure is the one the author calls efficiency. The difference between the two definitions matters whenever different studies are set side by side, because they do not all divide by the same thing.
Where the three-quarters figure comes from
The value quoted for the target recurve began life as a calculation parameter. The reference model, checked against high-speed footage of a leading Polish archer, sets efficiency at 0.75 and builds the rest of the calculation on it. Another model of the modern recurve derives it instead of assuming it and finds 0.729. Two models from different groups thus reach the same order of magnitude, but they remain calculations, which is why in the registry the claim that a target bow transfers about three quarters of the drawing work to the arrow is transferred.
Models, for their part, have a history of optimism. Those that treated the string as a thread that does not stretch predicted efficiencies above 90%, while bows measured around 1960 gave between 70 and 85%. An analysis with an elastic string showed where the gap came from: on the longbow of the early calculations efficiency fell from 92 to 78%, because the arrow leaves while the limbs and string are still moving, whereas air resistance accounted for less than 2%. The registry rates as contested the claim that the mathematical model of the bow predicts draw weight well and efficiency less well: the half about efficiency holds, because models put it higher than the measurement, while the half about draw weight applies to the longbow, and on recurve and compound the model has to be calibrated against the real bow.
Measurements on real bows are few. In kyudo, with ten archers and fifty shots per string, the Japanese bow passes between 67.7% and 69.4% of the energy of the drawn bow to the arrow. In the laboratory, a modern recurve mounted in a rack and loosed with a mechanical release ranges from 0.70 with a Dacron string to 0.762 with a Fast Flight one. The compound in the budget reaches 0.80 on the energy returned. These figures are close to three quarters, taken on different bows and with different definitions: they confirm the order of magnitude, not the value for a target bow.
Where the rest of the energy goes
The kick of the bow in the hand suggests that a lot of energy is scattered at release, but in the compound budget the recoil, measured with an accelerometer, came to 0.23 joules out of the 71.0 available; pulling the nock off the string cost about 0.05 joules; the arrow's oscillation, even with an amplitude of 2 centimetres, about 0.43; the shortening of the string between 0.2 and 0.3. Between the energy returned and the arrow's energy almost 14 joules remain, and the author attributes them to the parts of the bow that are still moving once the arrow has gone. He does not measure them on their own: he obtains them by difference, after showing that the other items are tiny.
To describe that share physicists use a virtual mass: it is as if the arrow, at launch, dragged along a few grams that do not leave with it. In the compound measured it came to about 5 grams, and half of that was in the string and what is attached to it, that is serving, D-loop and peep sight; the rest, according to the author, probably in the limbs and cams, which are heavier but move more slowly. The formula that follows, with efficiency equal to the arrow's mass divided by the sum of the two masses, holds up against the chronograph on three arrows of different weight. A model with an elastic string reaches the same picture by another route: when the arrow leaves, 11.0% of the energy is still in the motion of the limbs and 9.8% in that of the string. The registry rates as reported the claim that what the bow does not give the arrow stays in its moving parts, while recoil and de-nocking weigh almost nothing: the complete budget is a single one, on one bow, in a preprint.
It is tempting to think that lighter limbs shoot faster, but in a compound model the final speed does not change with the effective mass of the limbs or the inertia of the cams, only the way the arrow accelerates, and that model fixes the arrow's energy with a virtual mass taken from measurements. The claim that the mass of limbs and cams changes arrow speed little is transferred, because the question has only been put to models.
A heavier arrow takes a larger share
It follows from the virtual mass that the same bow is more efficient with a heavier arrow, because the grams left with the bow weigh less in proportion. The equipment tests all point this way. The most controlled one mounts a laminated recurve on a machine and shoots it at 40, 45 and 50 lb with three arrows identical except for an internal weight, from 23 to 33 grams, thirty shots per combination: at 50 lb the heavier arrow, timed 20 metres from the bow, is about 3 metres per second slower, but carries 32.4 foot-pounds of energy against 24.4. A second machine experiment repeats the comparison on two 34 lb bows, and with both the 30.35 g arrow carries off more energy than the 25.66 g one: 24.43 joules against 22.82 with the fibreglass-reinforced bow, 17.96 against 16.91 with the wooden one.
At the other extreme, a replica yew bow with a 9.7 g cane arrow transfers, on an estimate from the draw curve, only about a third of the stored energy, and the arrow leaves at between 33.5 and 39.2 metres per second; the registry rates as reported the claim that a wooden bow of moderate power with a very light cane arrow does not exceed forty metres per second and is useful only at close range. With the weight differences found between competition arrows, on the other hand, the effect does not show: with the same bow and the same archer, two shafts a quarter of a gram apart leave at 59.21 and 59.67 metres per second with practically the same energy. The claim that with the same bow a heavier arrow receives more energy is reported: the tests agree, but on an Olympic recurve with competition arrows the only comparison is the one between the two near-identical shafts, and nobody has followed the effect across the weight ranges used in competition.
String, limb shape and materials
If half the virtual mass is in the string, a lighter string ought to send the arrow off faster. On the compound measured this is a deduction from the formula, and the direct tests do not all point the same way. On material they agree: in kyudo the three synthetic strings pass about 69% of the energy to the arrow against 67.7% for hemp, a difference smaller than the variation from one shot to the next, and on the recurve loosed from the rack the same arrow leaves at 49.73 metres per second with Dacron and 52.09 with Fast Flight. On the number of strands they do not: in the same laboratory, removing up to six strands from a sixteen-strand string leaves the speed between 51.43 and 51.88 metres per second, with no trend, while on a shooting machine going from twenty strands to twelve gains half a percentage point at most. That is why the claim that the material and number of strands of the string change arrow speed is contested.
According to the models, the shape of the limbs matters in a less intuitive way. A model comparing six types of bow at equal draw weight, draw length, limb mass, arrow and string finds that storing more energy is not enough. Bows with rigid ears store more than a straight bow, but when the string leaves the bridges the push on the arrow drops suddenly and an oscillation starts that uses up energy, so a smaller share reaches the arrow; a recurve pushed to the extreme raises the stored energy from 0.434 to 0.810 and brings efficiency down from 0.770 to 0.417. In the end arrow speed changes little from one bow to another, and the modern recurve, which combines good storage with good efficiency, is the one that launches it fastest. That limb geometry decides what share of the stored energy passes to the arrow is transferred: it comes from the models of a single author, which compare historical types and not the limbs on sale today.
The advantage of composite bows, according to the same models, comes from the materials more than from the shape: per kilogram, sinew stores 2,500 newton metres and horn 1,500, against 1,100 for yew and 700 for maple, so a composite reaches the same draw weight with shorter, lighter limbs. There is a single direct measurement: at a ballistics station a 27 kg composite launches a 50 g arrow at the same speed as a 36 kg yew longbow. In a more recent machine test an elm bow is about 14% slower than a bow of similar shape reinforced with fibreglass, at the same draw weight and draw length, but that bow is not a horn and sinew composite. The claim that a horn and sinew composite bow launches the arrow better than a wooden bow of equal draw weight is reported.
In kyudo even the glue seems to count. Bows from a Kyoto workshop glued with traditional deer-hide glue, filmed at high speed, launch the arrow at almost the same top speed as those glued with synthetic glue, but with more push in the final stretch: the arrow accelerates sharply from a hundredth of a second after release. The study does not say how many archers or how many shots, and the claim that a Japanese bow glued with animal glue pushes the arrow harder in the final stretch, at the same top speed is reported.
Compound and recurve
For compound archers the question takes another turn, because the force rises steeply, stays almost constant for much of the draw and then drops sharply towards full draw. On the compound in the budget the let-off at full draw was about 65%, so about a third of the peak force was held on aim, and the bow stored 73.7 joules; two compounds in other studies stored 92.8 and 96.5. That a compound stores between 70 and 100 joules and lets off about two thirds of the force at full draw is reported, because each figure comes from a single bow and the let-off from only one. In a study on wind, a 37.1 lb recurve measured alongside a 53.5 lb compound stored 45.1 joules against 92.8, and its arrow left at 57.2 metres per second against 78.9. With draw weights and arrows this different the gap can be seen but not quantified, and the claim that compound arrows leave much faster than recurve arrows is reported.
At equal peak force, the only mechanical comparison comes from a model that sets a compound beside two recurves that exist only in the calculation: at equal maximum force the compound stores 75% more energy, and at equal energy it is held on aim with a force 2.3 times lower. The paper does not explain how the curves of the virtual recurves were built, and the claim that at equal peak force the compound stores much more energy than the recurve and is held on aim with much less force is transferred. For the opening question this means that, on a compound, the force held on aim and the work put into the bow are two separate things, and in the model the first goes down while the second goes up.
Draw weight and speed
The other lever on drawing work is draw weight. On the laminated recurve mounted on the machine the same 23 g arrow goes from 45.5 metres per second at 40 lb to about 54.3 at 50. Among archers there is a single study: across the 61 archers at a World Cup stage, declared draw weight correlates with speed at 0.854, but recurve and compound are analysed together, and much of the relationship is the difference between the two divisions. The claim that among archers, those who shoot more pounds launch the arrow faster is reported. How many pounds it makes sense to shoot is a matter of control more than of joules, and it is the subject of choosing your draw weight.
How far the measurements go
The measurements say with some confidence where the energy that does not reach the arrow goes and which way arrow weight and materials shift it, and less about the efficiency of your own bow. The three quarters for the target recurve is a model value, confirmed in order of magnitude by a kyudo bow, a recurve in a rack and a compound; the item-by-item budget comes from a single compound; limb shape is compared only by calculations, on historical bows. On an Olympic recurve shot by an archer, with that archer's arrows, there is no direct measurement of efficiency, and nobody has compared limbs on the market with everything else held equal.
What I take from it on the shooting line
These figures do not yield a setting to copy, but they do give three pointers for reading your own equipment.
The first concerns quoted speeds. Since the share that reaches the arrow depends on the arrow itself, a speed or an efficiency holds only for the arrow, draw length and draw weight it was measured with: judging by the machine tests, the same bow sends a lighter shaft off faster and gives it less energy. To compare two sets of limbs or two bows it makes sense to use the same arrow, the same draw length and the same draw weight, with a chronograph, bearing in mind that according to the models final speed varies much less between bows of different shape than stored energy does.
The second concerns the string and what is attached to it. On the compound measured half the virtual mass was there, but the tests on strands disagree, and in kyudo the difference between strings was smaller than the difference between one shot and the next. Before changing string to gain speed, I would check on a chronograph whether the difference with my arrow is larger than the variation in my own shots.
The third concerns dry firing. Energy that finds no arrow stays in the bow, and the first model of the modern recurve already used this to explain why dry firing often breaks the bow or the string: it is the limiting case of the budget, the one in which nothing goes to the arrow. For the rest, the choice between recurve and compound and the choice of arrows rest on other grounds, because none of these studies has linked the efficiency of the bow to how many points are scored.