Archer's Paradox,

by Ron Carmichael, USA Archery Level IV-NTS Coach

Note: there is no archer's paradox in association with COMPOUND bows/mechanical releases.. Archer's paradox exists only in those bows where the archer uses fingers to hold and release the bowstring.


    Archer (är-ch&r ) noun - one who uses a bow and arrow

    Par·a·dox (pr-dks)  noun

1.A seemingly contradictory statement that may nonetheless be true: the paradox that standing is more tiring than walking.
2.One exhibiting inexplicable or contradictory aspects: “The silence of midnight, to speak truly, though apparently a paradox, rung in my ears” (Mary Shelley).
3.An assertion that is essentially self-contradictory, though based on a valid deduction from acceptable premises.
4.A statement contrary to received opinion.

and in this case, my intent to explain:
5. An arrow in a bow does not point directly to the target, yet when loosed by the archer's fingers, consistently strikes that target.

[Latin paradoxum, from Greek paradoxon, from neuter singular of paradoxos, conflicting with expectation : para-, beyond; see para-1 + doxa, opinion (from dokein, to think.]

Also, I will define "loose"/"loosing" (luce/luce-ing): the act of an archer relaxing the fingers on the string of a drawn bow sufficiently that the bowstring and arrow leaves the fingers and is cast forwards, accelerated, towards a target by the bow. It is pronounced with an sss, not a z, sound. It is a more accurate term than "letting go" of the arrow, for several reasons. First, the archer simply cannot "let go" of the string my moving the fingers out of the way of the bowstring. The archer is not holding the arrow and in fact if even slightly touching the arrow with a finger, is less accurate. The act of loosing the arrow is highly practiced and developed by the dedicated archer to be a consistent and reproducible mechanism. "LOOSING" far better describes that mechanism, and is why it has been used for many hundreds of years, if not a millenium.

Layperson "logic" dictates that a straight and balanced arrow must be shot straight at a target in order to hit it.   In reality, the arrow must be aimed OFF of the target by a traditional archer in order to hit the target.  You do not point the arrow at the target, this is THE GREATER archer's paradox. 

Flaws of logical assumption

1. The arrow is obviously stiff and rigid, just hold it in your hands!  It doesn't bend much, if at all. After all, we've always heard, "straight as an arrow".
2. The string travels straight forward when released by the fingers because it pushes the fingers out of the way!  The string then must push the arrow STRAIGHT to the target!


The term "Archer's Paradox" was coined by Dr. Robert P. Elmer, a well-known archery author in the 1930s.  His hypotheses were confirmed YEARS LATER by others who used high speed photography to scrutinize the behavior of an arrow upon release by the archer to answer a paradoxical situation.  Initially he was not concerned (nor even aware) that the arrows flex like a snake upon loosing.

His concern was the question as to why an arrow will succeed in hitting a target even though the arrow is placed at an angle, aiming off to the side of the target, prior to shooting.  It is actually pointing away, off of the target to the side away from the bow. 

From appearances when properly placed on the bow of a right-handed archer, the arrow should strike well to the left of the target as it travels in a direct path to the target.  It does not.  This is a paradoxical situation!  (And is also present for left-handed archers, only to the opposite/right side of the target.)

What his reasoning dictated, and much later, subsequent high-speed, slow-motion photography revealed, was that the arrow flexes and the string does not travel straight forward upon loosing!  Even though to the naked eye an arrow seems a rigid and unbending shaft, in reality it bends and flexes when placed under the pressures of an accelerating bowstring and the lateral displacement caused by the bowstring sliding sideways from the fingers of the archer.   Even the string slides sideways from the fingers on it's way to the resting position after the arrow has flown.

Newton's Laws of Motion are fully in play here: For every action there is an equal and opposite reaction. The action is the bowstring accelerating in two dimensions - sliding sideways off of the fingertips and then forward towards the target.  The reaction is the arrow flexing in direct proportion to the force due to inertial resistance- the head of the arrow, aka the point or tip, is a heavier mass than any other element of the arrow.  

The arrow will bend, flex, in part due to the inertial resistance of the arrow point, typically in the general mass of 100 grains, which is at rest, in combination with the back end of the arrow (the much lighter nock, massing less than 6 grains) moving, accelerating, in two dimensions.

The arrow shaft bends in a balanced way to one side, and then bends the opposite way, and back, again and again, decreasing its oscillations slightly each cycle as it flies until it strikes the target.  The shaft bends due to the inertial resistance of the shaft and the (heavier) tip as well as the sidethrust of the string coming off of the fingers, and the manner of the bend is consistently the same because the bowstring slips off of the fingertips in the same sideways direction each time. (Assuming the archer is consistent in her/his release technique!)  The tail of the arrow gets pushed away from the bow at just the same time that the string is suddenly free to push the shaft towards the front of the bow.  The tip of the arrow gets no sideways push, but since it has mass it resists the forward motion for an instant while the nock is moving forward, and the shaft BENDS as a result.   The front portion of the arrow is also unable to move to the side due to the presence of the bow itself. (or attachments to the bow, such as the plunger). The plunger is a small spring-loaded piston which absorbs some of the lateral impetus of the arrow's intial sideways impetus, thereby decreasing the magnitude of the oscillations of the shaft.

The spine (stiffness) of the shaft together with the weight of the tip and the strength of the bow limbs, the tension of the plunger, and the fingers motion  all serve to determine how much bend will occur.  Practical experience and testing reveals that for maximum accuracy and for bow clearance the spine for a given bow must be neither too stiff NOR too weak. Otherwise the arrow's tail end where the fletchings/vanes are strikes the bow and is deflected.  The way the archer releases the string also controls how much bending occurs and that is one reason why it is so important that the archer be consistent  in how the arrow is "loosed" or released. 

Back to defining Archers' Paradox: 

Essentially, prior to release, the arrow must be pointed OFF of the target by a distance that equals proportionately the sideways deflection caused by the string sliding off of the fingers of the archer.  This Offset Distance, aka "Center-Shot" is subject to the SYSTEM.  let's define the SYSTEM as a combination of all the varying parts of the setup: the technique of the archer, the material of the finger tab, the spine of the arrow, the length of the arrow, the mass of the tip of the arrow, how far inside the arrow the tip extends, the nature of the bowstring (strands, composition, serving, length, twists), the mass and stiffness of the nock and its tension/grip on the string, the plunger button's spring resistance, the arrow rest and its flexibility, the settings of brace height, the tiller of the bow, the stability of the limb's tips, rigidity of the sight extension bar, the use of vibration dampeners including the stabilizers, and the degree of center-shot.    Change any one part of the SYSTEM and you alter the behavior of the rest of the parts.  They are all inter-dependent, in other words.

The arrow flexion, while unavoidable, is beneficial because it actually assists in accuracy.   Due to flexion the shaft's trailing end will not strike the arrow rest and the bow. Many mistakenly refer to this missing as the paradox. If it is anything of the sort, it is certainly the lesser of any paradoxes associated with archery.  Were it to hit or graze the bow's other components as it leaves the bow then the arrow's flight would be altered causing decreased accuracy (and torn or creased fletchings). This often happens when the arrow spine is too weak but can also happen if it is too stiff.  The spine must be ahhhhhh, just right. NOTE: primative bows, such as "stick bows", do not have these components. Instead the arrow typically rests on the index finger of the bowhand, up against the bow. The fletchings will always strike the bow as they pass by. The nature of the fletch, typically made of the feathers of a bird such as a turkey, inherently compress vertically and then spring back into form with little resistance. This durability and vertical softness coupled with a stronger lateral rigidity(to act/react to air resistance and rotate the arrow), allows for a consistent deflection that the archer can learn to compensate for and achieve reasonable accuracy. But what is accurate and acceptable for the stickbow archer is intolerable for the olympian.

My reasons for considering what is and is not Archer's Paradox includes the fact that not all arrow delivery systems rely on the arrow flexing to "get around" the riser of the bow (some actualy have the arrow's vanes hit the bow), but all such systems MUST have the arrow aimed "not at the target" but actually "off-target".

The cushion plunger or button is employed on a recurve bow to dampen the inital rebound reflection of the shaft, to absorb some of the energy of the flexion so the subsequent oscillations will be smaller. The button touches the arrow only prior to loose, and at the instant of loose. Once the arrow begins a forward acceleration it never again touches the plunger (if properly adjusted).   Fewer/smaller oscillations will keep the arrow closer to the center of the target during flight, resulting in more efficient flight and a more accurate shot.  The process of TUNING the recurve bow is making a series of tests and adjustments to the various components to arrive at just the right, minimal amount that the arrow need be aimed "off" of dead center and decrease the amount of flexion, for the system, to get the smallest groups of arrows at the target as possible.

THAT , in my opinion, is archer's paradox.  I'm amazed that people were able to deduce the causes prior to having ultra-high speed photography, and am amused that people today are unable to accept certain verifiable facts even with such photography. 

To see this flexion and paradox, you can check the various slow motion videos on the TSAA's High Speed Video Library page.

A.Ron Carmichael, 6/24/2001

For a more scientifically rigorous explanation, though emotionally less satisfying<G> please view Joe Tapley's page.

A related erroneous assumption: 
ROTATION of the arrow causes the arrow to hit the rest

It is impossible for an arrow to ROTATE sufficiently between the time the nock leaves the string and the nock passes the plunger/riser of the bow for this rotation to be worthy of concern.  If an arrow hits a rest or plunger then it is due to OTHER causes, such as being too weak/stiff of a spine, or improperly set to center-shot (see paradox above<G>), or the archer has failed to place the vanes properly relative to the plunger/riser..  Rotation of a shaft is so miniscule during the first 8 or 10 inches of arrow flight that it can only be estimated and calculated using advanced math, not through empiric measurements such as high-speed photography offer. I will concede, due to mathematics, it DOES rotate, and it is observable at times using high speed photography, but only a tiny amount of rotation can be seen and it is what we pharmacist-coaches call clinically insignificant.

Lacking a true airfoil, vanes of an arrow will cause rotation ONLY due to compression of air against an angular/curved surface of the vane.  This air compression can only come about by the arrow system traveling through air.   As the arrow starts moving forward the vanes will come under pressure.  Being flexible, they will "give" first, since the nock prevents the arrow from rotating while attached to the string.   Once the nock separates from the string, the mass of arrow begins to be overcome and rotation should begin, albeit slowly at first.  There is the inertia of arrow mass at rest (rotationally speaking) to be overcome.

The arrow system (point/insert/shaft/vanes/adhesive/nock) together all have mass of many grains, perhaps as much as 150, even more for some types of arrows.  This mass is rotationally at rest at the instant of release from the string, and cannot begin to change until acted on by the resistance of air, which will be compressed by the vanes.

During the initial milliseconds of acceleration, the vanes will begin to be pushed back by the air until compression equals their resistance to bending, and then the compressive force of the air will begin to be transferred to the mass of the shaft.  Once the compressive force begins to exceed the resistance of the mass THEN and ONLY then will the arrow begin to rotate. 

The bowstring will travel forward of the brace height (the position of the string on the bow when it is at rest) during release by perhaps an inch or two, making the distance from the point of nock release to the arrow rest/plunger perhaps 7 inches.  So the vanes would have to be compressed during this 7 inches AND reach maximum flexion AND overcome the mass of resistance AND rotate the shaft "significantly" in order for the vanes to be in danger of striking the bow DUE TO ROTATION.

Does the arrow rotate in that first 7 or 8 inches?   Possibly.   Even probably.  Is it a significantly large enough amount of rotation to cause contact in and of itself? No.  The rotation could only be measured in nano- or micro-meters, if it could be empirically measured at all.

For some pictures that help view the  behavior of the arrow during acceleration at the bow, click here.  In particular, this AVI file (alt) clearly shows the lateral displacement of the bowstring coming off of the fingers AND it shows that the spinwing does NOT rotate noticeably prior to clearing the riser.  You can use your right and left arrow keys to step through the video ONE FRAME at a time to see it more clearly.

Why are there vanes/fletches on an arrow?

In a perfect system: There is no wind. The bow is in perfect tune with the archer and the arrow, and all parts of the bow are set appropriately including the sight. The arrow has no structural imperfections (in other words, it is composed of man-made materials, not organic such as wood, feathers, flint tips, etc.), and the arrow is also an "X-10" type shaft which has a barrel shape that lends aerodynamic characteristics to the flight behavior. The archer has superbly developed, entirely consistent release technique (loosing of the arrow is 100% reproducible). NO VANES ARE NECESSARY. The archer will be able to group many if not all arrows shot in the 8" bullseye at 70 meters under these conditions. (Typical Olympic distance and target size).

It's not a perfect world. Especially with organic arrows, such as wood shafts, fletches made of turkey feathers, etc., the grain of the wood in the arrow shaft will have a weakness, a prediliction to bend one way more than any other, and will always want to go in one particular direction, say 90 degrees using a 360 degree circle with 0 at high noon. To the left, in other words. Early man graduated from spears (limited distance and accuracy, even with an atlatl) to bows and small pointy sticks, and somehow learned that putting part of a bird on their pointy sticks (perhaps to make the gods look upon their invention with favor) caused the sticks to go more accurately. We can look upon 10,000 year old relics and marvel at the craftsmanship used to adhere those feathers to the stick. But why?

If that arrow without the fletching wants to zoom left at 90 degrees, then a way must be found to neutralize that tendency. Vanes (aka fletchings) cause the arrow to rotate continually during flight. So, the urge to go left is actually rotated so that the arrow ends up going towards all points in the 360 degree circle. That weakness is now averaged out. So instead of flying to the target like it is sliding down a straight wire between the bow and the target, it's actually describing a hollow cylinder, with the arrow shaft becoming the wall of the cylinder. So where it lands, while on the target, MIGHT be anywhere along that edge of the circle (which is the diameter of the cylinder). You can call this space encircled, the GROUP. The arrows will tend to land in this GROUP. Beginners have the biggest group (lowest score) while elite archers have the smallest group (greatest potential for high score).

Now, remember the flexing of the arrow that always happens? It is flexing not at the precise middle of the shaft. Say for a 24" long shaft, it is NOT flexing/bending precisely at 12" from either end.

If you balance an arrow across your outstretched index finger, you will note that the pivot point is NOT the spot equiidistant from each end. It will be closer to the point of the arrow, because the point is heavier than the nock. This balance point is called "Forward Of Center", or "FOC" (eff oh see). (get your mind out of the gutter!) where center is that measured middle of the arrow - if the arrow is 24 inches long, center would be at 12 inches from either end. FOC is likely to be more like 6 or 8 inches from the front tip of the arrow. This is a good thing. FOC needs to be a certain percent closer to the front, to enable that cylinder that the arrow's flight describes to be smaller. If you hold an arrow between thumb and index finger, pinched right at the FOC, and with the other hand move the nock in a circle, you will see that the front part of the arrow goes in a much smaller circle. The back half is describing a CONE (rather than a cylinder) and the tip a much smaller CONE. So thanks to the FOC's effect on the cylinder, the GROUP becomes much smaller because the arrow will tend to follow the smaller cone than the larger cone - the point LEADS, and the Vanes FOLLOW.

While the arrow is rotating, it is also flexing. A lot, at first, but in ever-diminishing amounts. So as it rotates, it flexes, and the FOC point becomes a sort of pivot point. The distance from the FOC to the point, being shorter, describes a smaller cone than the back end(nock) of the arrow, and as the flight progresses, the flexing shrinks, and so shrinks the cones. As the cones shrink, the GROUP shrinks, and yes,the goal is to shrink the group. Everything the archer does in tuning a bow is done in the hopes to shrink that group. The proper size and design of vanes, the length of arrow, the weight of tip, and the spine of the arrow together are the primary step in reducing the group. Then comes adjustments to the various parts of the bow. The tiller, brace height, string size, length, and material, serving material, serviing diameter, nocking point material, nocking point position, finger tab size/material, plunger/center shot, resistance, spring strength, clicker rigidity, arrow spine, arrow length, point weight, nock type and size, vane selection and size, position of vanes on shaft, FOC, sight position/extension bar, stabilizers, vibration absorbers and placement, grip style and customization/angle/texture, arrow rest angle and height and type, all these are in play when tuning a bow. Throw in the infinite variables that are a human to shoot that arrow, and it is a marvel that one can place the arrow ROUTINELY into an apple hanging from the goalpost while standing on the far 30 yard line of a football field, using nothing more to aim with than a 1/2" metal circle held a yard away from the archer's eye.

So vanes are to allow for that gust of wind, that inconsistency in the archer's technique, that slightly bent arrow, and still be able to hit the bullseye. Now, this was all described rather crudely and in very general terms. Joe Tapley's site, on the other hand, will give you all of the mathematical models that describe this much, much more precisely. And Joe will argue with my own visualizations of "why".

Shrug. Sometimes it is better just to shoot the arrow and see what happens. Or, as some archers say, "grip it and rip it".