Joe Tapley
There have been innumerable descriptions over the years of the Archers
Paradox problem but surprisingly I can't recall seeing one that wasn't
just an arm waving exercise. No one ever seems to address the Archers
Paradox as a simple mechanical problem. I would guess that using basic
mechanics was exactly what those guys in a bar in the US back in 1912 or
whenever did and came up the requirement that an arrow should come out
of a bow with a horizontal velocity component as well as the component
in the direction of aim. I.e. the arrow should come out of the bow at an
angle to the horizontal direction the bow is pointing.
The
Archers Paradox problem arose because of two basic assumptions made
about the behaviour of an arrow being shot from a bow, Both assumptions
were perfectly reasonable (though both assumptions turned out to be
wrong). The first assumption was that the arrow did not bend during the
bow power stroke. (If you hold an arrow by the nock end and jiggle it
around sideways with your fingers then there is no way you can bend the
arrow; it is so light that it just rotates. The same would be true for
any sideways nock movement resulting from the action of the bow string).
The second assumption was that during the power stroke the bow string
(and hence the arrow nock) travelled in the plane of the bow from the
full draw position to the point at which the nock separated from the
string, at around the bracing height. This second assumption pretty much
follows from the first, if the arrow doesn't bend then there is no
reason why the string should move off the bow plane.
The following figure illustrates the behaviour of the arrow during the power stroke based on the above two assumptions.
As the arrow moves forwards the arrow center of mass (orange dot) is accelerated sideways by a force reaction between the arrow and the bow handle. If you assume the lateral acceleration of the arrow shaft is a constant then all you need to determine the resulting arrow lateral velocity component is the time duration of the handle-arrow force interaction (the time the arrow is under acceleration on the bow string) and the distance the arrow center of mass (COM) has moved laterally (distance D) in this time. The distance D can be calculated (knowing the position of the arrow COM, the handle width and the handle to brace height and full draw distances), though far easier to just measure it if you have the bow and arrow to hand. The time the arrow is on the bow string "T" can be measured or estimated in a number of ways e.g. from the arrow launch speed.
The lateral velocity of the arrow center of mass goes from zero at full draw to V at the point the nock separates from the string. So the average velocity (constant acceleration) is just V/2. Velocity is just distance divided by time so the lateral arrow velocity component is just V = 2xD/T.
For example if D was measured at 0.01 meters and T was measured at 0.015 seconds then the arrow lateral velocity component at launch would be 2*0.01/0.015 = 1.3 meters/second. Such a high lateral velocity would be obvious to the archer. The arrow would miss the target off to the side at even moderate distances. The Archers Paradox was that this horizontal velocity component was not visible even when by theory it should be obvious. Arrows went straight, left or right depending on the bow set up and how the arrow was shot. As already mentioned what was adrift with the theory was the two incorrect assumptions being made.
It's worth noting that you get an "Archers Paradox" effect even with a modern Olympic bow. This is because the Paradox effect is down to a force interaction between the arrow and the bow handle (or pressure button) and you still have this interaction even though the reasons for it may be different. With an Olympic recurve bow you have a force interaction between shaft and button of around 5 Newton say (defined by the button spring setting) and the force duration is of the order of 0.003 seconds (theory and experiment). Newton's second law tells you that velocity equals force*time/mass so assuming an arrow with a 0.02 kg mass with an Olympic recurve bow you get an "Archers Paradox" lateral velocity of around 5*0.003/0.02 = 0.75 meters/second.