When traveling through the atmosphere, and assuming no gyroscopic effect
from spinning, a projectile can maintain a particular orientation with
respect to its path if the center of drag is behind the center of
gravity. If the projectile is launched with the center of drag ahead of
the center of mass, the projectile will reverse its orientation after
launch and thereafter "fly" with its center of drag behind its center of
mass. If the center of drag is at the same location as the center of
mass, the projectile can tumble. The tail feathers make sure the center
of drag is well separated from the center of mass. The feathers would
not be necessary if the arrowhead is massive enough and small enough
that the center of mass is moved ahead of the center of drag even in the
absence of feathers.
Aside... I learned a lot of this from my younger days as an archer
(making both the arrows and the bows) and as a rocket builder. In a
propelled rocket, as opposed to a coasting arrow, getting the center of
drag ahead of the center of mass is disastrous because the rocket goes
berserk, whereas a coasting projectile simply reorients until the center
of drag is behind the center of mass.
Other than this, the feathers do not provide any "lift" as Frank
Cardulla seemed to be asking. Their effect is simply making sure the
center of drag is sufficiently behind the center of mass that the arrow
is stable. Therefore, the presence of feathers reduces the range
(because of air friction) compared to an arrow without feathers that is
stable because of a dense tip.
In vacuum, as Franks stated, the feathers do not affect the flight.
Whether the arrow tumbles depends on whether it was launched with any
torques around its center of mass. If the accelerating force was
applied in-line with the center of mass, then all the energy given to
the arrow ends up as linear kinetic energy. This arrow will go the
farthest. It will not stick in the ground because it will land with the
same orientation as it was launched.
If the arrow is launched with a torque, then the arrow launched in
vacuum will "tumble." This means some of the energy ends up as
rotational kinetic energy rather than linear. This means the velocity
of the center of mass is less. This arrow will not go as far as a
vacuum arrow that is given a greater center-of-mass velocity.
Finally, to answer the question... if the identical arrows are given
identical launches such that their center-of-mass launch velocities are
the same, then the arrow in atmosphere suffers drag and will fall short
of the range predicted by the usual textbook projectile-motion
equations. The center of mass of the arrow in vacuum will follow the
textbook projectile-motion equations and will go farther. The launch
angle changes the range, but does not change the fact that the vacuum
arrow wins the range contest.
Michael D. Edmiston, Ph.D.
Professor of Physics and Chemistry
Chair of Sciences
Bluffton College
Bluffton, OH 45817
(419)-358-3270
edmiston@bluffton.edu