The Mathematics of
the ATLATL
Topic Classification: algebra and trigonometry, the physics of projectile motion,
archaeology and human history, modern sports records.
Applicable Courses: MAT 114, 125, 136
Proposer: Dr.
Kiisa Nishikawa, Department of Biological Sciences,
Northern Arizona University
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Definition: An atlatl is an ancient dart-throwing tool
designed to increase distance and accuracy for hunting or warfare. The atlatl
is basically a stick with a hook on one end to hold a dart and a handle on the
other end that is grasped by the thrower, see illustration below. Atlatls were
developed independently by native peoples of North America and Europe. Atlatls
were eventually replaced by bows and arrows, but were used in North America for
more than 10,000 years. Ancient atlatls were generally made from hardwood and
measured approximately 24 inches (60 cm).
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Produced for NSF CCLI Grant #9980883 "Enhancing Quantitative Reasoning Using Visualization".
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Copyright: Kiisa Nishikawa and Michael I. Ratliff, 2001
Activities: It is hoped that this exercise could be used
in a variety of ways to instruct students in using math and physics, either as
an in-class example using algebra and trigonometry, as a self-study or homework
exercise, or as an out-of-class project for which students might find the
information for themselves. For the in-class demonstration, the instructor
might provide the math and physics background in class. Homework or self study
might involve rearranging the equations to solve different word problems, for
example deriving the best departure angle (45 degrees), or calculating the
maximum height of a throw with a given departure angle and initial velocity.
Skills such as graphing could be incorporated by having students plot the range
of an atlatl throw as a function of the departure angle (this would be a sine
function). Out-of-class projects might involve sending students to the Museum
of Northern Arizona to measure atlatls, searching for information about ancient
or modern atlatls on the web, making one’s own atlatl and throwing it, or
finding the appropriate equations of motion on the web or in a textbook or
encyclopedia.
Discussion: Consider what happens when you throw a dart
with your arm. As you throw the dart, your upper arm rotates at the shoulder,
your lower arm rotates at the elbow, and your hand rotates at the wrist. The
tangential velocity or linear velocity of the dart is equal to the sum of the
angular velocities at each of the joints times their radius of rotation.
1)
tangential
velocity = radius of rotation x angular velocity
v = r
w
The distance and
accuracy of a projectile are related to the velocity of the projectile when
released by the thrower. After the projectile leaves the thrower’s hand, gravity
determines the resulting path that the projectile will take. The challenge for
the thrower is to achieve the highest possible initial velocity.
With simplifying assumptions (such as no air
resistance, constant gravitational force, etc.), the distance, or range, that a
projectile will travel depends upon the initial tangential velocity of the
projectile (v), the departure angle (q), and gravity (g). If we assume a departure
angle of 45 degrees (for maximum performance), then the range is equal to the
tangential velocity squared divided by gravity (32 ft/s2).
2) range = (v2sin2q)/g
The atlatl works
by increasing the radius of rotation of the wrist, from about 2.4 inches (6 cm)
for an unaided throw to about 24 inches (60 cm) for a throw with the atlatl.
Thus, when the same angular rotations are applied at the shoulder, elbow and
wrist, the atlatl increases throwing distance. For an experienced atlatler, the
mechanical advantage of the atlatl over throwing a dart with the arm alone is
about 6:1. Thus, if you can throw a dart 50 feet with your arm alone, you could
throw it 300 feet with the atlatl.
Modern Day Atlatl
Record: Modern
day atlatls are designed by computers and are made of aluminum, fiberglass and
carbon fiber. The modern record atlatl throw, 848.5 feet, was set in July, 1995
in Aurora, Colorado. For more information about modern atlatls, visit the
website of the World Atlatl Association (use any search engine to find).
Related equations: Students may perform calculations on other
aspects of projectile motion, for example, maximum height, departure angle,
position of the projectile as a function of time, or kinetic energy. Some of
these equations are given below. The equations assume a reference frame in
which x is the horizontal dimension, y is the vertical dimension, and the
origin is the point at which the projectile begins its free flight. Time
(t) = 0 when the projectile begins its
free flight.
kinetic energy =
1/2mv2 (where m =
mass of the projectile)
horizontal
velocity (vx) = v cosq
vertical velocity
(vy) = v sinq - g t
horizontal
position (x) = v cosq t
vertical position
(y) = v sinq t -1/2g t2
Questions for
students:
1) If we assume a departure angle of 45 degrees, what was the
initial velocity of the dart for the world record atlatl throw?
2) How might you expect the world record atlatl thrower to differ
in his/her physical characteristics relative to the general population?
3) What was the maximum height achieved by the dart in the world
record atlatl throw? (Hint: At the highest point, the vertical velocity is
zero. Solve for t when vy equals zero, then solve for y using that
value of t.)
4) Find the range as a function of v and q.
5) Use equation 2 to prove that 45 degrees is the angle of
departure that maximizes the range.
6) For a given target distance and a given initial velocity, what
departure angle should a hunter use?
7) What factors limit the length of the atlatl, and therefore the
tangential velocity of the dart?
8) What might explain the observation that European darts averaged
7 feet in length, whereas North American darts averaged only 6 feet?
9) Assume the v is 100 ft/sec, eliminate t from the equations for
x and y. Then for several values of
Web sites: For more information about atlatls, ancient
and modern, check out the following: