- The document discusses the forces of drag and lift that act on a boomerang during flight and how its design dictates its motion. It aims to develop a physical understanding of these forces and apply mathematical models to analyze what causes a boomerang's angular precession and return. - The dominant torque is due to the unbalanced lift forces acting on the rotating boomerang. This torque causes the boomerang's principal axis to change direction during flight, which results in its precession and return path. - A mathematical model is derived and used with MATLAB to simulate the theoretical flight path of a boomerang based on its design parameters and motion characteristics. The model considers the boomerang's position,

1. −1
−0.5
0
0.5
1
0
0.5
1
1.5
2
0
0.5
1
1.5
2
2.5
3
displacement (x)displacement (y)
Altitude
RELEASE POINT
Acknowledgments
Thanks to our professors for the interest and support, Jenifer
Mellott for assisting in boomerang construction.
PROJECT GOALS
- Develop a physical understanding between the forces of
drag and lift acting on the body in motion and how design
parameters dictate this motion.
- Understand and apply the motion of the boomerang to
mathematical models of motion and analyze what causes the
angular precession, and ultimate return of the body.
.
Figure 1. Symmetric 4-Blade Constructed Model
References
1. John Taylor, Classical Mechanics Textbook, Ch. 10
2. Ernie Esser, UCLA Powerpoint, Web Access,.bashaar.org.il/files/boom
THEORY
The dominant torque is due to the unbalanced force of lift
acting on the rotational body. This resulting non-conservative
force of lift is determined by both the forward and rotational
speeds of the boomerang arms, acting in the the direction of
angular momentum. The torque is in a direction opposite to
the forward (thrown) velocity and perpendicular to the
angular momentum vector along the axis of rotation. This
resulting magnitude and direction of torque is responsible for
the steady change in the direction of angular momentum and
overall precession of the principal axis during flight for the
returning boomerang. The torque causes only a change in the
direction of the angular momentum, not the magnitude,
because the is perpendicular in direction1.
Figure 2. The rotation of the body about the principal axis (e3)
causes a difference in the lift forces which is dependent on the
translational velocity. A resulting torque forces the principal axis
to change in direction during flight. The overall lift also acts
against gravity to keep the boomerang in the air.
CONCLUSION
The overall flight pattern of a boomerang is determined
by several key forces. The lift forces, torque, and resulting
precession all depend on the geometry of the body as well as
the initial conditions as determined by the thrower.
MATHMATICAL MODEL
Given the design parameters and assumed physical
characteristics of motion, a differential equation is derived
to show the theoretical path for a returning boomerang.
The position and velocity of the body is mapped with
MATLAB software using an integration function to solve
the above equation for the position and velocity of the
center of mass at a given space coordinate (x,y,z).
Inputs:
Figure 3. Shown above is the mathematical model
for the path of a 4-blade boomerang under the
influence of only air lift and the force of gravity.
FLIGHT ANALYSIS & RESULTS
For boomerangs with more symmetric (and shorter)
blades the torque will be smaller and the rate of
precession more gradual, leading to a more circular
path. Boomerangs with 2 or 3 blades that are quite
longer will tend to have greater torque acting on them
and will curve around much sharper, creating a more
elliptical path. Several parameters dictate this motion in
which the boomerang returns, although they may take
different flght paths due to their design.
Assumptions:
- Constant Rotation Speed
- Constant Bank Angle
- Drag Force is Negligible
- Bernouli Lift Equation Applies
Side View Release View