−1−0.5

00.5

1

00.5

11.5

20

0.5

1

1.5

2

2.5

3

displacement (x)displacement (y)

Altit

ude

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.

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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

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Side View Release View