Sports Engineering

, Volume 16, Issue 2, pp 99–113

# Aerodynamic effects on the accuracy of an end-over-end kick of an American football

Original Article

## Abstract

The effects of initial conditions on the flight of an American football kicked in an end-over-end manner were investigated using a dynamic simulation employing the quaternion method. The effects of initial tilt and spin about the longitudinal axis of the ball were considered. For the most accurate kick, our simulations show that the ball should be vertical leaving the kicker’s foot, and have no angular velocity about the longitudinal axis of the ball. A case study was performed for which experimental data were available, showing the trends of the flight of the ball captured in our simulations in actual game situations.

### List of symbols

$$O = \left\{ {\vec{i}_{O} , \vec{j}_{O} ,\vec{k}_{O}^{1} } \right\}$$

Inertial coordinate system, point O at the spot of the kick

$$B = \left\{ {\vec{i}_{B} , \vec{j}_{B} , \vec{k}_{B} } \right\}$$

Body-fixed coordinate system, point B at CG of the ball

$$m_{B}$$

Mass of the ball (0.411 kg)

$$^{O} \vec{v}_{CG/O}$$

Inertial velocity of the ball

$$\vec{M}_{CG,body}$$

External torque applied to ball about CG

$$^{O} \vec{h}_{CG,body}$$

Angular momentum of the ball about CG w.r.t O-frame

$$\vec{r}_{CG/O}$$

Position vector of the CG of the ball w.r.t O-frame

$$v_{ij} ,i = x,y,z;j = O,B$$

i-Component of velocity in j-frame

$$^{O} \left[ C \right]^{B}$$

Matrix of directional cosines between O- and B-frames

$$^{O} \vec{\omega }^{B}$$

Angular velocity of B-frame w.r.t. O-frame

$$\omega_{i} ,\quad i = x,y,z$$

i-Component of angular velocity in B-frame

$$I_{i} ,\quad i = x,y,z$$

Moments of inertia of the ball w.r.t. B-frame axes

$$e_{i} ,\quad i = 0,1,2,3$$

Components of unit quaternion

$$\vec{e}_{yaw}$$

Unit vector of yaw force, pitching moment

$$\vec{e}_{lift}$$

Unit vector of lift force

$$\vec{e}_{drag}$$

Unit vector of drag force

$$\vec{F}_{aero}$$

Aerodynamic force vector

$$\vec{F}_{lift}$$

Aerodynamic lift force, magnitude $$F_{lift}$$

$$\vec{F}_{yaw}$$

Aerodynamic yaw force, magnitude $$F_{yaw}$$

$$\vec{F}_{drag}$$

Aerodynamic drag force, magnitude $$F_{drag}$$

$$\vec{F}_{aero,iB} ,\quad i = x,y,z$$

Aerodynamic force components in B-frame

$$\vec{F}_{g,iB} , \quad i = x,y,z$$

Gravitational force components in B-frame

$$M_{roll}$$

Aerodynamic roll moment magnitude

$$M_{pitch}$$

Aerodynamic pitch moment magnitude

$$M_{yaw}$$

Aerodynamic yaw moment magnitude

$$M_{aero,iB} ,\quad i = x,y,z$$

Aerodynamic moment components in B-frame

$$\theta_{wt}$$

Angle of attack (degrees)

$$\omega$$

Longitudinal spin rate (rev/s)

$$C_{D}$$

Aerodynamic drag coefficient

$$C_{L}$$

Aerodynamic lift coefficient

$$C_{Y}$$

Aerodynamic yaw coefficient

$$C_{M}$$

Aerodynamic pitch moment coefficient

$$\rho$$

Density of air (1.184 kg/m3)

$$V_{B}$$

Volume of ball (4.0 × 10−3 m3)

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