In a recent paper, Price, Moss, and Gay have given a simple explanation of a paradox in the flight of a football, why the long axis of the football, contrary to intuition but in good agreement with experience, turns so that it is tangent to the path of the football. Here, we add to the analysis the assumption of only first-order differences between the direction of the velocity, the orientation of the long axis, and the direction of the total angular momentum. The result is a closed-form solution that is particularly useful in revealing the way in which nutation and precession are mixed.

## INTRODUCTION

This short paper provides a supplementary view of the analysis described in a recent paper by Price, Moss, and Gay^{1} (hereafter PMG). The target of that paper, and of this paper, is the rotational kinematics and dynamics of a spiraling football during its trajectory. The particular focus is the clarification of how the symmetry axis of the axisymmetric American football remains closely tangent to the path of the football center of mass (CM).

In comparison with PMG, in this paper, there is a trade-off. At the cost of an additional (but very well justified) approximation, we achieve a closed form (“analytic”), rather than numerical, solution. As is usually the case, the closed form solution gives insights not available from a numerical solution. For the spiral pass, the insights are particularly useful in distinguishing precession and nutation.

We use the same notation as in PGM, notation that is similar to that of Soodak^{2} and that is briefly reviewed as follows:

= Football angular momentum= $ L l \u0302$, where*L**L*= magnitude of,*L*$ s \u0302$ = Unit vector aligned with the symmetry axis of the football,

$ v \u0302$ = Unit vector in the direction of the velocity of the football CM,

*I*= Moment of inertia about an axis through the CM and perpendicular to $ s \u0302$,_{t}$ \tau *$ = Magnitude of the aerodynamic torque in a simple model introduced by Soodak, experimentally verified by Rae and Streit,

^{3}and presented below.

## ANALYSIS

^{2}and is repeated in the appendix of the PMG paper. The second is a model of aerodynamic torque, the “pitch torque,” which again is given by Soodak and discussed in PMG.

^{4}

**with the second of Eqs. (1) and using $ L \xb7 L \u0307 = 1 / 2 \u2009 d ( L 2 ) / d t$,**

*L**L*is second order in our perturbation quantities, and henceforth, we can consider

*L*to be constant in time. We can thus rewrite Eqs. (1) as

*x*,

*y*,

*z*} coordinate system used in the PMG paper. (The football trajectory is confined to the

*x*,

*z*plane, with

*x*directed vertically upward.) The component equations are

**,**

*S***, in that plane as the vectors with the components,**

*E***is the same as in PMG and that**

*S***is the analogous vector for**

*E***. From Eqs. (8)–(13), we arrive at the equations for the time derivatives of these quantities,**

*L**y*component of $ v \u0302 .\xd7 v \u0302$ and represent the rate of rotation of the football velocity in the

*xz*plane.

*S*and

*E*components with

**, and $ s \u0302$, four functions because there are four degrees of freedom in the kinematics: The velocity is a specified vector, and the magnitude of**

*L***is fixed by its initial value, so only the four degrees of freedom of $ s \u0302$ and $ l \u0302$, in the plane perpendicular to $ v \u0302$, need to be computed. If we are given the initial values of $ s \u0302$ and $ l \u0302$, their values at any subsequent time will be the superposition of the four solutions in Eq. (23).**

*L*With Mathematica, the solutions of Eq. (21) with the velocity in Eq. (22) are easily found to be as follows:

^{3}.) In the above equations,

Figure 1 shows the result of the closed-form solution in Eqs. (24)–(33) for a “long bomb” spiral pass and compares it with the numerical solutions of Eqs. (1) for the long bomb spiral pass of PMG (Table I), with its true parabolic trajectory, solved as a coupled set of differential equations. In view of the fact that the initial misalignment is not particularly small (10°) and that the rotational rate of the true long bomb is not constant, the agreement is remarkable.

## CONCLUSIONS

From these results, we see that the fast precession solutions F1 and F2 describe a motion in which $\epsilon $ is smaller than $\delta $ by the ratio $ \omega gyr / \omega wob$, which means that the alignment of $ l \u0302$ and $ v \u0302$ is much better than that of $ s \u0302$ and $ v \u0302$. This is just what we expect for quasi torque-free precession.

When the motion is that of S1 and S2, we see that ** S** and

**are the same, meaning that $\delta $ and $\epsilon $ are the same. Thus, for $ \omega gyr \u226a \omega wob$, the symmetry axis remains well aligned with the angular momentum as they both rotate at $ \omega gyr$ due to the aerodynamic torque.**

*E*One of the advantages of a closed form solution is the insight into the interaction of the “torqued precession” frequency $ \omega gyr$ and the “nutation” frequency $ \omega wob$. The fast and slow frequencies of Eqs. (33) show that for $ \omega gyr \u226a \omega wob$, as in the case of our “long pass,” the “fast” oscillation is slightly less than $ \omega wob$ and the “slow” oscillation is slightly greater than $ \omega gyr$.

## ACKNOWLEDGMENTS

The author thanks the coauthors of PMG, William Moss and Timothy Gay, for their patience and tolerance.

## REFERENCES

The current paper can be considered to be an extension of the “Simplified Analytical Model” in Sec. VI of PMG, with the assumption dropped that $s\u0302$ and $l\u0302$ differ negligibly, so that there is no difference between $\delta $ and $\epsilon $. This reduction of the number of degrees of freedom preserves the resolution of the spiral pass paradox but eliminates the high frequency nutation.