In physics we define projectile motion as the motion of the center of mass of an object through a region of space where the object is subject to constant acceleration. A football moving through the air near the surface of the earth is subject to the constant gravitational acceleration of magnitude g = 9.8 m/s2, directed downward. If we neglect air resistance, then the football is a projectile, executing projectile motion.
We have studied projectile motion in the projectile motion lab. Open the linked spreadsheet to review what we have learned. (This is an Excel spreadsheet containing macros. Macros must be enabled. On Excel's menu bar click Tools, Macro, Security, and choose Security Level, Medium.)
The center of mass of the football follows a parabolic trajectory. The football's horizontal velocity is constant, while its vertical velocity changes by -9.8 m/s every second. The range of the football depends on its initial speed and its launch angle. For a given initial speed the football travels farthest if it is launched at an angle of 45 degrees with respect to the ground.
| Gravity does not exert a torque on the football and the football's angular momentum L does not change. The direction and magnitude of the angular momentum L are conserved. If a quarterback throws a perfect spiral and the football is spinning about its long symmetry axis with angular velocity w, then its angular momentum L = Iw points along that symmetry axis and the direction of this symmetry axis stays fixed in space. |
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| Often the quarterback throws a wobbly spiral. He gives the ball spin about an axis that makes a small angle with the long symmetry axis of the ball. The angular velocity w now has two component, one along the long symmetry axis wx, and one perpendicular to it, wy, as shown in the figure on the right.. |
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| The angular momentum L therefore
also has two components. But the
moment of inertia
Ix of the ball for rotating about
its long symmetry axis is different then the moment of inertia Iy for rotating
about a perpendicular axis.
The farther the mass of the football is from the axis
of rotation, the greater is the moment of inertia
of the football for rotation about this axis.. For our football we have Ly = Iywy, Lx = Ixwx, and Iy is greater than Ix. Therefore L and w do not point in the same direction. |
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| Angular momentum is still
constant, the direction of
L is fixed in
space. But the direction of the angular velocity
w is not, and
w precesses (i.e. rotates) about the direction of
L. As
w precesses, so does the long symmetry axis, and we have a wobbly spiral. This is called
torque-free precession, it occurs even in the absence of air resistance.
For torque-free precession the ratio of the wobble frequency to the spin
frequencies of a football is about 4/3. This ratio is determined by the
mass distribution (shape) of the football.
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In the projectile motion lab we have also explored the motion of a non-spinning ball in the presence of air drag. Open the linked spreadsheet to review what we have learned.
The center of mass of the football no longer follows a perfectly parabolic trajectory. The football's horizontal velocity decreases, and its vertical velocity no longer changes at a constant rate of exactly -9.8 m/s every second. The range of the football still depends on its initial speed and its launch angle, but for a given initial speed the football travels farthest if it is launched at an angle a few degrees less than 45 degrees with respect to the ground.
But what is the actual trajectory of a football? Watch the video clips linked below. You can press the step-up button any time during the play to step through the clip frame by frame.
In these clips you have probably observed, that when a pass is perfectly thrown and the football spins about its symmetry axis, the football tilts during flight, so that its symmetry axis is tangent to its trajectory. Why?
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Does a ball thrown by a right-handed quarterback really veer to the right? |
We have invested a lot of
time talking about the flight of a football after a pass.
What about kicks and
punts?
