Here, straight from the Wikipedia article, is a lovely picture of a basketball in a free-flight trajectory.

You probably expect a parabolic trajectory, and we do get pretty close. There are some deviations. The resistance of the atmosphere is the largest, and the rotation of the ball will itself result in aerodynamic effects that distort the flight of the ball from its idealized trajectory.

But in fact even in a perfect vacuum with no external forces but gravity we *still* won’t get a parabola. We’ll get a section of an ellipse.

Why? Newton’s laws tell us that if you’re in the gravitational field of a spherically symmetric object like the earth, it’s mathematically identical to a situation with an orbit about a point mass located at the center of that object. Without air resistance or other perturbations, as far as that basketball is concerned it’s in a long, thin, very eccentric orbit about the center of the earth. In a very, very exaggerated visualization of the effect the actual trajectory will look like the lower one here and the parabolic trajectory is the upper one (arbitrary units):

Let’s try to put some numbers to how big this effect will be. The shape of an ellipse is characterized by its *eccentricity*. An ellipse with an eccentricity of 0 is a circle, and it gradually gets more and more squashed as it approaches the maximum possible value of 1. At 1 and beyond it’s no longer a closed curve; it’s a parabola or hyperbola. The equation for the eccentricity of an ellipse in central force motion is

Plugging in the values for E, L and k, I find that for a normal gravitational potential the eccentricity is:

That’s after dropping terms in the fourth or higher powers of v, but for velocities lower than the kilometer per second range this is a fine approximation. Plugging in a test figure of v = 10 m/s, I see that we get an eccentricity of about e = 0.9999984. Pretty eccentric. And since e = 1 corresponds to a parabola exactly, it follows that the trajectory we see will in fact be a parabola to a very close approximation. How close? My rough order-of-magnitude estimate with these numbers is that the deviation of the ellipse from the parabola will be nanometers at typical basketball trajectories. Almost every other effect from the moon to local gravitational variations will probably swamp that so it’s probably permanently beyond the capabilities of tabletop experiments.

Still, interesting to think about!

[Exam report from last week: These are estimates as they haven't been passed back yet, but I think my classical mechanics exam went very well, but my E&M II exam was probably pretty sketchy. I can live with that though, there's always the final.]