We All Learned Physics' Biggest Myth: That Projectiles Make A Parabola

Anyone who’s ever taken a physics course has learned the same myth for centuries now: that any object thrown, shot, or fired in the gravitational field of Earth will trace out a parabola before striking the ground. If you neglect external forces like wind, air resistance, or any other terrestrial objects, this parabolic shape describes how the center-of-mass of your object moves extremely accurately, no matter what it is or what else is at play.

But under the laws of gravity, a parabola is an impossible shape for an object that’s gravitationally bound to the Earth. The math simply doesn’t work out. If we could design a precise enough experiment, we’d measure that projectiles on Earth make tiny deviations from the predicted parabolic path we all derived in class: microscopic on the scale of a human, but still significant. Instead, objects thrown on Earth trace out an elliptical orbit similar to the Moon. Here’s the unexpected reason why.

If you wanted to model the gravitational field at Earth’s surface, there are two simplifying assumptions you could make:

1. the Earth, at least in your vicinity, is flat rather than curved,
2. and that Earth’s gravitational field points straight down relative to your current location.
3. Any time you throw and release an object, therefore, it enters a situation known as free-fall. In the directions that are parallel to the Earth’s surface (horizontal), the speed of any projectile will remain constant. In the directions that are perpendicular to Earth’s surface (vertical), however, your projectile will accelerate downwards at 9.8 m/s²: the acceleration due to gravity at Earth’s surface. If you make these assumptions, then the trajectory you calculate will always be a parabola, exactly what we’re taught in physics classes around the globe.
   

   An illustration of “Newton’s Cannon,” which fires[+]

   WIKIMEDIA COMMONS USER BRIAN BRONDEL
   But neither one of these assumptions are true. The Earth may appear flat — so indistinguishable from flat that we cannot detect it over the distances most projectiles cover — but the reality is that it has a spheroidal shape. Even over distances of just a few meters, the difference between a perfectly flat Earth and a curved Earth comes into play at the 1-part-in-1,000,000 level.
   This approximation doesn’t matter so much for the trajectory of an individual projectile, but the second approximation does. From any location along its path, a projectile isn’t truly accelerated “straight down” in the vertical direction, but towards the center of the Earth. Over the same distance of a few meters, the difference in angle between “straight down” and “towards the center of the Earth” also comes into play at the 1-part-in-1,000,000 level, but this one makes a difference.