There’s many, many sports out in the world that involve sharply hitting a ball with something. Baseball, tennis, golf, cricket, polo, you name it. After being hit, a ball describes a trajectory determined by the gravity of the earth and the interaction of the ball with the atmosphere. This can be exploited in many sports, since different trajectories can be useful in different situations. Golf and baseball especially are built on tweaking air flow around the ball in order to make it do precisely what the athlete wants. Tennis too, which has been drawing a lot of interest with the dramatic underdog victories in the French Open.

In the absence of an atmosphere (or local gravitational variations and things of that nature) it’s generally said that a projectile will follow a parabola. This is true in a constant gravitational field, but the earth is a sphere of finite size and thus does not in fact have a uniform field. It’s very close to uniform near the surface and so a parabola is an excellent approximation. But formally it’s not quite right. Atmosphere aside, a thrown projectile is in a very extended orbit about the center of the earth, and thus the trajectory is a section of an *ellipse*.

So here’s a challenge problem that’s not so hard, but requires a little bit of mathematical fluency. It’s a Fermi problem, so estimates rather than exact numbers are what we’re looking for.

*Assume a baseball is hit into the outfield with a typical angle and initial speed. What is the order-of-magnitude difference in range between the trajectory approximated as a parabola and the trajectory approximated as an ellipse?*

As a hint that you don’t need, the difference is very, very small.