The complexities of falling freely

TL;DR

This work addresses the time of free fall under inverse-square gravity, replacing the standard constant-$g$ assumption. It derives an exact time formula via energy conservation and a trigonometric substitution, yielding $T = \dfrac{R_0^{3/2}}{\sqrt{2GM}} \left( \dfrac{\pi}{2} - \sin^{-1}(\sqrt{K}) + \sqrt{K(1-K)} \right)$ with $K=R_1/R_0$, and shows that the collapse time to the center is $T_C = \dfrac{\pi}{2} \dfrac{R^{3/2}}{\sqrt{2GM}}$. The paper also demonstrates that for small fall distances the exact formula recovers the familiar $T \approx \sqrt{\dfrac{2h}{g}}$ by using $g = GM/R_0^2$, and connects the appearance of $\pi$ to orbital dynamics, linking vertical free fall to circular and elliptical orbits. Overall, it provides a unified view that bridges elementary kinematics and celestial mechanics, clarifying the limits of the simple formula and highlighting geometric underpinnings of gravitational time scales.

Abstract

Suppose you drop a coin from 10 feet above the ground. How long does it take to reach the ground? This routine exercise is well-known to every AP physics and calculus student: the answer is given by a formula that assumes constant acceleration due to gravity. But what if you ask the same question in the more realistic scenario of non-constant acceleration following an inverse square law? In this article, we explain the analysis of this realistic scenario using freshman-level calculus and examine some implications. As a bonus, we also answer the following intriguing question: Suppose the Earth were to instantaneously collapse to a mathematical point at its center. How long would it take for us surface dwellers to fall to the center?

Figures (1)

• Figure 1: Ellipses with increasing eccentricity $e$. Circle on the top with $e=0$, and a skinny ellipse at the bottom with $e$ close to 1.

Theorems & Definitions (1)

• Remark 2.1