How long would it take for a person to fall through the Earth?

A 42-Minute Plunge: Falling Through the Earth

The Earth. A sphere of rock, metal, and mystery, a world we inhabit but rarely consider from a purely physics-based perspective. One such thought experiment, often posed in introductory physics courses, is this: how long would it take to fall through the Earth, ignoring the inconvenient realities of molten rock, immense pressure, and the rather significant problem of air resistance?

The answer, remarkably, is a little over 42 minutes. This seemingly simple question belies an elegant solution derived from a surprising application of simple harmonic motion.

Ignoring the Earth’s inhomogeneities – its layered structure of crust, mantle, and core – we can model the Earth as a perfect sphere with uniform density. This simplification allows us to utilize Newton’s Law of Universal Gravitation. As an object falls towards the Earth’s center, the gravitational force acting upon it changes. It’s not constant as it is on the surface; instead, it decreases linearly as the object approaches the center. Once past the center, gravity reverses direction, pulling the object back towards the opposite side.

This setup perfectly describes simple harmonic motion – a repetitive back-and-forth oscillation around a central point. The period of this oscillation, the time it takes for one complete cycle, can be calculated using a formula that depends only on the Earth’s mass and its radius. The result? Approximately 84 minutes.

But remember, we’re only interested in the time to fall through the Earth, not a complete oscillation. Therefore, we halve the period to arrive at our final answer: a little over 42 minutes.

It’s crucial to reiterate the idealized nature of this calculation. In reality, the Earth’s non-uniform density, the intense heat and pressure in its interior, and the ever-present (and lethal) force of air resistance would render this 42-minute journey impossible. The hypothetical faller wouldn’t simply oscillate; they’d be vaporized long before reaching the Earth’s center.

This thought experiment, however, provides a fascinating glimpse into the power of simplified models in physics. By stripping away the complexities of the real world, we reveal a surprisingly simple and elegant solution – a 42-minute freefall through the planet, a journey possible only in the realm of theoretical physics. It serves as a reminder that even seemingly impossible questions can yield surprising, and elegantly simple, answers when approached with the right tools and assumptions.