Falling Through Earth Calculator

🌍 Earth's Internal Structure

The Earth is not a uniform ball — it has distinct layers with very different densities, which profoundly affects the gravity train calculation.

📐 The Physics Behind the Gravity Train

The gravity train is a classic thought experiment in physics. Imagine drilling a frictionless, evacuated tunnel straight through the Earth and dropping an object into it. What happens?

Uniform Density Model: Inside a uniform sphere, only the mass closer to the center than you contributes to gravity (Shell Theorem). This gives a linear gravity profile:

where \(g_0 = 9.81\) m/s² is surface gravity, \(r\) is distance from center, and \(R = 6{,}371\) km is Earth's radius.

This creates simple harmonic motion with angular frequency:

The one-way travel time is half the oscillation period:

PREM Variable Density Model: The real Earth has a dense iron-nickel core (13 g/cm³) surrounded by a lighter rocky mantle (3–5 g/cm³). This means gravity actually increases as you descend through the mantle (peaking at ~10.68 m/s² at the core-mantle boundary at 2,891 km depth), then decreases through the core. The result: stronger initial acceleration and a shorter travel time of about 38 minutes.

💨 Maximum Speed at the Center

At the Earth's center, all gravitational acceleration has been converted to kinetic energy. The maximum velocity is:

This is approximately Mach 23 — 23 times the speed of sound! It's also exactly equal to the orbital velocity at Earth's surface, which is not a coincidence: the gravity train is mathematically equivalent to a degenerate (flattened) orbit.

📐 Chord Tunnels: The Surprising Shortcut

A chord tunnel connects two points on Earth's surface without passing through the center. For a chord subtending angle \(\theta\) at the center:

• Tunnel length: \(L = 2R\sin(\theta/2)\)
• Maximum depth: \(d = R(1 - \cos(\theta/2))\)
• Maximum speed: \(v_{max} = \omega R\sin(\theta/2)\) (lower than diametric)
• Travel time: Still \(\pi\sqrt{R/g_0} \approx 42\) minutes!

The equal travel time for all chords is a direct consequence of the isochronous property of simple harmonic motion — the same property that makes a pendulum's period independent of amplitude (for small swings).

🛠 Why Can't We Actually Build This?

While the gravity train is a beautiful theoretical construct, several practical obstacles make it impossible with current technology:

• Temperature: The Earth's core reaches 5,500°C (as hot as the Sun's surface). No known material can withstand these temperatures.
• Pressure: At the center, pressure exceeds 360 GPa (3.6 million atmospheres). The tunnel walls would need to resist enormous crushing forces.
• Air Resistance: Even if evacuated, maintaining a perfect vacuum over 12,742 km is impractical. Any air would create drag and heating.
• Coriolis Effect: Earth's rotation would push the falling object against the tunnel walls, requiring magnetic levitation or a curved tunnel.
• Tidal Effects: The Moon and Sun would create slight variations in the trajectory.

Nevertheless, the concept has inspired real proposals for "gravity trains" between nearby cities using shorter, shallower tunnels — essentially a high-tech version of a roller coaster!

📜 Historical Background

The gravity train concept has a rich history in physics and science fiction:

• 1638: Galileo Galilei first considered the problem of falling through the Earth.
• 1687: Isaac Newton's Principia provided the Shell Theorem needed to solve it.
• 1966: Paul Cooper published "The Gravity Train" in the American Journal of Physics, popularizing the chord tunnel result.
• 2015: Alexander Klotz published a refined calculation using the PREM model, finding the ~38 minute travel time.