Falling Through The Earth Just Got Quicker
For decades, it’s been accepted that jumping in a hypothetical hole through the earth would take 42 minutes to traverse the entire diameter. A new, more realistic calculation however, taking into account varying earth densities, shows the trip would actually take only 38 minutes.
This common physics classroom and cocktail party question is as impractical as it is audacious. Imagine a tunnel through the core of the earth connecting opposite sides of the planet; London to the Anitpodes islands for example. What would happen if something went down that ultimate rabbit hole? The somewhat troubling engineering and physics problems are of course ignored. The 3 million atmospheres of pressure and 6,000 degrees of celsius at the core are all but insurmountable if not completely so. To simplify matters further it is also generally assumed that all air has been evacuated from the tunnel as well.
Despite all these caveats, the answer though is still instructive and interesting. The trip to the other side of the planet would clearly be fast as hell and it would require no power source at all other than the ubiquitous and free gravity to pull it off (except of course the force field and super-air-conditioners). For years the standard calculation revealed the following:
Gravity would accelerate you, unhindered by air friction, to about 29,000 kph (18,000 mph).
Gravity would then start slowing you down after you passed the core since it would be at that time that the majority of earth’s mass would be increasingly behind you.
You would stop your deceleration and hang momentarily in the air (like bugs bunny I suppose) just outside the tunnel exit on the opposite side of the planet.
The entire trip would take 42 minutes
The other side of the planet in only 42 minutes. This is faster than the time it takes the International Space Station to complete half an orbit. A jet would have to fly ummm…. really really super fast to get you there in that amount of time (Mach 25 actually).
So why has this thought experiment remained popular if you have to suspend disbelief so much? First of all, it’s cool to contemplate, ok? If that’s not good enough for you then it also can help teach physics principles like the simple harmonic motion of a weight bouncing on a spring or the back and forth movement of a pendulum. We’ve all seen the experiment of the pendulum being released downward starting from right in front of someone’s face, only to return, risk free, almost to the point it was let go.
The new calculation of this hypothetical tunnel trip came from the mind of Canadian scientist Alexander Klotz which is published in the American Journal of Physics. All previous accepted calculations assumed a uniform density of earth like it was a meatball. Klotz however used seismic data from what’s called the Preliminary Reference Earth Model. This shows a change in density of the interior of the earth ranging from 1,000 kg per cubic meter at the surface to 13,000 kg per cubic meter at the core. There’s also a significant jump in density near the outer core. Plugging these numbers into the now more realistic equation showed a result of 38 minutes 11 seconds compared to the usual 42 minutes and 12 seconds.
Surprisingly, it was noticed that if you perform this calculation and ignore the seismic data and just assume that the force of gravity is constant throughout the trip, you’d end up with a trip-time of exactly 38 minutes, only eleven seconds off the new result. This very simple assumption of a constant force during the fall is more accurate than the original similarly simple assumption of a uniformly dense earth (the meatball). Why does this work so well? Because the real-world increasing density of the earth actually keeps the force of gravity fairly constant as you fall through the tunnel. Once you cross the boundary of the core, this changes and the pull of gravity weakens but by the time you enter this part of the earth’s interior, you are going so fast that you spend relatively little time there. The end result then is that the constant-force-assumption results in an answer very close to the 38 minute 11 seconds solution.
It has often been said that falling down a tunnel takes the same time whether it’s the longest distance (through the center) or a shorter one to a location, say, only a quarter of the way around the planet. This was said to happen because even though you don’t travel as fast, you have less distance to travel which balances out to the same travel time regardless of tunnel length. This symmetry was true with the original solution but the new one shows that this balance is lost causing shorter tunnel trips to balloon back to 42 minutes.
These shorter tunnel trips always seemed weird to me since I’d expect gravity to cause you to unavoidably drift towards and impact the wall of the tunnel that was facing the largest portion of the earth.
Oh well, one more thing to suspend my disbelief about.