March 26, 2015 weblog

# New way to calculate how long it would take to fall through a hole in the Earth

*The American Journal of Physics.*

If someone were to drill a hole all the way through the planet, and then somehow manage to fall into it, how long would it take them to arrive on the other side? That is a physics question put to students every year, and those who give it expect the answer to be 42 minutes. But is that answer correct? Klotz says no and has the math to prove it, *Science* reported.

The accepted answer of 42 minutes takes into account the constantly changing impact that gravity will have (and ignoring drag due to the presence of air) on the person falling, becoming less and less of a factor as the center of the Earth is approached then growing stronger and stronger as the person heads "up" against gravity on the other side. It is accepted that the speed attained during the descent on the first half of the journey would be significant enough to cause the person to continue moving against gravity on the other side of the planet, right up until the surface is reached.

But Klotz argues that it is time to start taking the different densities of the Earth's layers into consideration—after all, a lot of research has shown that our planet is a lot denser at the center than at the crust for example—and that of course would have an impact on the person falling through. He has used seismic data to calculate the different densities at different depths and has used that data to give a more accurate answer to the falling man question, stating that it would in fact, take just 38 minutes (and 11 seconds) to fall all the way through, not 42 and (12 seconds).

Interestingly, Klotz also notes that if gravity were to be assumed to be at a surface level constant throughout the duration of the trip, the math shows it would take just 38 minutes as well.

Explore further

**More information:**The gravity tunnel in a non-uniform Earth,

*Am. J. Phys.*83, 231 (2015); dx.doi.org/10.1119/1.4898780 . On

*Arxiv*: arxiv.org/abs/1308.1342

**Abstract**

This paper examines the gravity tunnel using the internal structure of Earth as ascertained from seismic data. Numerically, it is found that the time taken to fall along the diameter is 38 min, compared to 42 min for a planet with uniform density. The time taken to fall along a straight line between any two points is no longer independent of distance but interpolates between 42 min for short trips and 38 min for long trips. The brachistochrone path (minimizing the time between any two points) is similar in shape to the uniform-density solution but tends to reach a greater maximum depth and takes less time to traverse. Although the assumption of uniform density works well in many cases, the simpler assumption of a constant gravitational field serves as a better approximation to the true results.

© 2015 Phys.org

**Citation**: New way to calculate how long it would take to fall through a hole in the Earth (2015, March 26) retrieved 21 May 2019 from https://phys.org/news/2015-03-fall-hole-earth.html

## User comments

billpress11ExpiorerYou will be exposed to larger mass at center longer time and reach center faster.

billpress11I would agree with that, but wouldn't the deceleration, once beyond the center, be equal to the acceleration gained approaching the center?

TaterbugScience OfficerTheGhostofOtto1923If the person were allowed to oscillate back and forth, eventually he would come to rest at the center.

TheGhostofOtto1923"this is a perfect case of a so called perpetuum mobile. It would respresent a perfect (ideal) non-dissipative system where entropy production diS/dt=0, in accordance with the 2nd law of thermodynamics. Indeed the first law of thermodynamics (energy conservation) does not say much about this, except that no term for energy loss included."

-But in a less perfect case I would think that the movement would somehow affect the earths rotation and thus be slowed.

gkam---------------------------------------

Wouldn't it then be boosted by the approximately the same amount when it reversed direction?

adam_russell_9615True, so consider only how long it takes to reach center then multiply by 2.

As you proceed toward center the rate of acceleration diminishes until it reaches zero at the center. Consider a case where most of the mass is very close to the center. Then your acceleration wont diminish much until you are almost all the way to the center. Your trip from surface to center will be much quicker.

billpress11Raygunnerwalter_marvinMaydayThe pencil on my desk is drawn toward the Earth's center of mass with a force equivalent to the mass "below" it minus any other commingled mass attractions(the Moon comes to mind, for one), not toward some magical point at the Earth's center that exerts a continuous attractive force no matter how close it is to it.

If there is air in the tube, all bets are off. Then even the diameter of the tube plays a part.

A better thought experiment might be to let the tube fill with air and calculate the depth at which the air reaches maximum density. And what might that air density be? Any takers?

gkambillpress11adam_russell_9615:-)

shaveraMore-or-less, since so much of the mass is in the very dense center, and one travels fairly quickly through that center, the gravitation near the surface (on either end) has the longest time to act (and thus a larger impulse on the momentum), and so the acceleration being *roughly* the same as surface acceleration is a fairly decent estimate.

Victor G_baby-pandaFrank99As you fell, more and more mass would be above you countering any pull 'down' until you eventually came to a stop at the center, completely weightless because you're being pulled equally in all directions.

TopCat22... based on this observation it would be obvious that if using the gravity at the surface of both ends whatever exists in between the ends would be averaged out to be zero effect since whatever is happening going down happen coming back up the other side thus the 38 minutes would be shown to be more correct than the official 42.

ilya_simkhovichROBTHEGOBZ99impatrickoI'm dying for someone to tell me how this train of thought is wrong.

OceanDeepSeriously, though, it's good to see someone take a problem and think it through in this much detail. It's a nice counterpoint to the typical thoughtlessness of most behavior these days.

OceanDeeploneislanderIf one were to mathematically 'build' another planet with an arbitrary availability of material densities, it seems somehow intuitive that they ought to show a similar correlation. One ought to be able to prove an upper limit in the variance between the two. (a ball of pure hydrogen would leave the two equations with precisely the same result)

jonesdensonTo assume that you could reach the other side of the planet is an argument for perpetual motion, which is impossible.

Also in the case of falling down the first half, friction is working to resists the fall, while gravity is working to propell you to the other end; but after the first half both forces are working to keep you down.

vidyunmayaAppeal : Please do no dig a Graveyard hole in the earth to verify which calculations are right or wrong.

Please put Energies on "How to save earth planet and life Support'

adam_russell_9615If you tried it at the equator you would hit the side of the hole as the earth rotates and you do not.

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