Q: If you could drill a tunnel through the whole planet and then jumped down this tunnel, how would you fall?

Physicist: This is a beautiful question, in a small part because it’s an interesting thought experiment with some clever math, but mostly because of all the reasons it couldn’t be done and wouldn’t work.  Right off the bat; clearly a hole can’t be drilled through the Earth.  By the time you’ve gotten no more than 30 miles down (less than 0.4% of the way through) you’ll find your tunnel filling will magma, which tends to gunk up drill bits (also melt everything).

Jumping into a hole drilled through the Earth. What’s the worst that could happen?

But!  Assuming that wasn’t an issue, and you’ve got a tube through the Earth (made of unobtainium or something), you still have to contend with the air in the tube.  In addition to air-resistance, which on its own would drag you to a stop near the core, just having air in the tube would be really really fatal.  The lower you are, the more air is above you, and the higher the pressure.  The highest air pressure we see on the surface of the Earth is a little under 16 psi.  But keep in mind that we only have about 100 km of real atmosphere above us, and most of that is pretty thin.  If the air in the tube were to increase in pressure and temperature the way the atmosphere does, then you’d only have to drop around 50 km before the pressure in the tube was as high as the bottom of the ocean.

Even worse, a big pile of air (like the atmosphere) is hotter at the bottom than at the top (hence all the snow on top of mountains).  Temperature varies by about 10°C per km or 30 °F per mile.  So, by the time you’ve fallen about 20 miles you’re really on fire a lot.  After a few hundred miles (still a long way from the core) you can expect the air to be a ludicrously hot sorta-gas-sorta-fluid, eventually becoming a solid plug.

But!  Assuming that there’s no air in the tube, you’re still in trouble.  If the Earth is rotating, then in short order you’d be ground against the walls of the tunnel, and would either be pulverized or would slow down and slide to rest near the center of the Earth.  This is an effect of “coriolis forces” which show up whenever you try to describe things moving around on spinning things (like planets).  To describe it accurately requires the use of angular momentum, but you can picture it pretty well in terms of “higher things move faster”.  Because the Earth is turning, how fast you’re moving is proportional to your altitude.  Normally this isn’t noticeable.  For example, the top of a ten story building is moving about 0.001 mph faster than the ground (ever notice that?), so an object nudged off of the roof can expect to land about 1 millimeter off-target.  But over large changes in altitude (and falling through the Earth counts) the effect is very noticeable: about halfway to the center of the Earth you’ll find that you’re moving sideways about 1,500 mph faster than the walls of your tube, which is unhealthy.

The farther from the center you are, the faster you’re moving.

But!  Assuming that you’ve got some kind of a super-tube, that the inside of that tube is a vacuum, and that the Earth isn’t turning (and that there’s nothing else to worry about, like building up static electricity or some other unforeseen problem), then you would be free to fall all the way to the far side of the Earth.  Once you got there, you would fall right through the Earth again, oscillating back and forth sinusoidally exactly like a bouncing spring or a clock pendulum.  It would take you about 42 minutes to make the trip from one side of the Earth to the other.

The clever math behind calculating how an object would fall through the Earth:  As you fall all of the layers farther from the center than you cancel out, so you always seem to be falling as though you were on the the surface of a shrinking planet.

What follows is interesting mostly to physics/engineering majors and to almost no one else.

It turns out that spherically symmetric things, which includes things like the Earth, have a cute property: the gravity at any point only depends on the amount of matter below you, and not at all on the amount of matter above you.  There are a couple of ways to show this, but since it was done before (with pictures!), take it as read.  So, as you fall in all of the layers above you can be ignored (as far as gravity is concerned), and it “feels” as though you’re always falling right next to the surface of a progressively smaller and smaller planet.  This, by the way, is just another reason why the exact center of the Earth is in free-fall.

The force of gravity is F = -\frac{GMm}{r^2}, where M is the big mass, and m is the smaller, falling mass.  But, since you only have to consider the mass below you, then if the Earth has a fixed density (it doesn’t, but if it did) then you could say M = \rho \frac{4}{3}\pi r^3, where ρ is the density.  So, as you’re falling F = -\left(\frac{Gm}{r^2}\right)\left(\rho \frac{4}{3}\pi r^3\right) = -\left(\frac{4}{3}G\rho \pi\right) mr.

Holy crap!  This is the (in)famous spring equation, F = – kx!  Physicists get very excited when they see this because it’s one of, like, 3 questions that can be exactly answered (seriously).  In this case that answer is r(t) = R\cos{\left(t\sqrt{\frac{4}{3}G\rho \pi} \right)}, where R is the radius of the Earth, and t is how long you’ve been falling.  Cosine, it’s worth pointing out, is sinusoidal.

Interesting fun-fact: the time it takes to oscillate back-and-forth through a planet is dependent only on the density of that planet and not on the size!

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80 Responses to Q: If you could drill a tunnel through the whole planet and then jumped down this tunnel, how would you fall?

  1. alan morriss says:

    if the hole was drilled exactly from pole to pole rotational forces would be zero. the difficulties posed by this would be insignificant compared to the rest.

  2. In a YouTube video Neil deGrasse Tyson is shown falling through a hypothetical Earth-tunnel:

    https://www.youtube.com/watch?v=NOHBDiR5urE

    Tyson verbally repeats the predictions of the equations provided by “The Mathematician” (i.e, 84 minute oscillation period, dependent only on density, etc.).

    But we DON’T NEED to drill a hole through the EARTH to find out if the equations are correct. It is important to realize that we CANNOT KNOW whether the equations are correct or not until we actually DO a scaled down version of the experiment. As I’ve posted above, such a test is quite feasible.

    The STORY of falling through the Earth is so common that even physicists fail to acknowledge its MYTHICAL character. We do not yet have any empirical evidence to back it up.

    If we are only interested in ENTERTAINING a mostly rather gullible audience, then we will be satisfied with equations, hand-waving, video simulations and the like. Whereas if we abide by the EMPIRICAL IDEALS of science, then we will set out to actually PERFORM a scaled down version of this experiment that Galileo proposed so long ago.

  3. Kay says:

    @ Richard Benish
    Two problems with a straight hole from pole to pole is the Earth wobbles on its axis and so there still would be rotational forces. The other problem is convincing the Flat Earth lunatics there is a South Pole.

  4. gary peters says:

    Sounds just like religion.

  5. Cd says:

    I am a third generation driller..so the drilling part I understand. One the rig u would need would have to be about the size of Colorado. For the gear ratio and motor to turn the pipe with weight at let’s say 20 miles ( over 2x the deepest hole drilled) one it would take probably hundred years just to drill that depth for remember when making a hole ur just dispersing the matter to a different area. So getting this material above ground at that depth would I would guess 1ft of hole at 10k would take the material about day and a half to come to the surface.now question of the drill pipe 🙂 ..to withstand that pressure and torque to keep from shaping off well I know of no such materials.

  6. Kay says:

    Really, the hole wouldn’t be dug conventionally, everyone knows you would use a particle beam plasma rig.

  7. The gravitational principles that would be revealed by doing a scaled down version of the experiment are the same as what would be revealed by digging a hole through a planet. As the last few posts indicate, the planet idea involves lots of fantastic stuff. Whereas the scaled down version is easily within the scope of feasibility.

    So why not just do the scaled down version? It has not yet been done. Galileo proposed the idea 383 years ago. The experiment is not only feasible, it would be cheap compared to many other experimental adventures that physicists sign up for and taxpayers pay for. Why keep the spirit of Galileo waiting any longer?

  8. ron jamed says:

    I just wanted to know how many miles it would be if a hole was drilled through the earth from pole to pole

  9. Kay says:

    @ron jamed,
    C = 25,000 miles; Earth’s approximate circumference
    d = C/π
    d = 25,000/π
    d= 7961.783439490446 [approximate diameter in miles]

  10. Stephen Bryant says:

    Hail a neutrino taxi; you’ll be there in no time (approximately)

  11. anantbir says:

    gravity is a straight line in a curved space time therefore the equation

    f=-kx is not possible

  12. Talk, talk, talk.

    Why not let’s hear what Nature has to say?

    A scaled-down version of the experiment is well within our technological grasp.

  13. anantbir says:

    if a person jumped into the tunnel there wouldn’t be much gravitational force attracting the person falling down because he wouldn’t be attracted by earth’s gravity (at least not to the ground) as he has drilled a tunnel passing through the earth.

  14. Stephen Bryant says:

    I have a related question. Suppose it was not a person passing through the hole, but a small black hole, for example one 100 times the mass of the earth. What then?

    Note that the black hole would make its own tunnel, and I think it would be about a meter in size. But instead of the black hole falling through the earth, the earth would oscillate around it, at least it would as long as the planet survived.

  15. Would shmould.

    All these guesses need to be checked against Nature.

    A scaled down version of the experiment is well within our technological grasp.

    Does reality matter? Or do we just do math here?

    The configuration may vary. At the bottom of it all, the unanswered question remains: Gravity-induced free-falling radial oscillation–does this happen in physical reality, or is it only in our minds?

    We do not yet know, but are entirely capable of finding out. Why don’t we?

  16. Michael Fuller says:

    Does the 37 or so minutes it is said to do this, to fall through the Earth, account for all of the gravitational differences and nuances throughout the fall and return from the fall during the oscillation, when presuming near the center of the core it is weightless allwhile on the surface is normal gravity?

    Hmm

  17. The Physicist The Physicist says:

    @Michael Fuller
    Yup!

  18. jesus varela says:

    You guys should read a book named The Krone Experiment. You will like it.

  19. Gravity Levels says:

    But if there are rotational forces, wouldnt you just go a little past the core and come back a little past the core forever? Sorry if someone already said this,

  20. Robert says:

    So, if we ignore friction and air, but keep the earth rotating and started digging right on the equator… what curve would the hole have to follow so that the jumper always stayed in the middle of the shaft?

  21. Buzz says:

    Interesting thing is that it would take 42 minutes to go between any two points on Earth if you dug a straight tunnel and were able to slide without friction.

  22. ANDREW says:

    I Am Ready To Fall. Where Is The Hole Then

  23. Chris says:

    a question for all you physicists out there, if you drilled a hole into the side of a mountain when you got to the center going sideways, would the air pressure be greater in the middle of the mountain than the outside, assuming you went L to R not up or down from the base?

  24. Robert Stevenson says:

    No – identical. The air pressure is a function of the amount of air pushing down from above and it is isotropic (same in all directions at any single point). Therefore the pressure at any point in a tunnel that is open to the atmosphere is not dependent on rock mass above it but only on the mass of air above it (which is largely dependent on that point’s distance from the edge of the atmosphere.)

  25. Kay Gee says:

    You didn’t stipulate conditions: A tunnel with a draft (windy day) will act somewhat act as a venturie; in such a case, the air pressure will be less inside the tunnel than the outside. Any convergence or divergence of the tunnel will cause differences in pressures. Temperature and humidity also changes air density. As a former coal miner that has punched in one side of a mountain and out the other side, I know that differential and equal pressures will exist between the inside and outside of a mine at different times of the day; with either pressure being higher, lower, or equal. Yet coal mines do force airflow with an exhaust fan.`To answer your question correctly, you need to first provide all specific details that determine a correct answer.

  26. Embarae says:

    So all the theorist ideas aside…..if one fell though the hole in the earth, would you come out on the other side, feet first or head first? Simple question. I would like to know.

  27. Robert says:

    Quick answer would be if you got yourself into a fall where you were continually going feet first and not tumbling, then (if you came out, which you wouldn’t) you would come out feet first. If you dived in and managed to stay head first, you would come out head first.

    The only thing that would turn you would be inertia (i.e. if you had a tumble going you would keep tumbling) and the force of air on your body. The force of air would be bay far the dominant force in this situation. It would tend to turn you around if you changed your position as you would get different drag forces on different parts of your body.

    If you were in a vacuum and you were tumbling, you would just keep tumbling unless you could work out some complex arm rotations that counteracted the angular momentum of your body.

  28. tarun says:

    as the above topic if we really jump into the tunnel then after getting deeper and deeper do we come up back to earth surface because of escape velosity or does our earth floats in orbit because of gravity which implies by the tunnel do we enter space? if no what happens to us

  29. hb says:

    Does scientists have proof that the earth’s core is solid?

  30. Fergal reid says:

    If earth was cold and solid I think it never could b to drive a tunnel through the earth . The pressure of the earth would crush us in and if you could go through the centre you would stall at the centre gravity would pull back to the centre . Once you track the centre the force of gravity will reverse you will head on a upwards direction it’s an amazing question to answer .

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