Awesome question! For now, let's assume that the earth is solid all the way through so you actually can drill a hole through it (there's really tons of liquid mantle in the center which would just fill in any hole once it gets deep enough). For now, let's also assume that there's no air in the hole, so that the stone doesn't feel any resistance as it falls.

So, what happens? Well, you might know that as you get closer to the center of the earth, the force of gravity actually decreases - that's because once you're underground, there's some mass above you that exerts an upward gravitational pull on you. At the center of the earth, the net force of gravity is zero, because the earth's mass is pulling on you equally in all directions (that means you'd be weightless at the center of the earth!).

So when you drop the stone, it'll start accelerating downwards, but as it goes deeper down the hole its acceleration gradually decreases (though its speed keeps growing). Once it gets to the center of the earth, the stone's acceleration reverses - it starts to slow down and accelerate back towards the end of the hole you're standing on. Once it gets to the other end of the hole (on the other side of the world), it slows down to the stop, and starts falling back down the hole again, back towards you. It repeats the process, and you'd see the stone come back up to you from the hole, exactly back to where you dropped it from!

If you do out the math, it turns out it takes a little less than 90 minutes for this cycle to happen. If you don't catch the stone, it'll just keep falling up and down over and over again, taking 90 minutes for each trip.

What if there's so air in the hole? Then the stone would feel some resistance from the air, which would slow it down. It still oscillates back and forth, but each time it comes up from the center of the earth, it gains less height than it did the previous time. Eventually, it ends up stopping right at the center of the Earth, where it stops moving and just floats there (where it's weightless, remember!).

Of course the earth heats up as we go down and the core is at a temperature of about 8000 Kelvin degrees!! So, this is STRICTLY a thought experiment; but if you could do this, the stone would start at surface and accelerate downwards. However its rate of acceleration would gradually decrease so that when it was at the center it would have a high velocity that would be constant. Then as it moved through the other side, the stone would decelerate until it had zero velocity at the other side… only to fall BACK down the hole and accelerate again!! So, it would be like a pendulum going back and forth from the north pole to the south pole and back again forever.

Good question! We would have to make some serious approximations for a reasonable (and perhaps wanted) answer to formulate. Let’s assume we are digging through the earth’s axis of rotation, because otherwise the Coriolis Effect will occur. Let’s also ignore the massive engineering project that would enable this hole-digging project, because digging through thousands of miles of super high-pressure, high-temperature molten core is no easy business. Finally, let’s just assume a vacuum inside of that hole. If we take into account these approximations, then you would expect a rock dropped down this shaft to reach all the way through to the other side of Earth, and keep oscillating back and forth. Here is a good explanation video by Minute Physics on YouTube.

Assuming no air resistance and the hole staying a hole (part of the interior of the earth is molten), the stone would oscillate back and forth through the center of the earth to either end of the hole, much the way that a pendulum oscillates (but with friction).

This is a cool question about gravity! Let us start by assuming Earth has a solid core. In reality, the earth has a liquid mantle layer beneath the crust so we wouldn't actually be able to dig a hole through the center of the planet. Furthermore, let's assume no air resistance once we dig this hole to the other side of the world that we've thought of. Let's also say we drilled the hole so it basically makes a "tunnel" from the north pole to the south pole.

If we drop a stone through the tunnel from the north pole, it will initially accelerate towards the south pole at a decreasing rate as it moves, until it reaches the center of the Earth. Once it reaches the center of the earth, it will decelerate (have a negative acceleration) at an increasing rate as it moves, but still be moving toward the south pole. The reason it will start decelerating is because gravity is weaker closer to the center of the earth. As a rough approximation for why this happens, we can look at the equation for gravitational acceleration as a function of radius:

g=G M/r²

where G is the gravitational constant, andM is the mass contained within the radius r. If we further assume that earth has a constant density:

M = (4/3)*pi*density*r³.

Plugging this in for M in the equation for g would give

g=(4pi/3)G*density*r

As you can see, g is proportional to the radius from the center of the earth. As we approach the center of the earth, r gets smaller, and so does g, the gravitational acceleration. Thus, when the stone reaches the center of the earth, it will actually start to decelerate, even though it is still moving toward the south pole. Once it reaches the south pole, the stone will start accelerating at a decreasing rate toward the north pole. When it reaches the center of the planet again, it will begin to decelerate again, and continue this oscillating motion between the north and south poles!