Let’s say you managed to dig to the center of the Earth, took some photos for Instagram, then decided you might as well continue, having come this far, and dug to the other side as well. First of all, you’d drown, because the antipode of most locations is sea: for Eindhoven, it’s a spot a few hundred kilometers south of New Zealand.
Assuming you didn’t, and you decided to come back, jumping into the hole you just dug — what would your journey down look like? This is interesting because we clearly can’t assume g(r) in the equations of motion is constant: it goes to zero in the center of the Earth, then gradually increases as you get to the other side.
Plugging in the solution we found previously:
But this we recognize! It’s a kind of Hooke’s law, so we can expect an oscillatory solution. Still, let’s proceed from basic principles for the fun of it. The second derivative of r is -r, with some constants in front. In cases like this, we usually have either an exponential function, or a sinusoidal one. Since we flip the sign, it’s definitely sinusoidal.
To find A, ω and δ, we can consider the initial and boundary conditions. We jump into the hole, so at t=0, we’re at the radius of the Earth. That means we should have used a cosine (so δ=π/2), and that A is radius. Also at t=0, the acceleration is the acceleration at the surface of the Earth, so we know ω as well.
If we put all of this together, we get the following:
This has the correct behavior: at the surface, we have normal gravity. Over time, you fall down, pop up on the other side, then fall down again, indefinitely.
The first root is at 1267.342 seconds, so it takes about twenty minutes to get down there, then another twenty minutes to get up again.
We can reuse the solution to also find the velocity:
It turns out you go pretty damn fast. 8000 m/s at the center of the Earth is about Mach 23, or 28.800 km/h. You better hope there is no air resistance, otherwise you’re getting a little toasty — if you run the numbers, the stagnation temperature is around 23.138 K.
Across (or through) the Earth in forty minutes: definitely beats Verne’s 80 days! Better hope you put some handles on the other side though, otherwise you’re going to be going back and forth for quite a while.