I do something difference today, ha.

The problem is simple, How long does it take to tunneling through the center of the earth to the other side?

Assuming uniform density of the earth (), the acceleration inside the earth at radius is

in 1/sec^2.

because only the mass within the radius matter. Thus, the equation of motion is

the solution is

Thus, the time for a trip is = 2530.5 sec or 42 min and 10.5 sec.

The maximum speed when passing the core. The speed is

How about we use a realistic earth density?

The density, and acceleration can be found in the web, for example, here.

The travel time is 2291 sec, or 38 min 11 sec.

It is interesting that, in the uniform density calculation, the travel time is independent of the radius, but density. The peak velocity is

Let compare with Schwarzschild Radius:

The Schwarzchild density is half of the maximum density by classical argument.

In case of point mass.

The equation of motion is

change of variable

The above solution assume . We can see that, at , the speed go to infinity. Something wrong…..

Update: that is not wrong at all. imagine two neutrino with head-on collision, the released gravitational energy will be infinity! But, because of uncertainly principle, their separation distance can never to be zero.

Taking account of the earth rotation with the centripetal force, the equation of motion becomes,

where is the angular velocity of earth, and is the polar angle from the north pole. The travel time would be

rad/s

Thus,

1/sec^2.

which is a very small correction.

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John Compter

Aug 23, 2018@ 02:07:47Dear Sir,

Can you give the background behind the first equation, giving the acceleraion entering the earth?

GoLuckyRyan

Sep 05, 2018@ 01:53:35This is because the gravitational force inside a spherical shell (or hollow sphere) is cancelled. This can be proved easily by Gauss’s Law. Thus, when we are in radius r, we only need to take into account of the mass smaller than r but not larger. https://en.wikipedia.org/wiki/Shell_theorem

knowing that, the gravitation acceleration is

GM(r)/r^2

write M(r) = 4 pi / 3 * rho * r^3.