If I was an indestructible superhero in the exact center of the core of the Earth and I spit, which way would it go?
As soon as you spit the *cough* projectile (spittle/ phlegm/ sputum) would leave the centre and be travelling at a certain speed and direction into an increasing gravitational field. (This is assuming you spat, rather than merely dribbled... Please Register or Log in to view the hidden image!) So it would continue in that general direction, tending to become more and more "vertical" (normal to the tangent) as the increase in gravity raised its speed toward the inside of the Earth, shedding some speed in the original direction due to air resistance (assuming there's air there, of course).
LOL, I have no idea what you said. I never knew there was a gravitational field in the exact center of the earth. Thanks!
Not as such at exact centre*, but there'd be gravity pulling "outwards" toward the inside of the shell. * Well yes there is, but at the exact centre it's pulling equally in all directions, so effectively it's cancelled.
In the direction you spat. Assuming there was nothing in the way, it would move away from you in that direction, slow down, stop, then come directly back towards you and hit you in the mouth.
When you throw a ball straight up, does it go sideways or come straight back down? This is the same thing.