[cripplecreek]

You accelerate all the way to the center then begin to slow as you pass it. You slow to the apex right at the surface on the other side.

Aside from the impossibility of it, it would be a great transportation method. Maybe it would work on geologically inactive moons.

[gorush]

"You accelerate all the way to the center then begin to slow as you pass it. You slow to the apex right at the surface on the other side."

As a retired skydiver, I must question that assertion. A skydiver accelerates at first, but then acceleration slows as one approaches terminal velocity. Terminal velocity is dependent upon altitude (actually the thickness of the air mass at any given altitude) Terminal velocity is at its slowest the closer you get to the earth and presumably slower yet at the earth's center. (assuming our magical tunnel has air but no magma)

[Vinnie]

You accelerate all the way to the center then begin to slow as you pass it. You slow to the apex right at the surface on the other side.

That was my thought also.
Kinda like stepping on an elevator on the ground floor and stepping off on the 26th floor.

[Hoodat]

You accelerate all the way to the center then begin to slow as you pass it.

Correct. Maximum velocity would be achieved as the object passes through the center. The maximum rate of acceleration (from x to y) would be realized at the start. The initial acceleration would be 9.8 m/s². At the earth's center, it would be zero, and at the opposite end of the earth, it would be -9.8 m/s².

The object would not begin to slow until acceleration goes negative which would occur just after passing through the center point.

[boop]

"You accelerate all the way to the center then begin to slow as you pass it. You slow to the apex right at the surface on the other side."

Yes, that's what our physics professor said in college. You'd (theoretically) oscillate from one end to the other.