I feared interest in this delightful problem had died so thank you.

All the necessary information is in the OP and in post#14

I gave function the formula he requested, which is easily goolgable or derivable as John Cuthber said.

You will also need this.

Is the length of the tunnel or its depth relevant?

Assuming that the tunnel is straight through the earth, and that both sides of the earth would be at the same level above sea level, and that it was all solid glass(picking a smooth near friction-less element.)

Rolling a ball into it would be equivalent as rolling it down a hole with a hill at the other end.

The effect of gravity acting on the ball would change, and would even itself out going both up and down.

So if I derive an equation from the conservation of energy(you said no friction, so I can do this without altering the problem) then this problem becomes elementary.

The equation I got was:

**Edit: **How the heck do you attach images???????

Edit 2: I'm now testing ways to insert equations. Do not quote this post.

\[\large gh_1+(v_1^2)/2=gh_2+ (v_2^2)/2\]

Edit 3:

\[\large gh_1+(v_1^2)/2=gh_2+ (v_2^2)/2\]

Edit 4:

[latex]
\[gh_1+(v_1^2)/2=gh_2+ (v_2^2)/2\]
[/latex]

Edit 5:

[**LaTeX Error:** Syntax error]

Edit 6:

\[gh_1+(v_1^2)/2=gh_2+ (v_2^2)/2\]

Edit 7:

[**LaTeX Error:** Syntax error]

Edit 8:

You are not allowed to use that image extension on this community.

Edit 9:

You are not allowed to use that image extension on this community.

What is the point of having the ability to imbed urls into images to put them in a post, if you're not allowed to do it .......

Edit 10:

CLOSE ENOUGH DAMN IT

Edit 11:

Edit 12:

You are not allowed to use that image extension on this community.

Edit 13:

Edit 14:

You are not allowed to use that image extension on this community.

I give up with this thread, on account of it being impossible to add images. The site doesn't trust me enough to add image extensions.

Edit 15:

You've got to be kidding me.............

Edit 16:

[**LaTeX Error:** Syntax error]

Edit 17:

Almost..........

Edit 18:

YESSSSSS!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! YES YES YES!!!!!!!

Now, taking that equation, we can calculate the velocity when it reaches the center of the earth.

It's velocity would be 0 when it reaches the other side. Considering there's no friction.

I'm not doing this tonight.

**Edited by Raider5678, 17 May 2017 - 12:12 AM.**