Question

To find max velocity when velocity at centre is given

Answer 1

This is probably a far more complex question than you may have considered.

When gravity “pulls you down” to the earth’s surface, it is the entire mass of the earth pulling at you. Some of it pulls off at an angle (that hill over there is attracting you with its minuscule gravity) but on the whole, you are pulled down.

Now, lets place you at the center of the earth. Roughly speaking, there is the same mass pulling at you from every direction, and you have no preferred direction to fall. If the earth was a perfect, uniform sphere, then at the center all of the gravitational pull from all of the mass around you balance out, and you would be effectively “weightless.”

So, while falling towards the center of the earth, the mass of the earth that you had already fallen past would be pulling you away from the center of the earth. The gravitational acceleration would be decreasing as you fell, and you would at some point start to fall slower and slower, as air resistance would not be overcome as easily. When you made it to the center of the earth, you almost certainly would not become motionless, because momentum would carry you past the balance point.

Basically there are two answers. One assumes that you ignore any effects of air resistance, in which case your velocity at the center should be about escape velocity for the earth. If you try to factor in air resistance, your speed is going to be somewhat below normal terminal velocity for a human body. My best estimate is about 20 mph, but the math is complex and I’m not sure that I did it correctly.