Re: accelleration/decelleration falling through to the antipodes

Dear Theo,

First of all, forget the 4,000 C. temperature at the core of the Earth. The theoretical hole to antipodes would have to be drilled to connect the north and south poles. Any other such hole would place side accelerations on anyone or anything falling through causing the person or object to slide down one side of the hole instead of performing a clean fall. This is due to conservation of roational momentum. Think of riding a playground merry-go-round and pulling yourself to the center. The overall momentum will stay the same only if the rotational speed of the entire system increases. A single person going through an antipodal tunnel would not speed up the Earth, but the Earth could slow you down by frictional forces, to dissipate the momentum you have gained by riding on the Earth's surface. If you drilled antipodal holes connecting the equator, you would would have to reduce the momentum associated with your body/capsule traveling approximately 1,000 miles per hour at the surface.

If on the other hand, you constructed a tube held vertically upright at the center of the merry-go-round, and climbed down a ladder placed inside it, you would simply turn around in a very tight circle and would not have any additional rotational momentum to dissipate.

Qualitatively, there would be no sensation to the soles of the feet. The person would be in free fall. The sensation would exist throughout the transit. A person in an evacuated tunnel previously described (and contained in a pressurized capsule for life support) would feel the initial loss of gravity as they were dropped. The capsule and its occupant would experience exactly the same acceleration (going down), and decelration (going toward the surface after the capsule passed the center of the Earth). The person would feel nothing but weightlessness. The capsule would reach very near the surface at the opposite pole of the Earth and would then reverse direction to continue bobbing through the center for a very long time, until slowed and finally stopped by the slight bumpings on the inside of the tunnel wall.

Formulas describing the accleration, velocity, and position of the capsule during its trip would most directly require a calculus approach. As the capsule fell into the hole, all of the mass in an increasingly thick shell of the Earth, equal in thickness to the earth above the capsule, would exactly cancel its gravitational attraction on the capsule. The force of gravity would therefore, continue to decline as the capsule approached the center of the Earth, fianlly disappearing completely as the capsule passed the exact center. This is the point in the transit of maximum velocity. As the capsule continued its upward half of the journey, the force of gravity would then increase at precisely the rate matter was collecting under the capsule, thus slowing the ascent toward the surface.

Set up your equations such that they describe the force of gravity at the initial conditions (approximately 4,000 miles to the center, with the entire mass of the earth concentrated at that point. Be sure to include factors that will subtract the mass of a shell of Earth that has been traversed, and finally, calculate the change in velocity as a function of this decreasing force. The acceleration is the first derivative of velocity and velocity is the first derivative of position. The simple linear equations of position, velocity, and acceleration are: X=1/2gt^2+Vot V=gt + Vo A=g. Yours will be somewhat more complex.

I wish you luck. This is not a straight-forward problem but I am sure that you will be able to solve it if you keep these simple descriptions in mind.