04 Oct '07 14:24

With our discussion of the idea of driving a tunnel through the center of the earth, evacuating the air for said tunnel, and dropping an object, maybe a train loaded with passengers and freight, it reaches some maximum velocity dead center of the earth and then is decellerated with the exact same value of force and it reaches the other end of the tunnel on the surface with zero velocity, so you apply the brakes, unload, reload, and you are good to go again. Ignoring the engineering problems in such an impossible venture, ignoring friction, since we specified an evacuated tunnel, what is the average acceleration value and thus what is the final velocity as it hits the center of the earth. I made one assumtion that allowed me to calculate a number but I don't know if my assumtion is correct:

I visualized a graph of the mass as you go down and you see that immediately you have mass above you and below you so the accel rate has to start going down, so I artificially pegged the average at 50% of surface acceleration numbers, 32 Ft/sec^2 or 9.8 M/sec^2 and divide those by two and you get 16 Ft/sec^2 or 4.9 M/sec^2 and simplifying the distance to the center of the earth to 4000 miles or 6400 Km, I came up with a velocity at the center of just under 8,000 meters per second or 26,000 F/sec or 17,000 miles per hour.

Is my assumtion of the average at 50% correct? It might be the square root of it or something, don't know. Any help? How would you calculate the total trip time, surface to surface?

I visualized a graph of the mass as you go down and you see that immediately you have mass above you and below you so the accel rate has to start going down, so I artificially pegged the average at 50% of surface acceleration numbers, 32 Ft/sec^2 or 9.8 M/sec^2 and divide those by two and you get 16 Ft/sec^2 or 4.9 M/sec^2 and simplifying the distance to the center of the earth to 4000 miles or 6400 Km, I came up with a velocity at the center of just under 8,000 meters per second or 26,000 F/sec or 17,000 miles per hour.

Is my assumtion of the average at 50% correct? It might be the square root of it or something, don't know. Any help? How would you calculate the total trip time, surface to surface?