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If tunnel is dug from North pole to equator, and rails installed, what will be minimal travel time?

Lets assume that Earth is spherical and has uniform density. If tunnel is dug from North pole to equator, and rails installed in the tunnel , what will be minimal travel time of a street car freely moving on the rails without friction? If the tunnel is straight, then travel time will be π√(R/g) = 42 minutes. Can we beat this time by choosing a tunnel of other non-straight shape?

Public Comments

1. Since you're assuming that the earth is spherical then the straight tunnel from the north to south pole would be the fastest.
2. You are talking about the concept of a "gravity train". The curious thing about this is that for half of the trip you will be accelerating to the "low point" in the tunnel (closer to the center of the earth), then decelerating until you arrive at the equator (with a velocity of zero). Total time is 42 minutes 12 seconds. This is good, but there is actually a better design for your tunnel. Taking advantage of the acceleration of gravity, it actually makes sense to have a tunnel that isn't so straight but is instead bent toward the center of the earth more. This would be a hypocycloid. The distance would be slightly more, but offset by the higher average velocity. I'd have to think about how to compute the actual time on this new tunnel but it would indeed be less. Check Wikipedia and read up on "gravity trains" and "hypocycloids" for more information.