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"This category [science fiction] excludes rocket ships that make U-turns, serpent men of Neptune that lust after human maidens, and stories by authors who flunked their Boy Scout merit badge tests in descriptive astronomy."
- Robert Heinlein

Gravity-assisted Subway  
  The fastest way between continents (that does not entail leaving the ground).  

This piece of technovelgy gives a great illustration of the fastest way to get from point A on the surface of the earth to another surface point B.

The car shot forward and down.

Corbell was on a roller coaster. He pulled out a chair arm and hung on. The car fell at a slant for what felt like half a minute. Then there was high gravity as car and tunnel curved back to horizontal.

Light inside, darkness outside. Corbell was beginning to relax when the car rolled, surged to the left; rolled, surged to the right, steadied. What was that? Changing tunnels?

His ears popped.

Peerssa spoke. "Your speed is in excess of eight hundred kilometers per hour and still accelerating. A remarkable achievement.

"How do they do it?"

"At a guess, you are riding a gravity-assisted linear accelerator through an evacuated tunnel. You are about to pass beneath the Pacific Ocean..."

Technovelgy from A World Out of Time, by Larry Niven.
Published by Random House in 1976
Additional resources -

The ingenious solution is to dig a tunnel from one point to the other, using the force of gravity to provide some of the acceleration. The beauty of this approach is that, in a frictionless environment, you can speed the train on its way without having to apply any energy - and slow it down the same way!

The questions (for you engineers) are: what is the best shape (or curve) for the tunnel, and how can you avoid those pesky underground temperatures and pressures? See this article on the brachistochrone problem, one of the earliest problems posed in the calculus of variations. Newton solved this problem from scratch in one day in 1696; got your pencil? Ready, set ... go!

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Additional resources:
  More Ideas and Technology from A World Out of Time
  More Ideas and Technology by Larry Niven
  Tech news articles related to A World Out of Time
  Tech news articles related to works by Larry Niven

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