# Mathematics: To Everywhere in 42 Minutes

Although he is now 36, and a mathematician for Sylvania, Paul Cooper has never lost his boyhood enthusiasm for the fanciful science-fiction stories of Jules Verne. While musing about Journey to the Center of the Earth several months ago, Cooper himself took off on a mathematical flight of fancy that more than rivals Verne's most imaginative work. By crisscrossing the earth with subterranean tunnels, the freewheeling mathematician proposes in the current issue of the American Journal of Physics, man could achieve intercontinental travel at ballistic missile speed.

Verne's underground hike would, have taken far less time, says Cooper, if he had simply fallen into a frictionless tunnel bored through the earth's center. Accelerated by the force of gravity on the first half of his trip, he would have gained just enough kinetic energy to coast up to the other side—against the pull of gravity—in a total time of only 42.2 minutes.

Universal Timetable. Fascinated by his initial calculation, Cooper worked out a formula for the time required for an object to fall through a straight-line tunnel bored between any two points on the surface of the earth. Surprisingly, no matter how close or far apart the two points were, the time turned out to be constant: 42.2 minutes.

According to Cooper's equations, by "dropping" in airless, frictionless, straight-line tunnels, passenger vehicles powered only by the pull of gravity could theoretically travel between Washington and Moscow, which are 4,850 surface miles apart, in the same time it would take them to travel from Washington to Boston, only 400 miles away. "One can envisage a transportation system without timetables," says Cooper, tongue in cheek, "with the world's cities linked by tunnels, the departure time universally on the hour, and the arrival time 42.2 minutes later."

Gravity-Powered Travel. To be sure, some formidable obstacles would have to be overcome before his scheme could become reality. At its midpoint, a Washington-Boston tunnel would be five miles below the surface of the eartha technically difficult and prohibitively costly bit of construction. In addition., the subterranean temperature at a five-mile depth might be as high as 265° F., and a passenger vehicle would need an immense cooling system. Finally, because a perfect vacuum could not be created within the tunnel, and because the vehicle would probably have to ride on some sort of rail, friction would slow it down—leaving it with insufficient kinetic energy to complete its trip without a source of additional power. In a long-distance Washington-Moscow tunnel, which at its midpoint would dip some 716 miles below the earth's sur face, the problems would surely be magnified beyond solution.

Undaunted by such practicalities, Cooper has also set up and solved by computer a set of differential equations for curved tunnels that would provide minimum gravity-powered travel time between any two cities on earth. These tunnels would swoop into the ground at steeper angles and penetrate to even greater depths. Though travel times would vary, all would be less than the 42.2 minutes required for straight-line trips.

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