No. 3219: GRAVITATIONAL TRANSPORT
by Andy Boyd
Today, a wild ride. The University of Houston presents this series about the machines that make our civilization run, and the people whose ingenuity created them.
Traffic. It's a perpetual problem.
This fact was again driven home to me in an article on the front page of the Houston Chronicle. A straight stretch of Interstate 10 connects the suburban city of Katy to downtown Houston some thirty miles away. For years the road was hopelessly congested. Daily commuters were nicknamed "Katy crawlers." So in the years leading up to 2010 a massive expansion of the road was undertaken. Yet less than ten years after completion all three segments that make up the commute were listed among the top twenty most congested stretches of road in Texas. The Katy crawlers were once again crawling.
Photo Credit: Wikipedia.
So the wheels in my brain clicked into gear. Armed with paper, pencil, and the internet I outlined an energy efficient solution that could deliver Katy residents to downtown in under three minutes.
The idea is remarkably simple. To appreciate the concept, imagine a roller coaster poised 100 feet in the air. By the conservation of energy, the coaster can roll down the track and rise back up to the very same height of 100 feet powered solely by gravity. In reality, friction and air resistance will steal some of the energy so the coaster may need a bit of a boost, but the basic idea is sound.
Photo Credit: Pixabay.
Now, here's the trick. We don't need to build a towering roller coaster. Instead we can dig a tunnel shaped like a roller coaster track underground. People board a train car in Katy, the car is let go, and minutes later it arrives in Houston - all powered by gravity. The same tunnel works for the evening commute home.
How long the trip takes depends on the tunnel. Do we make it shallow or deep? Shallow and the train moves slowly, deep and the train travels a longer distance. It turns out that the fastest shape is known as a cycloid, something that occupied the attention of an astounding list of early mathematical luminaries. A cycloidal path drops straight down before gradually curving to become flat halfway between Katy and Houston. The path to the surface mirrors the path down.
And what would the ride feel like? Well, identical to riding a roller coaster that, in this case, stands ten miles tall. You'd feel changes in your weight - always lighter than normal, weightless at the beginning and end. And your maximum speed would reach over 1200 miles per hour. That's over three times faster than the fastest experimental trains. Two minutes and fifty-five seconds later you'd reach your destination. I wouldn't recommend taking a cup of coffee along for the ride. For that matter, I wouldn't recommend eating breakfast until you got to the office.
Of course, building a ten-mile-deep tunnel is impossible in practice, and friction and air resistance become very real problems at such high speeds. But there are workable if not as energy efficient alternatives. They're called subways.
I'm Andy Boyd at the University of Houston, where we're interested in the way inventive minds work.
Among the mathematical luminaries who worked on various problems related to cycloids were Galileo, Newton, Huygens, Jakob Bernoulli, Johann Bernoulli, Euler, Lagrange, Leibniz, and l'Hôpital.
Brachistochrone Curve. From the Wikipedia website: https://en.wikipedia.org/wiki/Brachistochrone_curve. Accessed December 17, 2019.
Cycloid. From the Wikipedia website: https://en.wikipedia.org/wiki/Cycloid. Accessed December 17, 2019.
Cycloid. From the Wolfram Math World website: http://mathworld.wolfram.com/Cycloid.html. Accessed December 17, 2019.
Dug Begley. "Loop 610 Grabs 'Most Congested' in State for Fourth Straight Year: Houston Area Home to 12 of 20 Worst Freeways, Study Finds." Houston Chronicle, December 11, 2019.
Tautochrone Curve. From the Wikipedia website: http://mathworld.wolfram.com/Cycloid.html. Accessed December 17, 2019.
This episode was first aired on December 26, 2019