Citizens in Chattanooga, Tennessee, have access to one-gigabit-per-second Internet—that’s 100 times the U.S. national average, and fast enough to download a two-hour movie in about five seconds. The only question is: what to do with it?

The city is hoping a contest with $300,000 in prize money will help answer that question. Entrants are invited to come up with clever ways of making use of the city’s blisteringly fast Internet, which was installed in late 2010 with a $111 million U.S. Department of Energy grant, as part of federal stimulus efforts that also built out the city utility’s long-planned smart grid.

Some early entries include health-care applications, such as transferring larger files like CT scans in real-time so that specialists can serve a larger area. Ideas contributed by students include a platform for high-definition video debates, and international collaborations with students in Sri Lanka, London, Jamaica, and elsewhere.

But even if some great ideas come out of the contest, the fact remains that most people in the U.S. still have access to only relatively slow Internet connections. Late last year, the United States ranked 25th in the world for average available Internet speed. By the end of this year, South Korea, a world leader in Internet speed, will provide one-gigabit service nationwide for about $27 a month.

Furthermore, where superfast Internet is available in the U.S., it is typically prohibitively expensive. The Chattanooga service has been available for more than a year to 150,000 residential and commercial customers for $350 per month, but it has so far found only eight residential subscribers and 18 commercial ones.

Even so, in Tennessee they are optimistic that the contest will bring rewards. “Eventually, these fatter pipes will get filled with bandwidth-eating applications,” says Jack Studer, partner at the Lamp Post Group, a VC firm in Chattanooga that, along with companies including Alcatel, Cisco, and IBM, is sponsoring the contests.

“What we are trying to do is inject some capital into innovation, with the goal of driving demand for higher-bandwidth networks and jump-start adoption across the country and world,” Studer says. “We plan to do this for multiple years—in the second and third year, we may see a revolutionary jump to things we may not be thinking about now.”

The $300,000 prize money will be split among students and entrepreneurs. Ten startups will get $15,000 this summer to develop and test their gigabit business ideas. A local judging panel will give a $100,000 prize for the winner. A separate student contest will carry a $50,000 prize. The deadline for entries is March 1.

Chattanooga is the only place in the United States providing such high-speed service. But others are on the way: Google is going into the Internet service provider business, stringing fiber on telephone poles in Kansas City, Missouri, and adjacent Kansas City, Kansas. The first of its customers should get a connection by the middle of 2012, a spokesman says.

Video is the fastest-growing bandwidth-hogging app, and it could be an important driving force for faster Internet speeds. Google is, in fact, hoping to provide a TV service as part of its broadband efforts. Earlier this month, the company filed applications with the Missouri Public Service Commission and the Kansas Corporation Commission that would allow it to supply a TV service.

The U.S. Federal Communications Commission in 2010 defined “basic broadband” as at least four-megabits-per-second download speed and one-megabit-per-second upload speed. The efforts in Chattanooga and Kansas City are a step toward carrying out the FCC’s ambitious National Broadband Plan, which aims to not only provide this minimal level of service to every community, but also to achieve the more ambitious goal of providing a majority of households with 100-megabit-per-second service by 2020.

The Chattanooga network was built by the city-owned Electric Power Board. The utility uses the fiber partly for a smart electric grid that does things like detect overloads and reroute power on the fly to avoid costly brownouts.

History suggests that faster broadband spurs innovation and new business, says Rob Vietzke, vice president of network services at Internet2, a networking consortium that provides blazing fast Internet to research labs and government agencies. For example, in 2005, YouTube emerged with an application enabled by the growing availability of broadband in U.S. homes and businesses. “Projects like Chattanooga and Kansas City reopen the opportunity for innovation,” Vietzke says. “You can’t predict exactly what will happen, but it lays the groundwork for people to think differently about how they do their work.”

One possible application of one-gigabit service involves streaming super-high definition video at four or more times the resolution of current HD technology, Vietzke says. Such high-quality streams could be useful for telemedicine and realistic remote meetings, but would require at least 100-megabit service, he adds.

Internet2, for its part, is working on delivering 100-gigabit service, initially to research centers in Indiana and Ohio—useful for such applications as crunching data from genomics research and from particle physics experiments at the Large Hadron Collider. (Some scientific instruments dish out even more data—deep-space telescopes, for example, can generate one terabit a second.)

All of this is way beyond the perceived needs of the average Chattanoogan. “Anything that makes my Netflix streaming move faster is okay with me,” quips Tom Balázs, an assistant professor of English at the University of Tennessee in Chattanooga.