It took nine minutes from the time the Solar Impulse first appeared in the midnight sky, lit up along the entire elegant swoop of its Airbus-size wings, to the moment the plane glided slowly and almost silently to a stop on the runway of Dulles airport in Washington.

At 00:23 on Sunday, Bertrand Piccard clambered out of the cockpit after a 14-hour flight fuelled by nothing but the sun and the photovoltaic cells along that vast wingspan. But as Piccard admitted, the technical demands of his improbable journey – in the first solar-powered aircraft to fly by night as well as day – made for a strange picture. "I was flipping the landing lights here and there to make you believe it was a UFO," he joked.

That spectacle of the Solar Impulse comes to a much broader audience on Sunday as the plane goes on limited display at the Smithsonian's Steven F Udvar-Hazy air and space museum.

For the Swiss team of Piccard and André Borschberg, who take turns flying solo, the exhibit is a chance to showcase the potential of clean energy technology. It also offers a brief pause before the final leg of their American journey – the relatively short hop to JFK airport in New York.

Borschberg is already plotting how to line up the plane with the Statue of Liberty for photographs.

The cross-country voyage, which took $115m (£73m) in investment and was 10 years in the making from the early design phase of the aircraft to the first takeoff in the US, did not break any records for speed or luxury.

The ultra-light plane carries only one person in its cramped cockpit, and reaches a maximum speed of 50mph on its own power – though a strong tail wind could boost its performance up to 100mph.

By the time the Solar Impulse touches down in New York, the trip from San Francisco, via Phoenix, Dallas, St Louis, Cincinnati, and Washington, will have taken nearly two months. The team could have driven that distance by car several times over. But speed was not the point, said Piccard.

The project has shown for the first time that it is possible to fly long distances – day and night – fuelled entirely by solar power. "What people have to understand is that the airplane can fly day and night with no fuel just because we are using very efficient technologies," Piccard said.

For all the futuristic technology, however, the flight across America has in some ways been a throwback to the early years of aviation: a solo pilot alone in the cockpit relying on an ability to improvise in tricky situations.

The Solar Impulse has the wingspan of a Boeing 747 airliner, but it weighs only as a much as a mid-sized car. Its four engines are fuelled entirely by solar power, gathered by the 12,000 photovoltaic cells on the plane's fuselage, and stored in some 400kg of batteries.

On calm, clear days, the plane climbs up to a maximum altitude of 28,000ft (8,500 metres), and then enters into a slow, gliding descent, switching off the engines to save power.

It's a process that is repeated multiple times during the course of the long flights, turning a ride in the Solar Impulse into something like a slow-moving airborne rollercoaster.

Because of the slow speeds, the Solar Impulse can only come in to land late at night, once commercial flights have shut down.

As the pilots have learned, huge wingspan and light weight are a terrible combination during turbulence, and for the violent storms that are a part of an American spring. "The weather has not been very favourable over the central part of the United States," said Borschberg.

In Dallas, the headwinds were so strong the Solar Impulse, chugging along at 40mph or so, was travelling backwards for a time, before Borschberg, whose turn it was at the controls, could correct course. "The wind speed was the same as the maximum speed of the airplane, which means I could position the airplane above the airport and the airplane was not moving," he said.

In St Louis, the hangar they were expecting to use had been smashed to bits by a tornado. Luckily, the crew came prepared with an inflatable hangar, which they draped over the aircraft. Strong headwinds also forced the crew to make an extra overnight stop in Cincinnati on Friday.

The conditions did not relent when Piccard took off on Saturday morning for Washington, an hour late as the crew desperately tried to siphon out the moisture from the plane, after mechanisms became inundated in an early-morning fog. "I think it was the most stressful moment of our mission across America," Piccard said.

By the time, the plane was airborne, the sun was well up, and so were the thermal winds.

An hour into the flight, the Solar Impulse was already running into turbulence, Piccard said in an interview from the cockpit. "There are some quite strong crosswinds coming from the left, so we have to correct and fly a little bit sideways," Piccard said. He also dropped to below 4,000 metres. "Above that the wind might push me to Florida – not to Washington DC," he said.

But by the time the plane touched down in Washington DC, those tense moments were behind Piccard. He and Borschberg were looking ahead: to the final leg of this journey, and their next one: a trip around the world by solar plane in 2015.

The two men are under no illusions that solar-powered flight will soon be an option for mass travel, however. Their hope, for now, is that their flight will allow people to get a glimpse of the possibilities of solar power. "What we are doing with the Solar Impulse is not for the goal of making a revolution in air transport," said Piccard. "But the Solar Impulse does have a goal of having a revolution in the mindset of the people."