# Why did the Solar Impulse flight take so long?

The Solar Impulse just made big news for flying from Japan to Hawaii on a solar-powered plane in five days.

But five days seems really long. Let's do the math on that. The distance from Japan to Hawaii is about $6,611 km$, so $6,611 km/5 days/24 hours = 55km/h$

So it was flying an average of $55km/h$? That seems too slow to maintain lift.

• The plane didn't fly in a straight line, due to the need to evade weather and to deal with prevailing winds. It also needed to circle a bit before landing because it arrived in very early morning when it was too dark. Actual distance covered (and average speed) was therefore a bit higher but still under 75 km/h) – Tonny Jul 3 '15 at 19:31

No, it could not fly much faster with the available energy.

Lift is a question of wing area and dynamic pressure. Solar Impulse 2 has 269.5 m² wing area to carry its 2.3 tons of mass. This is a wing loading of just 8.53 kg/m²; much less than even gliders have (they start at around 30 kg/m²). This allows it to fly very slowly; if we assume it flies at the lift coefficient $c_L$ for optimum range for propeller aircraft $$c_L = \sqrt{\pi \cdot AR \cdot \epsilon \cdot c_{D0}}$$ and guess the zero-lift drag coefficient $c_{D0}$ to be 0.029 and the aspect ratio $AR$ to be 19.18, the result is 1.377. This is already close to the maximum lift coefficient, which I expect to be 1.6. Now we need the lift equation to calculate the speed which is needed to keep it flying at this lift coefficient:$$v = \sqrt{\frac{2\cdot m\cdot g}{c_L\cdot S\cdot\rho}}$$ Let's assume it flies in 9000 m altitude where air density $\rho$ is only 0.4671 kg/m³, the result is 16.13 m/s or 58 km/h. Now consider that it will fly at the beginning, late at night and at the end of the flight at lower altitudes, and may not be able to fly a straight course (maybe to fly around bad weather), and the 55 km/h look entirely plausible, if not outright speedy. At sea level, the optimum range speed is only 9.96 m/s or 35.86 km/h, just about its take-off airspeed.

Flying faster would need more energy and would not allow Solar Impulse 2 to recharge its batteries for the night hours. Solar flight must be slow to be possible at all.

From here

Solar Impulse Performance

Solar Impulse can fly at the same speed than a car, between 36 km/h (20 Kts) and 140 km/h (77 Kts).

At sea level: minimum speed of 45 km/h (20 Kts) and maximum speed of 90 km/h (49 Kts). At maximum altitude: from 57 km/h (31,5 Kts) to 140 km/h (77 Kts).

From here

Wired.Com article, interview with pilots

“You have to react quickly because if you go down 50 meters, you reach the maximum speed,” he says of the design limit of the airplane. “If you go up 50 meters you reach the stall speed.”

The calculated maximum speed is 44 knots (51 mph), but the pilots limit themselves to 35 knots (40 mph). The stall speed — the point at which the airflow separates from the wing and it no longer generates sufficient lift — is just 19 knots (22 mph).

• Thanks. I didnt realize that an aircraft could fly at such slow speeds. – David says Reinstate Monica Jul 3 '15 at 17:25
• You're welcome. It has massive wings. The wing area is a key factor in the lift equation. – Simon Jul 3 '15 at 17:35
• @David It turns out that all solar-powered airplanes are quite slow. I've even seen video of someone following one down a runway during its takeoff run... on a bicycle. Solar energy density is actually quite small. While peak solar power density when the sun is directly overhead and there are no clouds is about 1 kW per square meter, overall average solar power density at the latitudes where most humans live is only 100-200 W (0.1-0.2 kW) per square meter. For reference, that's almost enough to power my video card... and that's even if you have 100% efficient panels (which no one has.) – reirab Jul 3 '15 at 19:53
• @DavidGrinberg : there are even conventional aircraft which can fly that slow, even slower. The Antonov 2, an almost 70 year old plane, for example, can fly so slowly that its manual doesn't even specify a stall speed! You can fly controllably at 40 km/h, and if you fly any slower, you'll just descend like with a parachute. – vsz Jul 4 '15 at 15:23