The lift equation, rearranged

Your indicated stall speed does not change with height

This is the most confused point in the whole subject, and it is confused in both directions. Some pilots are sure the stall speed rises with height. Others are equally sure it cannot, without being able to say why. Both are right, and they are talking about different speeds without noticing.

Start from the equation. In steady flight the wing must produce L = nW, and that is fixed by the mass and the load factor, not by the air. So ½ ρ V² S C_L has to come out the same at 12,000 feet as it does at sea level. Thinner air means a smaller ρ, so V² must be larger. The true airspeed at which the wing reaches C_Lmax genuinely does rise, by 1 over the square root of the density ratio.

Now look at what the instrument is doing. An airspeed indicator is not a speedometer. It is a differential pressure gauge measuring ½ ρ V² — dynamic pressure — and reporting it on a dial marked in speed, using sea-level density to do the conversion. It never knows the true airspeed and never claims to.

So the wing and the instrument are reading the same quantity. Both respond to ½ ρ V². When the wing runs out of lift, it does so at a particular dynamic pressure, and that dynamic pressure always puts the needle in the same place. The indicated stall speed is the same on the ground at a sea-level field as it is at 12,000 feet over the mountains. The table above shows exactly that with your own numbers in it: the true column climbs steadily down the page and the indicated column does not move at all.

Which one should you fly? The indicated one, always, because it is the one on your instrument and it is the one the wing agrees with. But the true one is what governs your groundspeed, your rate of descent over the ground, your turning radius and how much energy you arrive with. That is why the same approach speed feels fast at a high field: the needle is right where it always is, and at 6,000 feet of density altitude you are crossing the grass about nine per cent faster than the same needle means at sea level. If you want that split worked out on its own, there is a true airspeed calculator at /tools/true-airspeed, and the density altitude page is linked above.

Two honest caveats, neither of which changes how you fly it. What your ASI shows is indicated airspeed, which is equivalent airspeed plus whatever position and instrument error your particular installation has; at gliding speeds that error is small and compressibility is negligible, so treating the two as the same is safe, and your flight manual will have a calibration table if it matters to you. And the constant column above assumes C_Lmax itself is constant, which is very nearly but not exactly true: at a fixed indicated speed the Reynolds number falls as you climb, and a wing at a lower Reynolds number reaches slightly less C_Lmax. The indicated stall speed therefore creeps up a little with height rather than staying put to the last decimal. It is a second-order effect, far smaller than the 1/√σ the true speed moves by, and much smaller than the spread between one pilot's stall and another's.

The two square roots worth carrying in your head

Rearranged for speed, the equation reads V_s = √(2nW / ρSC_Lmax). Everything a pilot changes in flight — mass, bank angle, height — sits under a square root, and a square root is a merciful function. It is why gliding tolerates quite large changes in loading without the handling falling apart.

Mass. Filling the water bags can add a quarter to your all-up mass. That is a big number and it sounds alarming, but the stall speed only goes up by the square root of it: √1.25 is 1.118, so twelve per cent. On a glider stalling at 75 km/h dry, full water moves it to about 84. Worth knowing, easily flown, and not the reason anybody has ever regretted taking ballast. Wing loading tells the same story: it rises the full twenty-five per cent while the stall speed moves twelve.

Load factor. Bank is harsher, because n itself climbs steeply. A level turn needs n = 1 / cos φ, and cos falls away fast past 45 degrees. Thirty degrees of bank is 1.15 g and costs seven per cent on the stall speed; forty-five is 1.41 g and costs nineteen; sixty degrees is exactly 2 g and costs forty-one. That last one is the number to remember, because sixty degrees is not an unusual angle of bank in a strong thermal, and because the classic stall and spin happens in a turn rather than in a glide.

The two compose. A ballasted glider in a sixty degree turn is carrying both square roots at once: √1.25 × √2 is 1.58, so a wing that stalled at 75 wings-level and dry now stalls at about 119 km/h indicated. The numbers a pilot habitually thinks in — the circling speed that has always felt comfortable — were learned on the light, unballasted aircraft, and they do not transfer. There is a load factor calculator at /tools/load-factor that takes the turn side of this further.

Read the g figure in the results the other way round and it becomes the more useful question. At any speed, the wing can generate load factor up to C_Lmax × ½ρV²S / W before it stalls, and that is the square of your speed margin: at twice the stall speed you have 4 g of wing available, at 1.4 times you have 2 g. It is the aerodynamic limit only. The structure has its own, it is usually the lower of the two once you are fast, and it lives in your flight manual rather than on this page.

Why there is no C_Lmax for your glider here

C_Lmax is the one term in the equation that is a property of the aircraft rather than of the situation, and it is the one this site will not supply. It is not a published figure in any flight manual I have read. It depends on the aerofoil, on the twist along the span, on the flap setting, on the Reynolds number and therefore on the speed, and on how clean the leading edge is — a wing with flies on it and rain in the joints does not reach the C_Lmax of the same wing polished. Two aircraft of the same type will not agree exactly, and the same aircraft will not agree with itself in April and August.

What you do have is a stall speed, measured on a real aircraft of your type by somebody who was paid to do it properly, printed in your flight manual against a stated mass. Run the equation backwards from that and you get a C_Lmax that describes your aircraft, with its wing, as it was certified. That is what the default mode here does, and it is why the tool asks for the mass the figure applies to as well as the speed. Quote a stall speed measured at 300 kg against a mass of 500 kg and the C_Lmax that comes out is two thirds too large, which puts every stall speed below it about twenty-two per cent too low. That is the dangerous direction to be wrong in.

Do not treat the result as a certificate. It is the value that reproduces the book stall speed on a clean, correctly rigged wing in still air with a pilot who lets the aircraft settle. Every real departure from that reduces it, which means every real stall speed is a little higher than this arithmetic says. Fly the numbers in the manual, not the ones you derived from them.

Wing loading, and why pilots compare it

W/S falls straight out of the rearrangement, and it is the number pilots actually trade in: not mass, not area, but the two divided. It is what the wing feels. Two gliders at the same wing loading and the same C_Lmax stall at the same indicated speed regardless of how big they are, which is why a fifteen-metre racer and a heavy two-seater can share a circuit speed.

It is quoted in kilograms per square metre nearly everywhere and in pounds per square foot in the United States, and the conversion catches people out: one kilogram per square metre is only about 0.205 pounds per square foot, so 38 kg/m² is 7.8 lb/ft². Both are shown above so a figure quoted in either can be checked without arithmetic in your head.

Wing loading is also the whole argument for water ballast, and this equation is only half of it. Adding mass raises every speed on the polar by the same square root, and every sink rate with it, which stretches the whole curve away from the origin without changing its shape: the best glide ratio is exactly what it was and it now happens faster. On a strong day the quicker cruise wins more than the slower climb loses. On a weak day it does not. That trade is worked out on the polar and speed to fly page, which is where to go once this page has told you what the ballast does to the slow end.