Also, a 3300' take-off roll doesn't mean you need a 3300' runway to operate--if you lose the engine just before rotation speed, you might need a few thousand more feet to stop the aircraft.

3300 feet in balanced density conditions. You’d better not bet on 3300 feet in Phoenix during the day time most of the year. Or Denver for that matter.

Sorry for the amateur question, but why does it take longer to take off in hot places? Doesn’t heat generate lift, like in gliders crossing hilly terrain? Or is it the heat offsetting with the cooler ground that helps, so you’d need a greater amount in hot places?

It has to do with air density.

Heavy, cold air will generate more lift since it's "thicker" for lack of a better term.

Hot air is less dense. This means it takes more power to move more air.

It's easier to maybe think of the air that you breath as a "liquid" like material comprised of a mix of mostly nitrogen, some oxygen and a little carbon dioxide and other gases. That mix changes with temperature and elevation. Just like the deeper under the ocean that you go, you have more pressure, the lower to the ground you are the more atmospheric pressure you will feel. And the higher that you go, there's less pressure.

A different way to think about it is the Newtonian approach: your wing is accelerating air particles in a downward direction, and the equal-and-opposite reaction is what (partially) generates the lift on the aircraft. Lower density means there’s fewer particles to deflect downwards in a given volume of air. In a similar vein, a rotation of the propellor accelerates air particles rearward, which results in a forward thrust on the aircraft; fewer particles to accelerate means less thrust.

> why does it take longer to take off in hot places?

When the air is cold the air is more dense. This has an impact both on how much lift is generated per amount of speed over the wings, and also impacts how much power the engine/carburetor are able to generate.

When it is hot out Air is less dense so the wings don’t generate as much lift.

I believe engines make less power when the air in is hotter too as there is less delta T. Something something Carnot cycle

The Carnot efficiency does drop as T_c increases, so all else being equal, for the same amount of fuel being burned, you'll get less energy. However, more importantly, at higher temperatures, a given volume of air at the same pressure contains less oxygen. All else being equal, at higher ambient temperatures, you're burning less fuel. For a turbocharged engine, you could increase boost to counter-act the reduced density, but that increases both mechanical and thermal stresses within the engine.

Anyway, I think the reduction of oxygen density is the primary driver for reduction in power as ambient temperatures rise.

The engine indeed makes less power, but since the air density is lower, you also have much less drag. (I know that for cars, the drag reduction dominates the power efficiency).

For airplanes take-off, the lift is the issue I'd say.

As a concrete example, current Formula 1 cars reach their fastest speed at the Mexico Grand Prix, held at Autódromo Hermanos Rodríguez in Mexico City. Current race record is 372.5km/h (231.5 mph)[1].

The elevation of the track is around 2200m, and as you say the loss of drag more than compensates for the loss of engine power. Though being turbo engines helps as I understand.

[1]: https://en.wikipedia.org/wiki/List_of_Formula_One_race_recor...

Yep it's lift and for some planes power loss due to less oxygen in the engine, but that highly depends on the kind of engine. You do get faster easier due to less drag but you also need to be faster until you have enough lift to take off. So maybe the takeoff run won't take that much longer, but you'll have covered a lot more distance.

The effect operating on gliders is from thermals - warmer ground heats the air which then rises relative to the cooler air around it. When the glider is in the thermal, it is still gliding downward through that air, but the whole column is rising faster than the glider's sink rate.

Warmer air reduces density, thus reducing lift and slightly reducing the amount of oxygen reaching the engine's combustion area, thus some power loss as well. In order to take off in the same distance, it must be operating at reduced takeoff weight.

Other answers already address the engine performance loss and lift loss, but about that:

> Doesn’t heat generate lift, like in gliders crossing hilly terrain?

That lift is generated by a temperature difference, not a globally high temperature. Thermals used by gliders are thermals because the air is locally hotter than the environment. If the whole environment gets warm, you don't get thermals.

The term you're looking for is "density altitude".

hot air rises but its less dense thats why it rises, so it has less lifting capacity thermal updrafts are different they are a larger scale phenomenon

This is a tangent, but are runways generally longer in hotter areas? Is, for example, the median length of all runways in Arizona longer than the median in North Dakota?

It seems like ideally this is the way it should be because it's a known factor, so they ought to just build in compensate for it / normalize around it from the start. But I could imagine things standing in the way of that, like cost or land availability or maybe people focus more on meeting some minimum standard length.

Aspen seems like a likely destination for this kind of craft. Is an 8000' runway at 9000' of elevation sufficient?