system
Given the following post, respond as if you're commenting on Reddit:
human
The average temperature outside airplanes at 30,000ft is -40° F to -70° F (-40° C to -57° C). The average causing speed is 575mph. If speed=energy and energy equals=heat, is the skin of the airplane hot because of the speed or cold because of the temperature around?
gpt
You have to be careful when saying things like "speed = energy" and "energy = heat"; those aren't really true in general.

But anyway, if you assume a steady, isentropic flow of ideal gas around the wings of the plane, we can say that c*_p_*T + v^(2)/2 is constant along any streamline.

c*_p_* is just the specific heat capacity of the air at constant pressure; you can just think of it as some constant that depends on the type of gas.

This says that the temperature along any streamline is maximized at points where the flow velocity is as small as possible. Particularly, somewhere on the leading edge of the wing, there will be a point where the flow is stationary. This is called the stagnation point. And the temperature at that point is maximal.

Taking realistic values for the heat capacity of air, the speed of a cruising airplane, and an ambient temperature of -40 degrees C, the stagnation temperature is just

T*_0_* = T + v^(2)/(2c*_p_*).

Or rearranged, T*_0_* - T is about 33 degrees C. The temperature at the stagnation point is 33 degrees C higher than the temperature of the ambient air.

So does being slammed into the wing cause the air in its vicinity to warm up pretty substantially? Yes. Can it still be very cold compared to everyday temperatures? Yes (in this case, it's -40 + 33 = -7 degrees C, still below freezing).