How can speed and power be made independent in an airplane? I heard a piece of information many years ago, and the more I think about it, the less it makes sense. It was like:

When airplanes what to raise altitude, they do not increase speed, but they increase power. It is like when going up a hill, you do not walk faster, you just push harder.

Now, I understand the statement at an intuitive level. I understand what it means for me to go up a hill, because I know what I feel.
But things are quite different with motorized vehicle in general, and maybe with airplanes in special.
My understanding is that the following equation is (relatively) true (at least as a principle):

power = rpm * torque

and my understanding is that

speed = rpm * coef

so in the end

power = f(speed)

or

speed = f-1(power)

where f is mostly (?!) multiplicative, or at least linear(-ish).
Now, how can be power increased, without increasing speed?
This statement comes to confirm that I miss some information, since the general knowledge defies my formulas :)

There are two parameters that an A/T can maintain, or try to attain: speed and thrust.
(I understand that thrust is power).


The only way I can explain it is like this (and I need some confirmation):
The (pilot) &quot;asks&quot; to increase the rpm (as a base parameter, speed and power being derived parameters). In vacuum, all the energy is used or increase RPM, so power and speed remain proportional. But in the real world, and especially during flight, a lot of the energy is used or lost in many places. So speed and power no longer remain proportional, the difference being lost (e,g, to friction).
Additionally, please provide a better explanation of what I just wrote. I am really interested to understand the basics of this topic.
Basic formulas are OK. (Complicated) Integrals and derivatives should be avoided please :)

Notes:

I am not (by far) a mechanical engineer whatsoever. As a hobby, I practice &quot;mental&quot; mechanical engineering - trying to understand how some things can be designed; or trying to understand how some things are implemented. My background is more related to electric / electronics, software, management.

In the example with a man climbing a hill, my explanation makes some sense (to me): I prepare to increase the speed by using more power, then I make a short break to rest the leg, and to move to the other leg, and then again use higher power and so on. It is mostly like a human-implemented PWM. This is how the speed can remain constant, while the power increases. But how does this work for the engines / airplanes?

I am quite sure that ultimately everything is related to how much fuel is burned in the unit of time. Unless I am wrong.

My knowledge about airplanes and flight are limited at best, even though I understand some principles.
The problem is your assumption of

speed = rpm * coef

This is only true as long as there is a rigid connection between the RPM and the medium against which the speed is measured.
Let us consider a car for a moment. In a fixed gear, there is a fixed relationship between the engine RPM and the RPM at which the tires rotate. As long as the tires are not skidding, this will directly translate to a motion w.r.t. the road (against which speed is typically measured for a car). This results in the linear equation you show.
This is however not true for an airplane, where the propeller RPM does not directly translate to a speed w.r.t. the air because air is not a solid (otherwise it would be true and you can think of the propeller as a a screw that drives into the solid). Air can flow through the propeller without moving the plane. Consider e.g. a plane that is tied down on the ground and therefore cannot move w.r.t. the ground. You can still start the engine and run the propeller, which creates an airflow behind the propeller, but does not cause any movement. The same could not work with a car, as long as the tires are not allowed to skid.
In flight, the accelerated airflow behind the propeller creates thrust, which is a force that can accelerate the aircraft forwards. The actual motion will also depend on the aerodynamic forces (lift and drag) as well as gravity of course. A higher propeller RPM (at the same blade angle) will give you more thrust, but there is no rigid connection to speed.