Propellers and the mathematics behind them are quite complex, but we can do a crude analysis.

An approximation of the thrust produced by a propeller can be made by computing:

For takeoffs, you want to accelerate to liftoff speed in as short a distance as possible, which means you want as much thrust as you can get.

Looking at the above equation, there are only 3 terms. Speed is very small as you begin the takeoff roll and the efficiency factor is determined by the design of the prop.

That leaves engine power as the only remaining variable we can manipulate. Since the engine develops more horsepower at high RPM than it does at low RPM, you want to run the engine at high RPM so that you can produce the most thrust.

Going a little deeper: Power is computed by multiplying shaft torque by rotational speed. So to increase power you want the prop to spin faster.

Therefore you want to go to fine pitch.

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I really dislike the "car analogy" that many make, but if it helps: In a car you accelerate from a standstill in lowest gear. This is akin to putting the prop at fine pitch.

Ehm... Sorry, Cheesehawk, but I'm sure Cpt.Doggles' formula is right :D. It's a simplified, basic formula for prop thrust, obtained from the definition of prop efficiency, which can be found in any book about props... or on the same page You linked to, just a couple of lines below the formula You quoted. Note the formula being used for thrust estimation in the numerical example at the bottom of the same page.

Cheers - Art

thanks a lot!!
when it comes to things that are unintuitive, looking at a math formula is very very helpful, because it gives u an idea of what is important and what is irrelevant. Good job with the condensed version of the formula and explanation!!

No need to apologize, cheesehawk. The formula I posted is perfectly valid (if a bit simplistic). You'll notice I used the phrases "a crude analysis" and "an approximation". In real life we must also account for components of thrust not in line with the direction of motion, rotation of the fluid within the slipstream, etc. The formula you posted doesn't account for these either. It all depends on how much effort we are willing to invest and how precise we want our approximation to be.

The concept of delta-v added to the flow is not easily discerned from in-cockpit instrumentation. Horsepower on the other hand is at least related to engine RPM, and we have a tachometer in the cockpit :cool:

The formula I posted DOES take diameter into account, but the diameter term is hidden away in the efficiency term. I type Δ by going into the character map. It's not in the ASCII character set so I don't think it has an ALT+XXXX code.

delta is a greek letter, so in windows, you would go to control panel and add the greek keyboard and Δ is the D key. After you add the greek keyboard, you can switch around different languages by pressing alt+shift.
i work with multiple languages all the time, this is the way I would recommend to type Greek letters.