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:
Code:
Thrust = [efficiency factor * power] / speed
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.