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In other words, I want to understand how gravity converts into thrust in respect to pitch. What is the general mathematical relationship? (For visuals, imagine a bird diving to gain speed.)

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Assuming this is the standard inclined plane task from physics textbook, as here for instance, the acceleration is

$a = g \cdot \sin(\theta)$

where $a$ is acceleration, $g$ is the free fall acceleration (9.8 m/s^2) and $\theta$ is the pitch angle down (0 assuming horizontal).

This equation applies does not take the air resistance into consideration so would only apply to relatively slow speeds. The equation also assumes that the plane orientation matches the flight direction, this is obviously simplification.

Interestingly, under these conditions the acceleration does not depend on the mass of the plane. The equation should also apply for the positive pitch angles, but the acceleration would be negative in this case (slowing down).

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