Average Landing Distance (Part 2)

A massive particle is launched from ground level with a velocity of magnitude \(v\) and a launch angle of \(\theta\) with respect to the ground.

Suppose a large (essentially infinite) number of launches take place. Over the many trials, \(\theta\) varies uniformly between \(0\) and \(\frac{\pi}{2}\), and \(v\) varies uniformly between 0 and \(v_\text{max}\).

If there is a uniform downward gravitational acceleration \(g\), the expected average distance of the landing point from the launch point (assuming level ground) can be expressed as \(\dfrac{a}{b} \dfrac{v_\text{max}^{2}}{\pi g}\), where \(a\) and \(b\) are coprime positive integers.

Determine \(a+b\).

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