Average Landing Distance (Part 2)

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

Suppose a large (essentially infinite) number of launches take place. Over the many trials, θ\theta varies uniformly between 00 and π2\frac{\pi}{2}, and vv varies uniformly between 0 and vmaxv_\text{max}.

If there is a uniform downward gravitational acceleration gg, the expected average distance of the landing point from the launch point (assuming level ground) can be expressed as abvmax2πg\dfrac{a}{b} \dfrac{v_\text{max}^{2}}{\pi g}, where aa and bb are coprime positive integers.

Determine a+ba+b.

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