A massive particle is launched from ground level with a velocity of magnitude and a launch angle of with respect to the ground.
Suppose a large (essentially infinite) number of launches take place. Over the many trials, varies uniformly between and , and varies uniformly between 0 and .
If there is a uniform downward gravitational acceleration , the expected average distance of the landing point from the launch point (assuming level ground) can be expressed as , where and are coprime positive integers.
Determine .