Aim high and go far—Optimal projectile launch angles greater than 45°

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For ideal projectile motion, which starts and ends at the same height, maximum range is achieved when the firing angle is 45°. If air resistance is taken into account, the optimal angle is somewhat less than 45° and this is often considered obvious. We show here that it is not obvious by considering drag forces with different dependence on projectile speed. In some cases maximum range is achieved for launch angles greater than 45°. Simple physical arguments are given which help explain results that were found by computing trajectories and ranges.