Projectile Range?

Question

To attain the maximum range, a projectile has to be launched at 45 degrees if the landing spot and the launch spot are at the same height (neglecting air-resistance effects). Explain how the relation between the vertical and the horizontal components of the initial velocity affects the projectile range.

Answer 1

It's a tradeoff. Shooting it straight upwards gives it the longest "hang time", but it goes nowhere down the range. Shooting it straight out gives it the maximum downrange velocity, but no hang time, and so it hits the ground right away. It turns out that shooting it at 45 degree angle is the best tradeoff for maximum downrange distance. It's analyzed by decomposing the velocity vector into a sum of vertical and horizontal vector velocities, and doing the math on each.