Independent Axes

A thrown ball is two motions at once: sideways along x and up and down along y. Nothing pushes sideways, so the horizontal velocity never changes. The vertical motion is free fall:

x = vx t, vy = v0y - g t

The two motions share only the time t: dropped and thrown sideways from the same height, two balls land together.

Same height at the same instant: the sideways motion does not change the fall.