Independence of Perpendicular Motions

Position, velocity, and acceleration are vectors, so you can split any 2D motion into x and y components and treat each one with ordinary one-dimensional kinematics. The link between the two problems is time: both components tick forward on the same clock, so whatever time you find from the vertical drop is the same time you plug into the horizontal equation.

This works because acceleration in the x direction and acceleration in the y direction are separate causes with separate effects. For a projectile near Earth's surface, gravity pulls straight down with acceleration g, and nothing pushes or pulls horizontally. That means horizontal velocity never changes — it stays whatever it was at launch — while vertical velocity changes at a constant rate of g, exactly as if the object had just been dropped. A bullet fired horizontally and a bullet dropped from the same height hit the ground at the same instant because their vertical stories are identical; the horizontal motion is a separate, parallel story that doesn't touch the vertical one at all.

This idea is why two-dimensional kinematics in this unit never actually requires new equations — it just requires running the same x(t), v(t), a(t) derivative and integral relationships twice, once per axis. It also sets up projectile motion with launch angles: once velocity has both x and y components from the start, you'll still split it the same way, find a shared time of flight from the vertical equation, and use that time everywhere else.