Drop a ball from a table and fire another horizontally from the same height at the same instant, and they hit the ground simultaneously — always. Horizontal and vertical motion are independent. Vertically both balls start with v₀ᵧ = 0 and fall under gravity alone, so Δy = ½gt² gives a fall time that depends only on the drop height and g. The fired ball's horizontal velocity carries it sideways but does nothing to speed up or slow down its fall.

On the vertical axis, once both balls leave the table gravity is the only force acting on them, assuming no air resistance. Both start with an initial vertical velocity of zero, so their fall time comes entirely from Δy = ½gt². Same height, same g, identical time — the horizontal launch never enters that equation.

On the horizontal axis, the fired ball keeps its initial velocity vₓ because no force acts on it sideways after launch. That constant speed sets the range, Δx = vₓ × t, which is how far it travels before landing. Range and fall time are set by different equations, which is exactly why changing one leaves the other untouched.

Launch the cannonball slowly or at blinding speed: the vertical vectors map perfectly onto each other and both objects strike the floor at the same moment. This is the result Galileo used to break the Aristotelian idea that heavier or faster-moving objects fall differently, and it is the foundation of every projectile problem you will solve — you always split the motion into two independent one-dimensional problems and share only the time between them.