The two independent motions

Fire a bullet horizontally and drop another from the same height at the same instant. Which hits the ground first? They tie. That surprising tie is the single most important fact about projectile motion.

One motion, two ingredients

A projectile's velocity has a horizontal part and a vertical part:

$v_x = v_0\cos\theta \qquad v_y = v_0\sin\theta - gt$

Look at what gravity touches: only v_y. The horizontal component has no force acting on it (ignoring air), so it simply coasts, unchanged, from launch to landing. The two motions run on separate tracks:

• Horizontal: constant velocity — equal distance every second.
• Vertical: free fall from the last lesson — the 1 : 3 : 5 : 7 gaps.

The shadows on the ground are the proof you can see: they're equally spaced, even as the arc curves. Horizontally, a projectile is a ball rolling at constant speed; vertically, it's the dropped ball from last lesson. Played together, they trace a parabola.

Why a parabola?

Horizontal position grows like t, vertical drop grows like t². A curve whose y depends on the square of its x is a parabola — the same shape as y = ax² + bx + c from algebra. The physics of the shape is just "t and t² at the same time."

Using the independence

The independence isn't philosophy — it's the solving strategy. Every projectile problem splits into two one-dimensional problems that share only the clock:

1. Vertical decides the timing: how long until it lands.
2. Horizontal spends that time: distance = v_x × time.

That's the whole method. The bullet ties the dropped bullet because step 1 is identical for both — same height, same g, same fall time.

You can watch the two components live in the projectile motion simulator: scrub the time slider and watch the velocity arrow — its horizontal part never changes.

Why this matters

Range, apex, flight time — every projectile formula in the next lesson is just these two independent motions asked a specific question.