Velocity Components of a Projectile

The key principle

Projectile motion is two completely independent motions happening simultaneously:

• Horizontal: no force acts horizontally (ignoring air resistance), so horizontal velocity never changes
• Vertical: gravity acts downward at $g = 9.8$ m/s², so vertical velocity changes continuously

This independence is the foundation of all projectile motion analysis. The horizontal and vertical components never affect each other.

Setting up

A projectile is launched from the origin with speed $u$ at angle $\theta$ above the horizontal.

The initial velocity has two components:

$u_x = u\cos\theta \quad \text{(horizontal)}$ $u_y = u\sin\theta \quad \text{(vertical, upward)}$

Derivation

Horizontal component:

No horizontal force acts, so horizontal acceleration is zero. By the first equation of motion with $a = 0$:

$v_x = u_x + 0 \cdot t$

$\boxed{v_x = u\cos\theta}$

The horizontal velocity is constant throughout the flight. It is the same at launch, at the top, and at landing.

Vertical component:

Gravity acts downward. Taking upward as positive, vertical acceleration is $-g$.

By the first equation of motion:

$v_y = u_y + (-g)t$

$\boxed{v_y = u\sin\theta - gt}$

The vertical velocity starts at $u\sin\theta$ (upward), decreases at rate $g$, becomes zero at the top, then becomes negative (downward) on the way down.

How the vertical component changes through the flight

What stays constant, what changes