Result

Projectile trajectory

x = v_{0} cos θ t, y = v_{0} sin θ t - \frac{1}{2} g t^{2}

Statement

Horizontal and vertical position as functions of time for a projectile launched at angle θ with speed v₀.

Symbols and units

$v_{0}$	launch speed	m/s
$θ$	launch angle	degrees
$g$	gravitational acceleration	9.8 m/s²
$x, y$	horizontal and vertical position	m
$t$	time since launch	s

When it applies

Neglects air resistance; g treated as constant over the trajectory.

Don't use when

A poor model for light or fast-spinning objects, or long-range trajectories where air resistance is significant.

Derivation

Apply the constant-acceleration equations separately to each axis: zero acceleration horizontally, −g vertically.

a_{x} = 0, a_{y} = - g

With initial components v_{0x}=v_0\cos\theta and v_{0y}=v_0\sin\theta, the position equations follow directly:

x = v_{0} cos θ t, y = v_{0} sin θ t - \frac{1}{2} g t^{2}

Worked examples

Range of a projectile

v_0 = 20 m/s, θ = 30°. Time of flight, from y = 0:

t = \frac{2 v _{0} sin θ}{g} = \frac{2 ( 20 ) ( 0.5 )}{9.8} \approx 2.04 s

Horizontal range:

R = v_{0} cos θ t \approx (20) (0.866) (2.04) \approx 35.3 m

Watch out for

This is a model that neglects air resistance — accurate for dense, compact projectiles at moderate speed, poor for light or fast-spinning ones.

Connects to