Projectile Motion

Introduction

Projectile motion describes the curved path of an object launched into the air and moving only under the influence of gravity. It is one of the most fundamental topics in classical mechanics, providing the foundation for understanding everything from thrown balls to artillery shells. Studying it reveals how two independent directions of motion — horizontal and vertical — can be analysed separately and combined to predict where an object will land.

The Physics Explained

When an object is launched at some angle above the horizontal, its motion splits cleanly into two components that never interfere with each other. Horizontally, there is no force acting (ignoring air resistance), so the object travels at a constant speed forever in that direction. Vertically, gravity pulls the object downward with a constant acceleration of 9.8 m/s², which means the vertical speed changes continuously — slowing on the way up, stopping briefly at the peak, then accelerating back down.

The combination of constant horizontal speed and changing vertical speed traces out a parabolic arc. The key insight is that both components evolve simultaneously over the same time interval — you do not need to choose between them. At any instant you can ask: how far horizontally? how high vertically? and answer each question independently.

The range — the horizontal distance when the object returns to its launch height — is maximised at a launch angle of 45°. Angles above and below 45° by the same amount (e.g. 30° and 60°) produce the same range but different flight times and peak heights.

Key Equations

vx = v₀ · cos(θ) (horizontal component of launch velocity)

vy = v₀ · sin(θ) (vertical component of launch velocity)

x(t) = vx · t

y(t) = vy · t − ½ · g · t²

Range R = (v₀² · sin(2θ)) / g

Max height H = (v₀² · sin²(θ)) / (2g)

Time of flight T = (2 · v₀ · sin(θ)) / g

Key Variables

Symbol	Name	Unit	Meaning
v₀	Launch speed	m/s	The magnitude of the initial velocity
θ	Launch angle	degrees (°)	Angle above the horizontal at launch
vx	Horizontal velocity	m/s	Constant horizontal speed throughout flight
vy	Vertical velocity	m/s	Vertical speed, changes due to gravity
g	Gravitational acceleration	m/s²	9.8 m/s² downward on Earth's surface
t	Time	s	Time elapsed since launch
x(t)	Horizontal position	m	Distance travelled horizontally at time t
y(t)	Vertical position	m	Height above launch point at time t
R	Range	m	Total horizontal distance when object lands
H	Maximum height	m	Highest point above the launch height

Real-World Examples

Sport: A football kicked at an angle flies a parabolic arc — coaches and players instinctively adjust angle and force to control range and height.
Ballistics: Cannonballs, mortar shells, and rifle bullets all follow projectile paths (until air resistance becomes significant at high speeds).
Space probes: When a spacecraft in a vacuum fires a short thruster burst perpendicular to its orbit, the resulting trajectory is governed by the same projectile equations applied in three dimensions.

How the Simulation Works

You can set the launch speed (m/s) and the launch angle (degrees) using the sliders on the left. Pressing Launch fires the projectile from the bottom-left of the canvas. The path is drawn in real time, and a faint trace remains so you can compare multiple shots. The simulation uses a fixed gravitational acceleration of 9.8 m/s² and ignores air resistance, so the trajectory is a perfect parabola. Watch the horizontal and vertical components of velocity displayed in the readouts — the horizontal one stays constant while the vertical one counts down, reaches zero at the peak, then grows again on the way down.